泛函微分方程的循环多步法

其中泛函F:1×C~1(I)×C~0(I)→R,I表示区间[0,T],满足下列条件: A_1.对于给定的x∈C~1(I),映射t→F(t,x(·),x′(·))在I上连续. A_2.算子F满足Lipschitz条件:     

再析一道以二进制为背景的高考题

题 (2011年湖南卷理16)对于n∈N+,将n表示为n=a0×2k+a1 ×2k-1 +a2 ×2k-2+…+ak-1 ×21 +ak×20,当i=0时,ai=1,当1≤i≤n时,ai为0或1.记I(n)为上述表示中ai为0的个数.(例如:1=1 ×20,4=1 ×22+0×21 +0×20,故I(1)=0,I(4)=2),则(1)I(12)=____;(2)127∑n=12I(n)=____.     

一类非自治离散周期系统的周期解

τ∈I={τ_0 i,τ_0>0,i=0,1,2,…},x∈R~n,A:I×R~n→R~n×n和b:I×R~n→R~n是连续的.设对所有的(τ,x)∈I×R~n有某个整数m>1,使得A(τ m,x)=A(τ,x),B(τ m,x)=b(τ,x),并记I_0={τ_0,τ_0 1,…,τ_0 m-1}.这时称系统(1)为离散周期系统,用x(τ,τ_0,x_0)表示系统(1)满足初始条件x(τ_0)=x_0的唯一解,并对初始值x_0是这续的,τ≥τ_0>0.利用Schauder不动点定理,可以证明如下的:     

Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Let I be a compact interval of real axis R, and(I, H) be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f : I → I be a continuous multi-valued map. Assume that Pn =(x_0, x_1,..., xn) is a return tra jectory of f and that p ∈ [min Pn, max Pn] with p ∈ f(p). In this paper, we show that if there exist k(≥ 1) centripetal point pairs of f(relative to p)in {(x_i; x_i+1) : 0 ≤ i ≤ n-1} and n = sk + r(0 ≤ r ≤ k-1), then f has an R-periodic orbit, where R = s + 1 if s is even, and R = s if s is odd and r = 0, and R = s + 2 if s is odd and r 0. Besides,we also study stability of periodic orbits of continuous multi-valued maps from I to I.     

BIFURCATION SOLUTION BRANCHESAND THEIR NUMERICAL APPROXIMATIONS OF A SEMI-LINEAR ELLIPTIC PROBLEM WITH TWO PARAMETERS

1.IntroductionConsidertheequationdependingontheparametersA,pER,wheref'R-Rands:R-RaresmoothoddfunctionandLetS'u(x)-u(T--x),r={S,I}.Then(l.l)isr-equivalent.Theequality(l.Za)isjustanormalizationoffatx=0.WeintroduceaSobolevspaceX:=Ha(0,7),anddefineamappingT'gEL'(0,T)u'=TaEXimplicitly'Aweakformof(1.1)inXxRZisDuetof(0)~0,theproblem(1.3)(resp.(1))hasatrivialsolutioncurveIfwerestrictp=0,then(l.3)reducestoaproblemwithsingleparameteranditsbifurcationsonthetrivialsolutioncurveCOarewellknown,…     

A MIXED SUPERLINEARLY CONVERGENT ALGORITHM WITH NONMONOTONE SEARCH FOR CONSTRAINED OPTIMIZATIONS

§ 1　 IntroductionConsider the optimization problemmin{ f(x) :gj(x)≤ 0 ,j∈ I,gj(x) =0 ,j∈ L,x∈ Rn} ,(1 )where f(x) ,gj(x) :Rn→R,j∈I∪L.I={ 1 ,2 ,...,m} ,L={ m 1 ,...,m p} .We know thatthe sequential quadratic programming(SQP) [1～ 4] is one of the mostef-ficient methods to solve problem(1 ) because of its superlinear convergence.In order toovercome the Maratos effect[5] ,SQP should solve two quadratic sub-programmings ateachiteration,which,however,causes the amountof computation…     

