p阶Feigenbaum映射

A continuous map f from the unit closed interval into itself is called a p-order Feigenbaum's map if fp(λx) = λf(x),f(O)=1 and f|[λ,1] is univallecular. In this paper, some characterizations of p order Feigenbaum's maps are discussed and the existence for both types of such maps is proven.     

ON THE FEIGENBAUM＇S FUNCTIONAL EQUATION f^P （λx）=λf（x）

The author considers the Feigenbaum‘s functional equation f^P(λx)=λf(x) for each p≥2.The existence of even unimodal C^1 solutions to this equation is discussed and a feasible method to construct such solutions is given.     

Likely Limit Sets of Feigenbaum''''s Maps and Their Hausdorf f Dimensions

A continuous map from a closed interval into itself is called a Feigenbaum's map if it is a solution of the functional equation f2(λx)=λf(x).In this paper, the likely limit sets of a type of Feigenbaum's maps are studied and their Hausdorff dimensions are estimated.As an application, we prove that for any 0相似文献   

A REMARK ON THE UNIQUENESS OF THE HARMONIC MAPS

This paper studies the uniqueness of the harmonic maps from R~n into S~(n-1). At first it is proved that the map ψ(x)=x|x|:B~n→S~(n-1) is the unique energy minimizing harmonic map. Then the uniqueness of the harmonic maps with identity boundary value from B~n to S~(n-1) follows.     

Frequency Assignment through Combinatorial Optimization Approach

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|(?)2 if d(x, y)=1 and |f(x)-f(y)|(?)1 if d(x,y)=2. The L(2,1)-labeling numberλ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v) : v∈V(G)}=k. We study the L(3,2,1)-labeling which is a generalization of the L(2,1)-labeling on the graph formed by the (Cartesian) product and composition of 3 graphs and derive the upper bounds ofλs(G) of the graph.     

泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)&gt;0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N&gt;0,k(s)&gt;0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

We study the existence of solutions to the following parabolic equation{ut-△pu=λ/|x|s|u|q-2u,(x,t)∈Ω×(0,∞),u(x,0)=f(x),x∈Ω,u(x,t)=0,(x,t)∈Ω×(0,∞),(P)}where-△pu ≡-div(|▽u|p-2▽u),1相似文献   

Construction of convex solutions for the second type of Feigenbaum’s functional equations

In this paper, convex solutions for the second type of Feigenbaum’s equation f (x) = λ1 f (f (λx)), 0 < λ < 1, f (0) = 1, 0 f (x) 1, x ∈ [0, 1] are investigated. Using constructive methods, we discuss the existence and uniqueness of continuous convex solutions, C1-convex solutions and C2-convex solutions of the above equation.     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

$L^p$函数的分布函数在端点处的极限性质

We consider the limiting property of the distribution function of L~p function at endpoints 0 and ∞ and prove that for λ 0 the following two equations limλ→+∞λ~pm({x : |f(x)| λ}) = 0, limλ→0+λ~pm({x : |f(x)| λ}) = 0hold for f ∈ L~p(Rn) with 1 ≤ p ∞. This result is naturally applied to many operators of type(p, q) as well.     

Non-wandering sets of the powers of dendrite maps

Let(X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f)and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold:(1) If x ∈Ω(f)-Ω(f~n) for some n ≥ 2,then x ∈ EP(f).(2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = ?0(the cardinal number of the set of positive integers).     

树映射的链回归点与拓扑熵

Let f be a tree map,P(f) the set of periodic points of f and CR(f) the set of chain recurrent points of f. In this paper,the notion of division for invariant closed subsets of a tree map is introduced. It is proved that: (1) fhas zero topological entropy if and only if for any x∈CR(f)-P(f) and each natural number s the orbit of x under f^5 has a division; (2) If f has zero topological entropy,then for any xECR(f)--P(f) the w-limit set of x is an infinite minimal set.     

A note on homoclinic orbits for second order Hamiltonian system

Some existence and multiplicity of homoelinic orbits for second order Hamiltonian system x-a(t)x f(t,x)=0 are given by means of variational methods, where the function -1/2a（t）|s|^2∫^t0f（t,s）ds is asymptotically quadratic in s at infinity and subquadratic in s at zero, and the function a (t) mainly satisfies the growth condition limt→∞∫^t 1 t a（t）dt= ∞,VI∈R^1.A resonance case as well as a noncompact case is discussed too.     

Fixed Points of Order Convex Maps in Ordered Banach Spaces

In this paper we discuss problems of the existence of fixed points of order convex maps. Lemma 1 Let (E,P) be an OBS,Suppose f: S_1→S_1 is a completely continuous map. Then f has fixed points in S_1. Lemma 2 Let (E,P) be an OBS whose positive cone is normal and has nonempty interior. Suppose f: P→P is a continuous, order increasing convex map, f(0)=0; such that. Then there exist a convex subset and a positive number r such that and.     

