Banach空间中渐近非扩张映射的不动点迭代

设D是一致凸Banach空间X的非空闭凸子集 ,T∶D→D是渐近非扩张映射且kn ≥ 1 ,∑ ∞n =1(kn- 1 ) <∞ .设T的不动点集F(T) ≠ ,T是全连续的 (X满足Opial条件 ) ,{xn},{yn},{zn}由定义 2给出 ,如果 ∑∞n =1cn <∞ ,∑ ∞n =1c′n <∞ ,∑ ∞n =1c″n <∞ ,且下列条件之一满足 :(i)b″n ∈ [a ,b] ( 0 ,1 ) ;b′n ∈ [0 ,β];bn ∈[0 ,α],αβ β <1 ;(ii)b′n ∈ [a ,b] ( 0 ,1 ) ;b″n ∈ [a ,1 ];bn ∈ [0 ,b];(iii)bn ∈[a ,b] ( 0 ,1 ) ;b′n ∈ [a ,1 ],则 {xn},{yn},{zn}强收敛于T的不动点 .( {xn}弱收敛于T的不动点 ) .     

树映射的非游荡集与拓扑混合性

设 T是个树 ,C0 ( T)表示 T上所有的连续自映射 (即 :树映射 )的集合 ,W={ fn:n≥ 2是自然数 ,f∈ C0 ( T) } .讨论了每一点都是非游荡点的树映射的性质 ,并证明了 :若混合映射 f∈ W( W在 C0 ( T)内的闭包 )且 T的每个端点都不是 f的不动点 ,则存在 g∈ C0 ( T)及自然数 k>1使 f=gk.     

插值多项式对解析部分的收敛估计

设D是单位圆{z||z|<1},T为单位圆周{z||z|=1}.对于f∈C(T),我们记L_n(f,z)为在n 1次单位根{e~(2kπ/n 1)i}~n_k=0上对f(z)的n次插值多项式.自然的L_n(f,z)在D内解析,因此,当f不能解析延拓到D内时,就不可能保证L_n(f,z)一致收敛于f.甚至,存在着f∈C(T),且f是某个D内解析函数的边值,但L_n(f,z)在T上发散.     

给定直径的树上的随机游动接触时（英文）

记T(n,d)为n个顶点且直径为d的树的集合.对T∈T(n,d),max_i∈V H(π,i)和min_i∈V H(π,i)分别表示从平稳分布出发、最终终止于T中某一点的最优终止规则所决定的那些随机游动中所走的平均步数的最大值和最小值.本文通过对树T进行一些图的变换来减小或增大max_i∈V H(π,i)与min_i∈V H(π,i),从而确定了T(n,d)中分别取到max_i∈V H(π,i)上界以及min_i∈V H(π,i)下界的极图.此外还确定了直径不大于4的树中分别取到max_i∈V H(π,i)下界和min_i∈V H(π,i)上界的图.本文的部分结果是对[Graphs Combin.,2013,29(4):757-772]中相应结论的推广.     

非齐次多重调和方程拟共形映射的一个Schwarz-Pick型不等式

令ψn∈φ((D)),ψj∈φ(T)和K≥1,其中n≥2为整数,j∈{1,...,n-1}.本文建立了单位圆盘D到自身且满足非齐次多重调和方程Δnf= ψn以及相应边界值条件:Δn-1f|T =ψn-1,...,Δ1f|T =ψ1和f(0)= 0的K-拟共形映射f的一个Schwarz-Pick型不等式.进一步地,我们证...     

Bernardi积分算子的星象性

设函数f(z)=z+a2z^2+…在单位园D内解析，常数c∈（-1,1]，定义Bernardi积分算子Fc如下Fc(z)=1+c/x^4∫0^zf(t)t^c-1dt,z∈D记S（c)=∞↑∑↑n=1(-1)^n/1+c+n,ρ=0.09032…，δ（c)=-[2ρ+1-c+2(1-c^2)S(c)/1+c-2(1-c^2)S(c)]。本文改进了有关Bernardi积分算子星象性的条件，得到Rcf(z)&;gt;δ(c)(z∈D)蕴涵着Fc(z)的星象性。     

ON THE FUNDAMENTAL FUNCTION FOR CARDINAL L-SPLINE INTERPOLATION

§1 IntroductionFor any set T={t_j}~n_(j=0) of n+1 real numbers, we consider theconstant coefficient differential operator L_(n+1) (D)=multiply from j=0 to n (D-t_j),D=d/dx and denote by S_n(T) the collection of all functions S∈     

一类T形树匹配唯一的充要条件

证明：若m∈Ze^ ,则T形树T(1,m,n）匹配唯一当且仅当n≠m,m 3,2m 5.     

自然数列对虾树及其串联树的强优美性

具有完美匹配M的n阶树T是强优美的,如果对任意的uv∈M,存在树T的一个优美标f,使得f(u)+f(v)=n-1.讨论了自然数列对虾树及其串联树的强优美标号.     

sin1/x为非周期函数的证明

在这篇短文里 ,我们将给出 sin1x为非周期函数的多种证法 ,这些证法是基于周期函数的下述定义 .定义　设 f (x)是定义在 D上的函数 ,若存在某个正数 T,使得 x∈ D,有 x± T∈D,且 f (x± T) =f (x) ,则称 f (x)是定义在D上的周期函数 ,并称 T为 f (x)的一个周期 .证法 1　因为 sin 1x 的定义域 D =(-∞ ,0 )∪ (0 , ∞ ) ,所以 0 D.若 sin 1x 以某个 T >0为一个周期 ,则T∈ D,且应有 T - T =0∈ D,矛盾 .这个矛盾表明 sin 1x 不是周期函数 .证法 2　假设 T >0是 sin 1x 的一个周期 ,则对 x∈ D,有 sin1x=sin 1x T,特别地 ,有…     

