关于具分形边界连通区域上的谱渐近及弱Weyl-Berry猜想

本文研究了连通分形鼓上的谱渐近，对满足“切口”条件的连通分形鼓以及一类自然连通的分形鼓，分别证明了弱Weyl－Berry猜想是成立的.     

Trace results on domains with self-similar fractal boundaries

This work deals with trace theorems for a family of ramified bidimensional domains Ω with a self-similar fractal boundary Γ^∞. The fractal boundary Γ^∞ is supplied with a probability measure μ called the self-similar measure. Emphasis is put on the case when the domain is not a −δ domain and the fractal is not post-critically finite, for which classical results cannot be used. It is proven that the trace of a function in H¹(Ω) belongs to for all real numbers p1. A counterexample shows that the trace of a function in H¹(Ω) may not belong to BMO(μ) (and therefore may not belong to ). Finally, it is proven that the traces of the functions in H¹(Ω) belong to H^s(Γ^∞) for all real numbers s such that 0s<d_H/4, where d_H is the Hausdorff dimension of Γ^∞. Examples of functions whose traces do not belong to H^s(Γ^∞) for all s>d_H/4 are supplied.There is an important contrast with the case when Γ^∞ is post-critically finite, for which the functions in H¹(Ω) have their traces in H^s(Γ^∞) for all s such that 0s<d_H/2.     

The interval function of a connected graph and road systems

Let V be a finite nonempty set. In this paper, a road system on V (as a generalization of the set of all geodesics in a connected graph G with V(G)=V) and an intervaloid function on V (as a generalization of the interval function (in the sense of Mulder) of a connected graph G with V(G)=V) are introduced. A natural bijection of the set of all intervaloid functions on V onto the set of all road systems on V is constructed. This bijection enables to deduce an axiomatic characterization of the interval function of a connected graph G from a characterization of the set of all geodesics in G.     

Schwarz problem for bi-analytic functions in multiply connected domains

The Schwarz problem for bi-analytic functions in unbounded circular multiply connected domains is considered. We combine constructive methods applied to boundary value problems for complex partial differential equations in simply connected domains and for the Riemann–Hilbert type problems in multiply connected domains. A general method is outlined and the case of doubly connected domains is discussed in details. Solution is obtained in the form of a series.     

Lipschitz images with fractal boundaries and their small surface wrapping

Assume and is a Lipschitz -mapping; and denote the volume and the surface area of . We verify that there exists a figure with , and, of course, , where depends only on the dimension and on . We also give an example when is a square and ; in fact, the boundary of can contain a fractal of Hausdorff dimension exceeding one.

     

Truncated fractal basin boundaries in the pendulum with nonperiodic forcing

Summary It is well known that oscillators such as the pendulum can have fractal basin boundaries when they are periodically forced with the consequence that the long term behavior of the system may be unpredictable. In engineering and physical applications, the forcing is often nonperiodic and eventually decays to zero, and simulation of the pendulum with decaying forcing (M. Varghese, J. S. Thorp, Physical Review Letters, vol. 60, no. 8, pp. 665–668, Feb. 1988) exhibits truncated fractal basin boundaries which also limit the system predictability. We develop a coordinate change for the pendulum with decaying forcing that allows us to apply standard qualitative methods to study the basin boundaries. We prove that the basin boundaries cannot be fractal and show by example how the extreme stretching and folding leading to a truncated fractal basin boundary may arise.     

Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics

In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.

     

Riemann 映射与单连通区域分类

设G为复平面上一个单连通区域及φ为G的Riemann 映射. 本文通过φ是否属于G上多项式在不同拓扑下的闭包的情况对G进行分类. 特别地, 我们对已知的几类单连通给出了刻画.     

On Seymour’s strengthening of Hadwiger’s conjecture for graphs with certain forbidden subgraphs

Let H be a set of graphs. A graph is called H-free if it does not contain a copy of a member of H as an induced subgraph. If H is a graph then G is called H-free if it is {H}-free. Plummer, Stiebitz, and Toft proved that, for every -free graph H on at most four vertices, every -free graph G has a collection of ⌈|V(G)|/2⌉ many pairwise adjacent vertices and edges (where a vertexvand an edgeeare adjacent if v is disjoint from the set V(e) of endvertices of e and adjacent to some vertex of V(e), and two edgeseandfare adjacent if V(e) and V(f) are disjoint and some vertex of V(e) is adjacent to some vertex of V(f)). Here we generalize this statement to -free graphs H on at most five vertices.     

Estimation of the L p -norms of stress functions for finitely connected plane domains

Let u(x, G) be the classical stress function of a finitely connected plane domain G. The isoperimetric properties of the L ^p -norms of u(x, G) are studied. Payne’s inequality for simply connected domains is generalized to finitely connected domains. It is proved that the L ^p -norms of the functions u(x, G) and u ^?1 (x, G) strictly decrease with respect to the parameter p, and a sharp bound for the rate of decrease of the L ^p -norms of these functions in terms of the corresponding L ^p -norms of the stress function for an annulus is obtained. A new integral inequality for the L ^p -norms of u(x, G), which is an analog of the inequality obtained by F. G. Avkhadiev and the author for the L ^p -norm of conformal radii, is proved.     

一个分形函数的分数阶微积分函数

Based on the combination of fractional calculus with fractal functions, a new type of is introduced； the definition, graph, property and dimension of this function are discussed.     

Monotone variations of the first eigenvalue for doubly connected domains

The first Dirichlet eigenvalue for the Laplace operator of a doubly connected plane domain varies with the relative position of the inner hole. Under certain mild hypothesis it decreases as the two boundary curves get closer along some specific direction. We made this observation by analyzing level curves and rearrangement technique.     

分形插值函数的矩量及扰动误差估计

研究分形插值函数的矩量积分问题.对于一类分形插值函数,给出了它在各阶区间上的(p,q)阶矩量积分的计算公式.此外,考虑含有扰动项的迭代函数系所产生的分形插值函数的矩量,讨论了扰动项对于矩量的影响,给出扰动前后矩量的误差估计式.     

Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l.     

Integrals and derivatives connected to the r-Lambert function

In this paper, we derive some formulas for higher order derivatives of r-Lambert functions. Moreover, an integration formula involving powers of r-Lambert function is obtained.     

Eigenvalues of the negative Laplacian for simply connected bounded domains

This paper is devoted to asymptotic formulae for functions related with the spectrum of the negative Laplacian in two and three dimensional bounded simply connected domains with impedance boundary conditions, where the impedances are assumed to be discontinuous functions. Moreover, asymptotic expressions for the difference of eigenvalues related to the impedance problems with different impedances are derived. Further results may be obtained.     

Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r−1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r=2, and very recently the conjecture was proved for the case where r=3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we present two proofs of this conjecture for tournaments with minimum in-degree at least 2r−1. In particular, this shows that the conjecture is true for (almost) regular tournaments. In the first proof, we prove auxiliary results about union of sets contained in another union of sets, that might be of independent interest. The second one uses a more graph-theoretical approach, by studying the properties of a maximum set of vertex-disjoint directed triangles.