Solutions of Equations Involving Analytic Functions in a Banach Algebra

By the properties of univalent analytic functions,we have discussed the exi-stence and uniqueness of eqation f(x)=a in a Banach algebra.We have the follo-wing fundamental lemmas. Lemma 1. Let A be a Banach algebra with identity e,W,U be two opensubsets of C, U W,f be an analytic function in W, and univalent in U, V=f(U),a∈A.If σ(a) V,then there exists unique x∈A such that σ(x) U {λ∈W|f(λ) V}and f(x)=a     

Inverse Functions of Number-Theoretic Functions (I)

If gf(x) =x for every x, then g is called a left inverse function of f and f is a right inverse function of g. If f is both left and right inverse function of g, then f and g are said to be mutually inverse to each other. We show that (§ 1) the following results hold. A function f has a left inverse if and only if f is univalent, a function g has a right inverse if and only if g is exhaustive, i. e., g takes every (natural) number as values. Hence f has both left and right inverse if and only if f is both univalent and exhaustive, i. e., f is a permutation on the domain of natural numbers. Let g_1 and g_2 be two left inverse functions of the function f. If for every left inverse g of f, we have $g_1(x) \leq g(x) \leq g_2(x)$, then g_1(x) is called the weak, and g_2(x) is the strong, left inverse function of f. Similarly we define the weak and the strong right inverse functions. We show that(§ 2) every strict increasing function f must possess weak and strong left inverse functions, and all of its left inverse functions must be exhaustive slow increasing (a function g(x) is slow increasing if and only if g(Sx) —Sg(x) =0, here s denotes the successor function). On the other hand, every exhaustive function g must possess weak and strong right inverse functions, and all of its right inverse functions must strict increasing. We show also that (§ 3): If f_1(x) and f_2(x) both take g(x) as their strong (weak) left inverse, then f_1(x)=f_2(x)(f_1(Sx)=f_2(Sx)). If g_1(x) and g_2(x) both take f(x) as their strong or weak right inverse, then g_1(x)=g_2(x). From these results we see that we may find a function from its strong (weak) left or right inverse function. Let there be f(c) \leq x 相似文献   

光滑函数类中一族无高阶导数的大范围收敛迭代法

The problem whether the iteration formula with the global convergence which does notneed to compute the second order derivative of the function can be found, raised in [7], issolved for f(x)∈C~1(R~1) in the present paper by using the methods of prior estimates andintroducing a parametric function. The main results are as follows: 1. For f(x)∈C~1(R~1), the families of iteration formulas of the global convergence,without derivatives of higher order, are suggested in the following formx_(n 1)=x_n±|f(x_n)|/|f'(x_n)| α(x_n)|f(x_n)|,(1)x_(n 1)=x_n-α|f(x_n)|/(α-1)f'(x_n)sgnf(x_0)±(f'2(x_n)αp(x_n)|f(x_n)|),(2)x_(n 1)=x_n±|f(x_n)f'(x_n)|/f'2(x_n) 1/2p(x_n)|f(x_n)|,(3)Where the real parameter a∈(0, 2] and the real parametric functions α(x)=α(f(x),f'(x)) (>0) and p(x)= p(f(x), f,(x)) (>0) with certain arbitrariness are continuous orpiecewise continuous. 2. The convergence order of the iteration sequence {x_n} generated by (1), (2) or (3)is 2 for a simple real zero of f(x), and is 1 for a multiple zero.     

Regular maps of graphs of order 4<Emphasis Type="Italic">p</Emphasis>

A 2-cell embedding f : X → S of a graph X into a closed orientable surface S can be described combinatorially by a pair M = (X;ρ ) called a map, where ρ is a product of disjoint cycle permutations each of which is the permutation of the arc set of X initiated at the same vertex following the orientation of S . It is well known that the automorphism group of M acts semi-regularly on the arc set of X and if the action is regular, then the map M and the embedding f are called regular. Let p and q be primes. Du et al. [J. Algebraic Combin., 19, 123-141 (2004)] classified the regular maps of graphs of order pq . In this paper all pairwise non-isomorphic regular maps of graphs of order 4 p are constructed explicitly and the genera of such regular maps are computed. As a result, there are twelve sporadic and six infinite families of regular maps of graphs of order 4 p ; two of the infinite families are regular maps with the complete bipartite graphs K2p,2p as underlying graphs and the other four infinite families are regular balanced Cayley maps on the groups Z4p , Z22 × Zp and D4p .     

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: c(x, y) = inf x(0)=x x(1)=y u∈U∫_0~1 L(x(s), u(x(s), s), s)ds, where U is a control set, and x satisfies the ordinary equation x(s) = f(x(s), u(x(s), s)).It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation:V_t(t, x) + sup u∈UV_x(t, x), f(x, u(x(t), t), t)-L(x(t), u(x(t), t), t) = 0,V(0, x) = Φ0(x).