Why is the Danger Cylinder Dangerous in the P3P Problem?

The PnP problem is a widely used technique for pose determination in computer vision community,and finding out geometric conditions of multiple solutions is the ultimate and most desirable goal of the multi-solution analysis,which is also a key research issue of the problem.In this paper,we prove that given 3 control points,if the camera's optical center lies on the so-called“danger cylinder”and is enough far from the supporting plane of control points,the corresponding P3P problem must have 3 positive solutions.This result can bring some new insights into a better understanding of the multi-solution problem.For example,it is shown in the literature that the solution of the P3P problem is instable if the optical center lies on this danger cylinder,we think such occurrence of triple-solution is the primary source of this instability.

具p-双调和算子的非局部椭圆方程Navier边值问题的广义解

利用变分方法和相应的临界点定理研究一类具有p-双调和算子的非局部椭圆方程Navier边值问题,在非线性项满足超线性条件时,得到了两个非平凡广义解的存在性定理.

一类p-Laplacian问题正径向解的存在性与多解性

研究了p-Laplacian问题{-div(|∇u|^(p-2)∇u)=q(|x|)f(u),|x|>1,x∈R^(N),u(x)=b,|x|=1,u(x)→a,|x|→+∞,其中,1<p<N,a,b为正参数,q∈L^(1)_(loc)((1,+∞),[0,+∞)),f∈C([0,+∞),[0,+∞))。运用锥上的不动点定理、上下解方法和拓扑度理论,获得了p-Laplacian问题正解的存在性和多解性结果。

一般有界区域上高维变权p-Laplacian问题保号解的存在性

研究一般有界区域上高维变权p-Laplacian方程-div(φ_(p)▽(u))=γm(x)f(u),u(x)=0,x∈∂Ω保号解的存在性.其中Ω是R^(N)上的一个有界且在其边界上光滑的区域,m(x)∈C(Ω),γ是一个参数,f∈C(ℝ,ℝ),对于s≠0满足sf(s)>0.当满足f_(0)■(0,∞)或f_(∞)(0,∞)(其中f_(0)=lim|s|→0 f(s)/φ_(p)(s),f_(∞)=lim|s|→∞f(s)/φ_(p)(s)),且γ≠0属于一定区间时,本文研究上述高维p-Laplacian方程保号解的存在性.我们用全局分歧技巧和连通序列集取极限的方法获得主要结果.

p-Laplace问题规范解的存在性

对一类包含p-Laplace算子的非线性问题规范解的存在性进行研究,用极小化方法研究了当微分方程连续且满足某些适当的Berestycki-Lions类型条件下,Nehari-Pohozaev流形的正规范解。

具p-Laplace算子的分数阶脉冲微分方程奇异边值问题的解

用Banach压缩映像原理和Krsnoasel’skii不动点定理证明一类具有p-Laplace算子的分数阶脉冲微分方程奇异边值问题解的唯一性和存在性.

一类带p-Laplace算子的非线性二阶m点共振边值问题解的存在性

基于拓扑横截方法连同障碍带技巧,得到一类带p-Laplace算子的非线性二阶m点共振边值问题解的新的存在性结果,并给出应用举例。

The Simplest Possible Fully Correct Solution of the Clay Millennium Problem about P vs. NP. A Simple Proof That P ≠ NP = EXPTIME

In the current paper, I present probably the simplest possible abstract formal proof that P ≠ NP, and NP = EXPTIME, in the context of the standard mathematical set theory of computational complexity and deterministic Turing machines. My previous publications about the solution of the P vs. NP with the same result NP = EXPTIME, to be fully correct and understandable need the Lemma 4.1 and its proof of the current paper. The arguments of the current paper in order to prove NP = EXPTME are even simpler than in my previous publications. The strategy to solve the P vs. NP problem in the current paper (and in my previous publications) is by starting with an EXPTIME-complete language (problem) and proving that it has a re-formulation as an NP-class language, thus NP = EXPTIME. The main reason that the scientific community has missed so far such a simple proof, is because of two factors 1) It has been tried extensively but in vain to simplify the solutions of NP-complete problems from exponential time algorithms to polynomial time algorithms (which would be a good strategy only if P = NP) 2) It is believed that the complexity class NP is strictly a subclass to the complexity class EXPTIME (in spite the fact that any known solution to any of the NP-complete problems is not less than exponential). The simplicity of the current solution would have been missed if 2) was to be believed true. So far the majority of the relevant scientific community has considered this famous problem not yet solved. The present results definitely solve the 3rd Clay Millennium Problem about P versus NP in a simple, abstract and transparent way that the general scientific community, but also the experts of the area, can follow, understand and therefore become able to accept.

p-CenterProblem的一个基于中心图合并操作的启发式算法

设计的启发式算法从图的合并操作出发 ,降低了计算复杂度 。

Existence of Multiple Positive Solutions for Higher-Order p-Laplacian Boundary Value Problems

The existence of multiple positive solutions for a class of higher order p Laplacian boundary value problem is studied. By means of the Leggett Williams fixed point theorem in cones, existence criteria which ensure the existence of at least three positive solutions of the boundary value problem are established.