On the Combinatorics of Rooted Binary Phylogenetic Trees

We study subtree-prune-and-regraft (SPR) operations on leaf-labelled rooted binary trees, also known as rooted binary phylogenetic trees. This study is motivated by the problem of graphically representing evolutionary histories of biological sequences subject to recombination. We investigate some basic properties of the induced SPR-metric on the space of leaf-labelled rooted binary trees with n leaves. In contrast to the case of unrooted trees, the number |U(T)| of trees in which are one SPR operation away from a given tree depends on the topology of T. In this paper, we construct recursion relations which allow one to determine the unit-neighbourhood size |U(T)| efficiently for any tree topology. In fact, using the recursion relations we are able to derive a simple closed-form formula for the unit-neighbourhood size. As a corollary, we construct sharp upper and lower bounds on the size of unit-neighbourhoods and investigate the diameter of . Lastly, we consider an enumeration problem relevant to population genetics.AMS Subject Classification: 05C05, 92D15.     

Ordering trees by their spectral radii

Let T be a tree with n vertices and let A（T） be the adjacency matrix of T. Spectral radius of T is the largest eigenvalue of A（T）. Wu et al. [Wu, B.F., Yuan, X.Y, and Xiao, E.L. On the spectral radii of trees, Journal of East China Normal University （Natural Science）, 3：22-28 （2004）] determined the first seven trees of order n with the smallest spectral radius. In this paper, we extend this ordering by determining the trees with the eighth to the tenth smallest spectral radius among all trees with n vertices.     

一类新的等可分树和化学树

设T是一个n阶树,e是它的一条边.用n1(e T)和n2(e T)分别表示树T中位于边e两侧的顶点的个数;n1(e T)+n2(e T)=n.设T和T′都是n阶树,e为T的一条边,f为T′的一条边,且n1(e T)=n1(f T′)或者n1(e T)=n2(f T′),则称e和f是等可分的边;如果能适当排列T的边e1,e2,…,en-1和T′的边e1′,e2′,…,en-′1,使得ei和ei′(i=1,2,…,n-1)都是等可分边,则称T和T′是等可分的树.等可分的化学树具有相同的W iener指数,因而有相似的物理化学性质.I.G u tm an等人给出了一些方法,构造等可分的树和化学树.本文给出了一种方法,构造出了一类新的等可分树和化学树.     

Some typical properties of large AND/OR Boolean formulas

In this paper typical properties of large random Boolean AND/OR formulas are investigated. Such formulas with n variables are viewed as rooted binary trees chosen from the uniform distribution of all rooted binary trees on m nodes, where n is fixed and m tends to infinity. The leaves are labeled by literals and the inner nodes by the connectives AND/OR, both uniformly at random. In extending the investigation to infinite trees, we obtain a close relation between the formula size complexity of any given Boolean function f and the probability of its occurrence under this distribution, i.e., the negative logarithm of this probability differs from the formula size complexity of f only by a polynomial factor. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 337–351 (1997)     

Some relations on the ordering of trees by minimal energies between subclasses of trees

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. A tree is said to be non-starlike if it has at least two vertices with degree more than 2. A caterpillar is a tree in which a removal of all pendent vertices makes a path. Let $\mathcal{T}_{n,d}$ , $\mathbb{T}_{n,p}$ be the set of all trees of order n with diameter d, p pendent vertices respectively. In this paper, we investigate the relations on the ordering of trees and non-starlike trees by minimal energies between $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ . We first show that the first two trees (non-starlike trees, resp.) with minimal energies in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same for 3≤d≤n?2 (3≤d≤n?3, resp.). Then we obtain that the trees with third-minimal energy in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same when n≥11, 3≤d≤n?2 and d≠8; and the tree with third-minimal energy in $\mathcal{T}_{n,8}$ is the caterpillar with third-minimal energy in $\mathbb{T}_{n,n-7}$ for n≥11.     

SOLUTION OF A RESEARCH PROBLEM

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一类树并的补图的色唯一性

彻底解决了一类不可约树并的补图是色唯一的 ,并得到了一些图的伴随多项式的最小根的重要规律 .     

T3C: improving a decision tree classification algorithm’s interval splits on continuous attributes

This paper proposes, describes and evaluates T3C, a classification algorithm that builds decision trees of depth at most three, and results in high accuracy whilst keeping the size of the tree reasonably small. T3C is an improvement over algorithm T3 in the way it performs splits on continuous attributes. When run against publicly available data sets, T3C achieved lower generalisation error than T3 and the popular C4.5, and competitive results compared to Random Forest and Rotation Forest.     

Mean-field lattice trees

We introduce a mean-field model of lattice trees based on embeddings into ^d of abstract trees having a critical Poisson offspring distribution. This model provides a combinatorial interpretation for the self-consistent mean-field model introduced previously by Derbez and Slade [9], and provides an alternative approach to work of Aldous. The scaling limit of the meanfield model is integrated super-Brownian excursion (ISE), in all dimensions. We also introduce a model of weakly self-avoiding lattice trees, in which an embedded tree receives a penaltye ^– for each self-intersection. The weakly self-avoiding lattice trees provide a natural interpolation between the mean-field model (=0), and the usual model of strictly self-avoiding lattice tress (=) which associates the uniform measure to the set of lattice trees of the same size.