When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points

There are two algebraic lower bounds of the number of n-periodic points of a self-map f : M → M of a compact smooth manifold of dimension at least 3: NF_n(f) = min{#Fix(g~n); g ～ f; g continuous} and NJD_n(f) = min{#Fix(g~n); g ～ f; g smooth}. In general, NJD_n(f) may be much greater than NF_n(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism,the equality NF_n(f) = NJD_n(f) holds for all n ? all eigenvalues of a quotient cohomology homomorphism induced by f have moduli 1.     

Least number of periodic points of self-maps of Lie groups

There are two algebraic lower bounds of the number of n-periodic points of a self-map f :M → M of a compact smooth manifold of dimension at least 3:N Fn(f) = min{#Fix(gn); g ～f; g is continuous} and N J Dn(f) = min{#Fix(gn); g ～ f; g is smooth}.In general,N J Dn(f) may be much greater than N Fn(f).If M is a torus,then the invariants are equal.We show that for a self-map of a nonabelian compact Lie group,with free fundamental group,the equality holds  all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.     

紧致流形上Lichnerowicz方程的梯度估计(英文)

Let(M,g) be a smooth compact Riemannian manifold of dimension n.Denote△f=△-▽f.▽ the weighted Laplacian operator,where f is a smooth real valued function on M.When N is finite and the N-Bakry-Emery Ricci tensor is bounded from below by a constant,we establish local gradient estimates for positive solutions of the following simple Lichnerowicz equation△fu+cu~(-α)=0 on a compact Riemannian manifold,where α is a positive constant and c is a smooth function.     

RESEARCH ANNOUNCEMENTS FIXED POINTS AND BRAIDS Ⅱ

Let X be a connected compact polyhedron and let f:X→X be a map.Then theNielsen number N(f) is always a lower bound to MF[f]:=Min{#Fix(g)|g≈f:X→X},the least number of fixed points in the homotopy class.(See [2] or [4].)It is known [1] that if X has no local cut points and X is not a surface ofnegative Euler characteristic,then N(f)=MF[f] for all maps f:X→X.We now     

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.     

ON EULER CHARACTERISTIC OF MODULES~(**)

This paper gives a characteristic property of the Euler characteristic for IBN rings. The following results: are proved. (1) If R is a commutative ring, M, N are two stable free R-modules, then χ(MN)=χ(M)χ(N), where χ denotes the Euler characteristic. (2) If f: K_0(R)→Z is a ring isomorphism, where K_0(R) denotes the Grothendieck group of R, K_0(R) is a ring when R is commutative, then f([M])=χ(M) and χ(MN)=χ(M)χ(N) when M, N are finitely generated projective R-modules, where.the isomorphism class [M] is a generator of K_0(R). In addition, some applications of the results above are also obtained.     

Approximation Solutions of Nonlinear Strongly Accretive Operator Equations by Ishikawa Iteration Procedure with Errors

Let 1＜ρ≤2,E be a real ρ-uniformly smooth Banach space and T:E→E be a continuous and strongly accretive operator.The purpose of this paper is to investigate the problem of approximating solutions to the equation Tx=f by the Ishikawa iteration procedure with errors (?) where x_0 ∈ E,{u_n},{υ_n}are bounded sequences in E and{α_n},{b_n},{c_n},{a_n~'},{b_n~'},{c_n~'} are real sequences in[0,1].Under the assumption of the condition 0＜α≤b_n c_n,An≥0, it is shown that the iterative sequence{x_n}converges strongly to the unique solution of the equation Tx=f.Furthermore,under no assumption of the condition(?)(b_n~' c_n~')=0,it is also shown that{x_n}converges strongly to the unique solution of Tx=f.     

A STATE SPACE ISOMORPHISM THEOREM F0R MINIMAL DYNAMICAL SYSTEMS

Consider a discrete time dynamical system x_(k 1)=f(x_k) on a compact metric space M, wheref: M→M is a continuous map. Let h:M→R~k be a continuous output function. Suppose that all ofthe positive orbits of f are dense and that the system is observable. We prove that any outputtrajectory of the system determines f and h and M up to a homeomorphism.If M is a compactAbelian topological group and f is an ergodic translation, then any output trajectory determinesthe system up to a translation and a group isomorphism of the group.     

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.     

Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

Let G be a simple graph with 2n vertices and a perfect matching.The forcing number f(G,M) of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G.Among all perfect matchings M of G,the minimum and maximum values of f(G,M) are called the minimum and maximum forcing numbers of G,denoted by f(G) and F(G),respectively.Then f(G)≤F(G) ≤n-1.Che and Chen(2011) proposed an open problem:how to characterize the graphs G with f(G)=n-1.Lat...     

On Robustness of Orbit Spaces for Partially Hyperbolic Endomorphisms

In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space(the inverse limit space) M~f of f is topologically quasi-stable under C~0-small perturbations in the following sense: For any covering endomorphism g C~0-close to f, there is a continuous map φ from M~g to Multiply form -∞ to ∞ M such that for any {y_i }_(i∈Z) ∈φ(M~g), y_(i+1) and f(y_i) differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense: For any pseudo-orbit {x_i }_(i∈Z),there is a sequence of points {y_i }_(i∈Z) tracing it, in which y_(i+1) is obtained from f(y_i) by a motion along the center direction.     

Boundability of Open Manifolds

An open manifold in question is assumed to bea noncompact manifold with compact boundary (probably empty). Let M be a connected open n-dimensional smooth manifold. Take the one-point-compactification of M, it will be denoted by M=M∪{*}, where * is the infinite point. Since M has countable basis, M is metrizable and {*} is a closed subset and a G_δset. By Urysohn lemma, there exists a continuous function f:→[O,1] with f~-1(O)=M and f~-1(1)=*.Choose a sui-     

ON THE L_1 EXACT PENALTY FUNCTION WITH LOCALLY LIPSCHITZ FUNCTIONS

In this paper,we discuss the following inequality constrained optimization problem (P) min f(x) subject to g(x)≤0,g(x)=(g_1(x),…,g_r(x))~T, where f(x),g_j(x)(j=1,…,r)are locally Lipschitz functions.The L_1 exact penalty function of the problem (P) is (PC) min f(x)+cp(x)subject to x∈R~n, where p(x)=max{0,g_1(x),…,g_r(x)},c>0.We will discuss the relationships between (P) and (PC).In particular,we will prove that under some (mild) conditions a local minimum of (PC) is also a local minimum of (P).     

Quasi-periodic solutions for the general semilinear Duffing equations with asymmetric nonlinearity and oscillating potential

In this paper,we prove the existence of quasi-periodic solutions and the boundedness of all the solutions of the general semilinear quasi-periodic differential equation x′′+ax~+-bx~-=G_x(x,t)+f (t),where x~+=max{x,0},x~-=max{-x,0},a and b are two different positive constants,f(t) is C~(39) smooth in t,G(x,t)is C~(35) smooth in x and t,f (t) and G(x,t) are quasi-periodic in t with the Diophantine frequency ω=(ω_1,ω_2),and D_x~iD_t~jG(x,t) is bounded for 0≤i+j≤35.     

映射的横截性定理

Hirsch conjectured: M、N、A are differential manifolds, g∈C^∞(A,N), then the set T = {f∈C^∞(M,N)|f∈g} is dense in C∞(M,N)and open if g is proper.In this paper, we prove the transversality theorem of map in the Jet bundle.Theorem 1 Let M, N. A be differential manifolds, g∈C^∞(A,J^τ(M、N)), then the set T{f∈C^∞(M,N)|f^τf∈g } is residual in C^∞(M,N) and open if g is proper.Theorem 1 contains Thom's transversal ity theorem as a special case. We can obtain Hirsch's conjecture by using theorem 1.     

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.     

CONVERGENCE OF NONLINEAR CONJUGATE GRADIENT METHODS

1. IntroductionConsider the unconstrained OPtbo8tion problem,min f(x), (1.1)where j is smooth and its gradient g is available. Conjugate gradieot methods are highly usefulfOr solving (1.1) especially if n is large. They are iterative methods of the formHere oh is a 8tepsbo obtained by a 1-dboensional line search and gk is a scalar. The chOiceof Ph is such tha (l.2)--(l.3) reduces to the linear cOnugate gradient method in the casewhen j is a strictly convex qUadratic and crk is the exact 1-…     

一类特殊拟可微函数最优化问题的线性化方法

In this paper,we consider the problem of minimizing a particular class of quasi-differentiable functions:min{f(x)=max min fij(x)}.An algorithm for this problem is giver.At each iteration by solving quadratic programming subproblems to generate search directions,its convergence is proved in the sense of inf-stationary points.     

依赖于导数项的非对称振子解的无界性（英文）

In this paper, we consider the unboundedness of solutions for the asymmetric equation x'+ax~+-bx~-+(x)ψ(x')+f(x)+g(x')=p(t),where x~+= max{x, 0}, x~-= max{-x, 0}, a and b are two different positive constants,f(x) is locally Lipschitz continuous and bounded, (x), ψ(x), g(x) and p(t) are continuous functions, p(t) is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case 1/a~(1/2)+1/b~(1/2)∈Q and the nonresonance case 1/a~(1/2)+1/b~(1/2)?Q     

Unb ounded Motions in Asymmetric Oscillators Dep ending on Derivatives

In this paper, we consider the unboundedness of solutions for the asymmetric equation x00+ax+?bx?+?(x)ψ(x0)+f(x)+g(x0)=p(t), where x+ = max{x, 0}, x? = max{?x, 0}, a and b are two different positive constants, f (x) is locally Lipschitz continuous and bounded,?(x), ψ(x), g(x) and p(t) are continuous functions, p(t) is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case √1a+ √1b ∈Q and the nonresonance case√1a + √1b /∈Q.