On Exponential Time Lower Bound of Knapsack under Backtracking

M.Aleknovich et al. have recently proposed a model of algorithms, called BT model, which generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model by Woeginger. BT model can be further divided into three kinds of fixed, adaptive and fully adaptive ones. They have proved exponential time lower bounds of exact and approximation algorithms under adaptive BT model for Knapsack problem. Their exact lower bound is $\Omega(2^{0.5n}/\sqrt{n})$, in this paper, we slightly improve the exact lower bound to about $\Omega(2^{0.69n}/\sqrt{n})$, by the same technique, with related parameters optimized.Comment: This paper supersedes the result of arXiv:cs/060606

Virasoro Constraints For Quantum Cohomology

Eguchi-Hori-Xiong and S. Katz proposed a conjecture that the partition function of topological sigma model coupled to gravity is annihilated by infinitely many differential operators which form half branch of the Virasoro algebra. In this paper, we give a proof to this conjecture for the genus 0 part.Comment: LaTex, 38 pages. Corrected some typos and minor mistakes in the first version of this paper posted on June 7, 199

Weinstein Conjecture and GW Invariants

In this paper, we establish a general relationship between the nonvanishing of GW invariants with the existence of the closed orbits of a Hamiltonian system. As an application, we completely solved the stabilized Weinstein conjecture

Conservation laws and symmetries of Hunter-Saxton equation: revisited

Through a reciprocal transformation $\mathcal{T}_0$ induced by the conservation law $\partial_t(u_x^2) = \partial_x(2uu_x^2)$, the Hunter-Saxton (HS) equation $u_{xt} = 2uu_{2x} + u_x^2$ is shown to possess conserved densities involving arbitrary smooth functions, which have their roots in infinitesimal symmetries of $w_t = w^2$, the counterpart of the HS equation under $\mathcal{T}_0$. Hierarchies of commuting symmetries of the HS equation are studied under appropriate changes of variables initiated by $\mathcal{T}_0$, and two of these are linearized while the other is identical to the hierarchy of commuting symmetries admitted by the potential modified Korteweg-de Vries equation. A fifth order symmetry of the HS equation is endowed with a sixth order hereditary recursion operator by its connection with the Fordy-Gibbons equation. These results reveal the origin for the rich and remarkable structures of the HS equation and partially answer the questions raised by Wang [{\it Nonlinearity} {\bf 23}(2010) 2009].Comment: 18 page

Family Seiberg-Witten invariants and wall crossing formulas

In this paper we set up the family Seiberg-Witten theory. It can be applied to the counting of nodal pseudo-holomorphic curves in a symplectic 4-manifold (especially a Kahler surface). A new feature in this theory is that the chamber structure plays a more prominent role. We derive some wall crossing formulas measuring how the family Seiberg-Witten invariants change from one chamber to another.Comment: 46 pages Typos corrected, references updated, Theorem 2.2 made more precis

An Improved Decision Procedure for Linear Time Mu-Calculus

An improved Present Future form (PF form) for linear time $\mu$-calculus ($\nu$TL) is presented in this paper. In particular, the future part of the new version turns into the conjunction of elements in the closure of a formula. We show that every closed $\nu$TL formula can be transformed into the new PF form. Additionally, based on the PF form, an algorithm for constructing Present Future form Graph (PFG), which can be utilized to describe models of a formula, is given. Further, an intuitive and efficient decision procedure for checking satisfiability of the guarded fragment of $\nu$TL formulas based on PFG is proposed and implemented in C++. The new decision procedure has the best time complexity over the existing ones despite the cost of exponential space. Finally, a PFG-based model checking approach for $\nu$TL is discussed where a counterexample can be obtained visually when a model violates a property

DNS Study on Vorticity Structures in Late Flow Transition

Vorticity and vortex are two different but related concepts. This paper focuses on the investigation of vorticity generation and development, and vorticity structure inside/ outside the vortex. Vortex is a region where the vorticity overtakes deformation. Vortex cannot be directly represented by the vorticity. Except for those vorticity lines which come from and end at side boundaries, another type of vorticity, self-closed vorticity lines named vorticity rings, is numerously generated inside the domain during flow transition. These new vorticity rings are found around the hairpin vortex heads and legs. The generation and growth of vorticity rings are produced by the buildup of the vortices according to the vorticity transport equation. On the other hand, vortex buildup is a consequence of vorticity line stretching, tilting and twisting. Both new vorticity and new vortices are generated during the flow transition. According to the Helmholtz vorticity flux conservation law, vorticity line cannot be interrupted, started, or ended inside the flow field, the newly produced vorticity has only one form which is the vorticity rings. In addition, an interesting finding is that a single hairpin vortex consists of several types of vorticity lines which could come from the side boundaries, whole vorticity rings and part of vorticity rings

Super-pixel cloud detection using Hierarchical Fusion CNN

Cloud detection plays a very important role in the process of remote sensing images. This paper designs a super-pixel level cloud detection method based on convolutional neural network (CNN) and deep forest. Firstly, remote sensing images are segmented into super-pixels through the combination of SLIC and SEEDS. Structured forests is carried out to compute edge probability of each pixel, based on which super-pixels are segmented more precisely. Segmented super-pixels compose a super-pixel level remote sensing database. Though cloud detection is essentially a binary classification problem, our database is labeled into four categories: thick cloud, cirrus cloud, building and other culture, to improve the generalization ability of our proposed models. Secondly, super-pixel level database is used to train our cloud detection models based on CNN and deep forest. Considering super-pixel level remote sensing images contain less semantic information compared with general object classification database, we propose a Hierarchical Fusion CNN (HFCNN). It takes full advantage of low-level features like color and texture information and is more applicable to cloud detection task. In test phase, every super-pixel in remote sensing images is classified by our proposed models and then combined to recover final binary mask by our proposed distance metric, which is used to determine ambiguous super-pixels. Experimental results show that, compared with conventional methods, HFCNN can achieve better precision and recall

On some properties of a class of fractional stochastic heat equations

We consider nonlinear parabolic stochastic equations of the form \partial_t u=\sL u + \lambda \sigma(u)\dot \xi on the ball $B(0,\,R)$, where $\dot \xi$ denotes some Gaussian noise and $\sigma$ is Lipschitz continuous. Here \sL corresponds to an $\alpha$-stable process killed upon exiting $B(0, R)$. We will consider two types of noise; space-time white noise and spatially correlated noise. Under a linear growth condition on $\sigma$, we study growth properties of the second moment of the solutions

Critical points of solutions for mean curvature equation in strictly convex and nonconvex domains

In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of $\mathbb{R}^{n}(n\geq2)$. Secondly, we study the geometric structure about the critical set $K$ of solutions $u$ for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of $\mathbb{R}^{n}(n\geq3)$, and deduce that $K$ exists (respectively does not exist) a rotationally symmetric critical closed surface $S$. In fact, in an eccentric spherical annulus domain, $K$ is made up of finitely many isolated critical points ($p_1,p_2,\cdots,p_l$) on an axis and finitely many rotationally symmetric critical Jordan curves ($C_1,C_2,\cdots,C_k$) with respect to an axis.Comment: 13 pages, 5 figure