THE HAUSDORFF DIMENSION AND MEASURE IN GENERALIZED GEOMETRIC GRAPH DIRECTED CONSTRUCTION

1GeneralizedGeometricGraphDirectedContructionAgenera1izedgeometricgraphdirectedconstructioninR"consistof(1)afinitesequence0fnonoverlapping,compactsubsetsofRm:J1,J2,''tJn.SuchthateachJihasanonemptyinterior,(2)adirectedgeaPh{G}withvertexsetconsistingoftheintegers1,'',n,andsiAn-laritymapsTf?)ofR".where(i,j)EG,withratiost::),suchthat(a)foreachi,1Si5n.thereissomejsuchthat(i,j)EG.(b)foreachi,{ri:)(J,)I(i,i)EG}isanon0verlappingfandlyand(c)ifthepathc0mponentofGr0tedattheyertexi1isacycIe:Fo…     

BIFURCATION SOLUTION BRANCHES AND ITS NUMERICAL ANALYSIS OF A SEMI-LINEAR ELLIPTIC PROBLEM

1　IntroductionConsider the parameter dependent equationu"+ (λ+ s(μ) ) f( u) -μsinx =0　 in ( 0 ,π)u( 0 ) =u(π) =0 ( 1 .1 )whereλ,μ∈R are parameters and f:R→R and S:R→R are smooth odd functions anda) f′( 0 ) =1 ,　　 b) f ( 0 )≠ 0 ,　　 c) s( 0 ) =0 ,　　 d) s′( 0 ) =1 . ( 1 .2 )Let S:u( x)→ u(π-x) ,Γ ={ S,I} ,then ( 1 .1 ) isΓ -equivariant.The equality ( 1 .2 a) isjust a normalization of f at x=0 .Otherwise,one may reseek the parameter x to ensure( 1 .2 a) .To simplify an…     

珠心算排积法论

1.5乘的本个数位乘数0 2 4 6 8本个数0 0 0 0 0 取O 取5双数位乘数其本个数是0,单数位乘数其本个数是2.5乘的进位规律(2)位积:5+后半 (本个数是单)例1 5 x 72。396 ,/'II>/1吲期朝例2 5 x 19,4231.6乘的本个数7进小半37本个5,2进半l,位积62本个0.3进小半1.位积l3本个5,9i!t小半4,位积99本个5,6进半3,位积86本个Ol进小半0l本个5.9进小半4,位积99本个5.4进半2.位积74本个O,2进半1.位积l2本个O,3进小半1。位积l3本个5位乘数0 2 4 6 8 1 3 5 7 9. . I I I I I I I I I I本个数0 2 4 6 8 6 8 0 2 4自己 替数本双数位乘数 OI 2I 4I 6I 8I…     

对带有盒约束的二次整数规划的一种线性化方法

本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合0-1整数规划问题,然后利用Ilog-cplex或Excel软件中的规划求解工具进行求解,从而解决原二次整数规划.     

Global optimality conditions for quadratic 0-1 optimization problems

In the present work, we intend to derive conditions characterizing globally optimal solutions of quadratic 0-1 programming problems. By specializing the problem of maximizing a convex quadratic function under linear constraints, we find explicit global optimality conditions for quadratic 0-1 programming problems, including necessary and sufficient conditions and some necessary conditions. We also present some global optimality conditions for the problem of minimization of half-products.     

Decomposition and Linearization for 0-1 Quadratic Programming

This paper presents a general decomposition method to compute bounds for constrained 0-1 quadratic programming. The best decomposition is found by using a Lagrangian decomposition of the problem. Moreover, in its simplest version this method is proved to give at least the bound obtained by the LP-relaxation of a non-trivial linearization. To illustrate this point, some computational results are given for the 0-1 quadratic knapsack problem.     