带有p-Laplacian算子的分数阶非线性积分边值问题的唯一正解与和算子方法

该文研究了一类带有p-Laplacian算子并且非线性项f中含有分数阶导数项以及边界条件中含有非线性积分项的分数阶边值问题唯一正解的存在性.通过构造适当的辅助边值问题和锥上的等价类,利用锥理论与和算子方法获得了该边值问题唯一正解存在的充分条件,建立了一致收敛于唯一正解的单调迭代格式,最后给出了一个具体的例子作为所获结论的应用.

P∗(κ)-线性权互补问题的一种全牛顿步可行内点算法

本文提出一种求解P∗(κ)-线性权互补问题(LWCP)的新全牛顿步可行内点算法.首先基于一个连续可微的核函数,构造新代数等价变换,得到光滑中心路径的等价形式.然后沿着搜索方向使用全牛顿步,无需进行线搜索,节省运行内存.最后分析算法的可行性及收敛性,并通过数值算例验证算法的有效性.

SOLUTIONS TO DISCRETE MULTIPARAMETER PERIODIC BOUNDARY VALUE PROBLEMS INVOLVING THE p-LAPLACIAN VIA CRITICAL POINT THEORY

In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [C. Bonanno and P. Candito, Appl. Anal., 88（4） （2009）, pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.

A General Sufficient Condition of Four Positive Solutions of the P3P Problem

In this paper, it is proved that, given 3 control points A, B and C, if the camera＇s optical center O lies on one of the three planes perpendicular to the plane ABC and going through one of the three altitudes of the triangle ABC, and additionally its projection on the plane ABC is within the circumscribed circle of the triangle, that is, O is within the so-called “danger cylinder”, then the corresponding P3P problem {O, （ABC）} must have 4 positive solutions. This result is purely geometrical, and more instructive. It can bring some new insight into a better understanding of multiple-solution problem in the PnP problem, and could be used as some theoretical guide to arrange control points in real applications.

Fast Algorithm for the Travelling Salesman Problem and the Proof of P = NP

In the theory of computational complexity, the travelling salesman problem is a typical one in the NP class. With the aid of a brand-new approach named “maximum-deleting method”, a fast algorithm is constructed for it with a polynomial time of biquadrate, which greatly reduces the computational complexity. Since this problem is also NP-complete, as a corollary, P = NP is proved to be true. It indicates the crack of the well-known open problem named “P versus NP”.

Analysis of Facility Systems’ Reliability Subject to Edge Failures: Based on the p-Median Problem

We view a facility system as a kind of supply chain and model it as a connected graph in which the nodes represent suppliers, distribution centers or customers and the edges represent the paths of goods or information. The efficiency, and hence the reliability, of a facility system is to a large degree adversely affected by the edge failures in the network. In this paper, we consider facility systems' reliability analysis based on the classical p-median problem when subject to edge failures. We formulate two models based on deterministic case and stochastic case to measure the loss in efficiency due to edge failures and give computational results and reliability envelopes for a specific example.

Positive Solutions for Fourth-order Delay Differential Equation of Boundary Value Problem with p-Laplacian

In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian（Φp（u000））0（t）＋a（t）f（t, u（t？τ）, u0（t））=0, 0〈t〈1;u000 （0）=u00 （0）=0, u0 （1）=αu0 （η）;u（t）=0, ？τ ≤t≤0. By using Schauder fixed-point theorem, some su？cient conditions are obtained which guar-antee the fourth-order delay differential equation of boundary value problem with p-Laplacian has at least one positive solution. Some corresponding examples are presented to illustrate the application of our main results.

Smoothing Newton Algorithm for Nonlinear Complementarity Problem with a PFunction

By using a smoothing function,the P nonlinear complementarity problem(P NCP)can be reformulated as a parameterized smooth equation.A Newton method is proposed to solve this equation.The iteration sequence generated by the proposed algorithm is bounded and this algorithm is proved to be globally convergent under an assumption that the P NCP has a nonempty solution set.This assumption is weaker than the ones used in most existing smoothing algorithms.In particular,the solution obtained by the proposed algorithm is shown to be a maximally complementary solution of the P NCP without any additional assumption.

THE UNIQUENESS OF THE L_(p) MINKOWSKI PROBLEM FOR q-TORSIONAL RIGIDITY

In this paper,we prove the uniqueness of the Lp Minkowski problem for q-torsional rigidity with p>1 and q>1 in smooth case.Meanwhile,the Lp Brunn-Minkowski inequality and the Lp Hadamard variational formula for q-torsional rigidity are established.

L^p-L^q decay estimates of solutions to Cauchy problems of thermoviscoelastic systems

L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained.