OSCILLATION CRITERIA FOR SECOND ORDER QUASILINEAR DIFFERENTIAL EQUATIONS

1 IntroductionThis paper is concerned with the problem of oscillation of second-orderquasilinear differential equationsFOr equation (1), the following conditions, which are referred to (FO), arealways assumed to be valid(a) a > 0 is a constant;(b) F: [to, co) x R x R x R x R - R is a continuous function;(c) sgn F(t, x) u? v3 w) = sgn x for each t 2 to and x, u? v? w E R,(d) r(t) E C([to, co); (0, co)) and jco ddt = co.' (e) T: [to, co) - R is continuous, T(t) 5 t for t 2 to and ill7T(t)…     

A common property of R(E,F) and B(R~n,R~m) and a new method for seeking a path to connect two operators

Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite rank.It is well-known that B(Rn) is a Banach space as well as an algebra,while B(Rn,Rm) for m = n,is a Banach space but not an algebra;meanwhile,it is clear that R(E,F) is neither a Banach space nor an algebra.However,in this paper,it is proved that all of them have a common property in geometry and topology,i.e.,they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces).Let Σr be the set of all operators of finite rank r in B(E,F) (or B(Rn,Rm)).In fact,we have that 1) suppose Σr∈ B(Rn,Rm),and then Σr is a smooth and path-connected submanifold of B(Rn,Rm) and dimΣr = (n + m)r-r2,for each r ∈ [0,min{n,m});if m = n,the same conclusion for Σr and its dimension is valid for each r ∈ [0,min{n,m}];2) suppose Σr∈ B(E,F),and dimF = ∞,and then Σr is a smooth and path-connected submanifold of B(E,F) with the tangent space TAΣr = {B ∈ B(E,F) : BN(A)-R(A)} at each A ∈Σr for 0 r ∞.The routine methods for seeking a path to connect two operators can hardly apply here.A new method and some fundamental theorems are introduced in this paper,which is development of elementary transformation of matrices in B(Rn),and more adapted and simple than the elementary transformation method.In addition to tensor analysis and application of Thom’s famous result for transversility,these will benefit the study of infinite geometry.     

SMOOTH AND PATH CONNECTED BANACH SUBMANIFOLD ∑r OF B(E,F) AND A DIMENSION FORMULA IN B(IRn,IRm)

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear oper-ators from E into F, ∑r the set of all operators of finite rank r in B(E, F), and ∑#r the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submani-fold in B(E, F) with given expression of its tangent space at each A ∈∑r. In this paper, the equality ∑#r = 1 is proved. Consequently, the following theorem is obtained: for any non-negative integer r, ∑r is a smooth and path connected Banach submanifold in B(E, F) with the tangent space TA∑r={B∈B(E,F): BN(A) R(A)} at each A∈∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = IRn and F = IRm, then ∑r is a smooth and path connected submanifold of B(IRn,IRm) and its dimension is dim ∑r = (m + n)r- r2 for each r, 0 ≤ r < min{n,m}.     

Limited memory quasi-newton method for large-scale linearly equality-constrained minimization

1. IntroductionConsider the following linearly constrained nonlinear programming problemwhere x e R", A E Rmxn and f E C2. We are interested in the case when n and m arelarge and when the Hessian matrix of f is difficult to compute or is dense. It is ajssumed thatA is a matrix of full row rank and that the level set S(xo) = {x: f(x) 5 f(xo), Ax ~ b} isnonempty and compact.In the past few years j there were two kinds of methods for solving the large-scaleproblem (1.1). FOr the one kind, pr…     

THE CONVERGENCE OF APPROACH PENALTY FUNCTION METHOD FOR APPROXIMATE BILEVEL PROGRAMMING PROBLEM

' 1 IntroductionWe collsider the fOllowi11g bilevel programndng problen1:max f(x, y),(BP) s.t.x E X = {z E RnIAx = b,x 2 0}, (1)y e Y(x).whereY(x) = {argmaxdTyIDx Gy 5 g, y 2 0}, (2)and b E R", d, y E Rr, g E Rs, A, D.and G are m x n1 s x n aild 8 x r matrices respectively. If itis not very difficult to eva1uate f(and/or Vf) at all iteration points, there are many algorithmeavailable fOr solving problem (BP) (see [1,2,3etc1). However, in some problems (see [4]), f(x, y)is too com…     

A Note on a Problem of M. S. Berger

In this paper, we discuss the problem concerning global and local structure of solutions of an operator equation posed by M. S. Berger. Let f : U (?)E→ F be a C1 map, where E and F are Banach spaces and U is open in E. We show that the solution set of the equation f(x)=y for a fixed generalized regular value y of f is represented as a union of disjoint connected C1 Banach submanifolds of U, each of which has a dimension and its tangent space is given. In particular, a characterization of the isolated solutions of the equation f(x) = y is obtained.     

环状硅酸盐图的友好指数集

Let G be a graph with vertex set V(G) and edge set E(G). A labeling f : V(G) →Z2 induces an edge labeling f*: E(G) → Z2 defined by f*(xy) = f(x) + f(y), for each edge xy ∈ E(G). For i ∈ Z2, let vf(i) = |{v ∈ V(G) : f(v) = i}| and ef(i) = |{e ∈ E(G) : f*(e) =i}|. A labeling f of a graph G is said to be friendly if |vf(0)- vf(1)| ≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as {|ef(0)- ef(1)|: the vertex labeling f is friendly}. This is a generalization of graph cordiality. We investigate the friendly index sets of cyclic silicates CS(n, m).     

Generalized Inverse Analysis on the Domain Ω(A,A+) in B(E,F)

Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B~+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B~+(E, F). In this paper we introduce an unbounded domain ?(A, A~+) in B(E, F) for A ∈ B~+(E, F) and A~+∈GI(A), and provide a necessary and sufficient condition for T ∈ ?(A, A~+). Then several conditions equivalent to the following property are proved: B = A+(IF+(T-A)A~+)~(-1) is the generalized inverse of T with R(B)=R(A~+) and N(B)=N(A~+), for T∈?(A, A~+), where IF is the identity on F. Also we obtain the smooth(C~∞) diffeomorphism M_A(A~+,T) from ?(A,A~+) onto itself with the fixed point A. Let S = {T ∈ ?(A, A~+) : R(T)∩ N(A~+) ={0}}, M(X) = {T ∈ B(E,F) : TN(X) ? R(X)} for X ∈ B(E,F)}, and F = {M(X) : ?X ∈B(E, F)}. Using the diffeomorphism M_A(A~+,T) we prove the following theorem: S is a smooth submanifold in B(E,F) and tangent to M(X) at any X ∈ S. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain ?(A, A~+). It seems to be useful for developing global analysis and geomatrical method in differential equations.     

SOME RESULTS ON UNIVERSAL MINIMAL TOTAL DOMINATING FUNCTION

1. IntroductionLet G = (V, E) be an undirected graph. The open neighborhood N(v) of vertex v E Vis given by N(v) = {u E V: tv E E}. A total dominating function (TDF) of G is afunction f: V - [0, 11 such that Z f(u) 2 1 for each vertex v. A TDF f is minimaladN(~)(MTDF) if no function g < f is also a TDF of G. The illteger valued TDFs are preciselythe characteristic functions of total dominating sets of G (i.e., subset X G V such that anyv is adjacellt to at leajst one x E X). T…     

一些超空间的非双Lipschitz 齐次性

A metric space(X, d) is called bi-Lipschitz homogeneous if for any points x, y ∈ X,there exists a self-homeomorphism h of X such that both h and h-1are Lipschitz and h(x) = y.Let 2(X,d)denote the family of all non-empty compact subsets of metric space(X, d) with the Hausdorff metric. In 1985, Hohti proved that 2([0,1],d)is not bi-Lipschitz homogeneous, where d is the standard metric on [0, 1]. We extend this result in two aspects. One is that 2([0,1],e)is not bi-Lipschitz homogeneous for an admissible metric e satisfying some conditions. Another is that 2(X,d)is not bi-Lipschitz homogeneous if(X, d) has a nonempty open subspace which is isometric to an open subspace of m-dimensional Euclidean space Rm.     

A feasible and superlinear algorithm for inequality constrained minimization problems

1. IntroductionConsider the optimization problemmin {f(x): gi(x) 5 0, j E I; x E R"}, (l)where f(x), gi(x): Rad - R, j E I ~ {1, 2,...,m}.It is well known that one of the most effective methods to solve problem (1) is thesequential quadratic programming (i.e., SoP) (see [1--6]), due to its property of superlinearconvergence. Especially in recent years, in order to perfect SoP both in theory and application, there have many papers, such as [7--10], been published. These papers focus mainly…     

Note on the Paper “A Note on the SVD of Matrices”

Let F be a field, R be an F-algebra having SVD property['], R+ = {a e Rla = a}. In[2], we proved thatLemma 1 Let R be an F-algebra which having SVD proPerty Then R+ is a sub6e1d inF, R+ is a fOrmaIly rea1 6eId, and R+ is Pythagorean with no zerChdivisor.In the proof of Lernma 1 of [2], thereis this step: "FOr any a E R+ l clearly, there ekistsd E R+ n F such that a2 = ad = a', thus a = la E F, is fOllOws that R+ C F".Recently, paper [11 shows that "The proof of this step is no…     

A QP-FREE AND SUPERLINEARLY CONVERGENT ALGORITHM FOR INEQUALITY CONSTRAINED OPTIMIZATIONS

1 IntroductionConsider tl1e optimizatioll problemndn{f(x): gj(x) 5 0, j e I, x E R"}, j1)where f(x), gj(x): R" - R, j E I = {l,2,...,m}.We know tl1e quasi-Newton meth.d[1]'[9]1[5]1[1O1 is one of the most effective methods to solveproblenl (1) due to its property of superlinear convergence and is still all hot topic at presenttime, which attracts a Iot of authors to make iInprovemellt both in theory a1ld app1ication.Fechinei and Lucidi[3] in 1995 proposed a locally superlinearly convergell…     

有关M.S.Berger问题的注记

In this paper, we discuss the problem concerning global and local structure of solutions of an operator equation posed by M. S. Berger. Let f : U 真包含 E → F be a C1 map, where E and F are Banach spaces and U is open in E. We show that the solution set of the equation f(x) = y for a fixed generalized regular value y of f is represented as a union of disjoint connected C1 Banach submanifolds of U, each of which has a dimension and its tangent space is given. In particular, a characterization of the isolated solutions of the equation f(x) = y is obtained.     

A note on the existence of fractional f-factors in random graphs

Let G=Gn,p be a binomial random graph with n vertices and edge probability p=p(n),and f be a nonnegative integer-valued function defined on V(G) such that 0a≤f(x)≤bnp-2np ㏒n for every x ∈V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0,1] so that for each vertex x,we have dh G(x)=f(x),where dh G(x) = x∈e h(e) is the fractional degree of x in G. Set Eh = {e:e ∈E(G) and h(e)=0}.If Gh is a spanning subgraph of G such that E(Gh)=Eh,then Gh is called an fractional f-factor of G. In this paper,we prove that for any binomial random graph Gn,p with p≥n-23,almost surely Gn,p contains an fractional f-factor.     

Oscillatory and Asymptotic Behavior for Second Order Quasilinear Difference Equations

1 IntroductionConsider the second order quasilinear difference equationA(g(Ay.--l)) + f(n,y.) = 0, for n E N(no), (l'l)where A is defined by Ay. = Vn+1--yn, n E N(no) = {no, no + 1,'' }, nO E N = {l, 2,'. }.The following hold throughout the paPer:(H0) (i) g: R-R is a continuous increasing fUnction with propertiessgng(y) = sgny) g(R) = R;(il) f: N(no) x R--+ R is continuous as a function of y E R;(iii) yf(n,y) > 0 for n E N and y / 0.By a solution of the equation (1.l) we mean a non…     

PICARD ITERATION FOR NONSMOOTH EQUATIONS

1. IntroductionConsider the following nonsmooth equationsF(x) = 0 (l)where F: R" - R" is LipsChitz continuous. A lot of work has been done and is bellg doneto deal with (1). It is basicly a genera1ization of the cIassic Newton method [8,10,11,14],Newton-lthe methods[1,18] and quasiNewton methods [6,7]. As it is discussed in [7], the latter,quasiNewton methods, seem to be lindted when aPplied to nonsmooth caJse in that a boundof the deterioration of uPdating matrir can not be maintained w…     

Extension of Floquet''''s theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

SOME PRIMITIVE POLYNOMIALS OVER FINITE FIELDS

1 IntroductionLet F, denote the finite field of order q wl1icl1 is a power of a prime. A polynomialf(x) E F,[x] of degree n 2 1 is called a primitive polynomial if it is tl1e 11tinimal po1ynomialover Fq of a primitive element of Fqn. Note that if f(x) is a primitive polyllontiaI of degree n alldf is a priniltive eIen1ent of F,n which is a root of f(x) t tl1en f(T) = (T--()(x--(q)... (x --(q"-- 1 ) =x" -- T(f)x"--' ' (--1)"N(f), where T and N are tl1e trase and the norm from F,n to F…