A new penalty parameter for linearly constrained 0–1 quadratic programming problems

The equivalence between the linearly constrained 0–1 quadratic programming problem and the continuous quadratic programming problem is studied in this note. Specifically, we show that the existing penalty parameter from the literature can be further improved.     

考虑协同效应的突发事件救援人员分组方法

针对突发事件的应急救援人员分组问题,提出了一种考虑人员间协同效应的应急救援人员分组方法。首先,对突发事件应急救援人员分组问题进行了描述。然后,基于系统协同理论的思想,提出了救援小组的协同度模型,定量化的度量救援人员间的协同效应。进一步地,综合考虑救援人员的基础效能与协同效应,基于协同度模型给出了救援人员完成不同任务的实际效能的表达式。在此基础上,以最大化各救援小组的实际效能为目标,构建了突发事件应急救援人员分组的优化模型,并给出了将这一多目标0-1二次规划模型转换为单目标0-1线性规划模型的方法,进而可求解并确定救援人员的最优分组方案。最后,通过一个实例分析说明了本文所提出方法的可行性和有效性。     

An improved linearization technique for a class of quadratic 0-1 programming problems

The recent research on linearization techniques for solving 0-1 quadratic programming problems focuses on providing concise models and tightening constraint bounds. In this paper, we propose a computational enhancement for a linearization technique to make the linearized model much faster to solve. We investigate the computational performance of the proposed approach, by comparing it with other linearization techniques on a class of 0-1 quadratic programming problems. We can further speed up the proposed technique by heuristically tightening the constraint bounds, as demonstrated by solving the uncapacitated single allocation p-hub median problem using the Civil Aeronautics Board data.     

Penalty Parameter for Linearly Constrained 0–1 Quadratic Programming

A linearly constrained 0–1 quadratic programming problem is proved to be equivalent to a continuous concave quadratic problem with an easily computed penalty parameter. Moreover, it is proved that the feasibility of the former problem can be checked by solving the latter.     

Tightening concise linear reformulations of 0-1 cubic programs

A common strategy for solving 0-1 cubic programs is to reformulate the non-linear problem into an equivalent linear representation, which can then be submitted directly to a standard mixed-integer programming solver. Both the size and the strength of the continuous relaxation of the reformulation determine the success of this method. One of the most compact linear representations of 0-1 cubic programs is based on a repeated application of the linearization technique for 0-1 quadratic programs introduced by Glover. In this paper, we develop a pre-processing step that serves to strengthen the linear programming bound provided by this concise linear form of a 0-1 cubic program. The proposed scheme involves using optimal dual multipliers of a partial level-2 RLT formulation to rewrite the objective function of the cubic program before applying the linearization. We perform extensive computational tests on the 0-1 cubic multidimensional knapsack problem to show the advantage of our approach.     

Quantitative stability analysis for vector problems of 0–1 programming

We consider multiple objective 0–1 programming problems in the situation where parameters of objective functions and linear constraints are exposed to independent perturbations. We study quantitative characteristics of stability (stability radii) of problem solutions. An approach to deriving formulae and estimations of stability radii is presented. This approach is applied to stability analysis of the linear 0–1 programming problem and problems with two types of nonlinear objective functions: linear absolute value and quadratic.     

Quadratic programming with one negative eigenvalue is NP-hard

We show that the problem of minimizing a concave quadratic function with one concave direction is NP-hard. This result can be interpreted as an attempt to understand exactly what makes nonconvex quadratic programming problems hard. Sahni in 1974 [8] showed that quadratic programming with a negative definite quadratic term (n negative eigenvalues) is NP-hard, whereas Kozlov, Tarasov and Haijan [2] showed in 1979 that the ellipsoid algorithm solves the convex quadratic problem (no negative eigenvalues) in polynomial time. This report shows that even one negative eigenvalue makes the problem NP-hard.This author's work supported by the Applied Mathematical Sciences Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013. A000 and in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS 8920550.