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 $O(n)$ time and space, where $n$ is the string size. However, their solution can only find the \emph{leftmost} longest repeat for each of the $n$ 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 $(M,\textit{g},\sigma)$ be a compact Riemannian spin manifold of dimension $m\geq2$, let $\mathbb{S}(M)$ denote the spinor bundle on $M$, and let $D$ be the Atiyah-Singer Dirac operator acting on spinors $\psi:M\to\mathbb{S}(M)$. We study the existence of solutions of the nonlinear Dirac equation with critical exponent $$D\psi = \lambda\psi + f(|\psi|)\psi + |\psi|^{\frac2{m-1}}\psi \tag{NLD}$$ where $\lambda\in\mathbb{R}$ and $f(|\psi|)\psi$ is a subcritical nonlinearity in the sense that $f(s)=o\big(s^{\frac2{m-1}}\big)$ as $s\to\infty$. A model nonlinearity is $f(s)=\alpha s^{p-2}$ with $2<p<\frac{2m}{m-1}$, $\alpha\in\mathbb{R}$. In particular we study the nonlinear Dirac equation $$D\psi=\lambda\psi+|\psi|^{\frac2{m-1}}\psi, \quad \lambda\in\mathbb{R}. \tag{BND}$$ 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 $(\lambda,\psi)$ of (BND) and (NLD) for every $\lambda>0$, even if $\lambda$ is an eigenvalue of $D$. For some classes of nonlinearities $f$ we also obtain solutions of (NLD) for every $\lambda\in\mathbb{R}$, except for non-positive eigenvalues. If $m\not\equiv3$ (mod 4) we obtain solutions of (NLD) for every $\lambda\in\mathbb{R}$, 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