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分形几何中的若干前沿问题
  • 项目名称：分形几何中的若干前沿问题
  • 项目类别：重点项目
  • 批准号：10631040
  • 申请代码：A011405
  • 项目来源：国家自然科学基金
  • 研究期限：2007-01-01-2010-12-31
  • 项目负责人：文志英
  • 负责人职称：教授
  • 依托单位：清华大学
  • 批准年度：2006
中文摘要：

分形几何研究不规则几何形体，是具有很强应用背景的交叉学科。本项目研究分形几何的一些难度很大但非常重要的前沿问题，它们涉及到分形几何的基本理论以及与其它学科的交叉，相对独立但密切相关，它们对分形几何自身的发展以及应用前景均有重要意义。这些问题包括分形几何理论的一些基本问题，如一些主要集类的维数与测度，带重叠结构的自相似集的结构，莫朗集类；贝努利 积与非吉布斯型测度的重分形机理；分形上的偏微分方程，典型的分形区域上的某些非线性反应扩散方程、非线性波动方程以及非线性椭圆方程全局广义解存在性和非存在性，以及这些解的数值模拟计算；非周期链的薛定愕方程的算子谱的分形结构；代换动力系统的因子性质与吸引子的分形结构；实数的各种展式的与相关的动力系统结构以及数论中例外集的分形维数，丢番图逼近的度量和几何性质，特别是正特征域上的雅利克定理与Duffin猜测；应用相关的一些问题如脱硫剂表面的分形结构等。

中文主题词： 分形，重分形分析，代换动力系统，分形上的PDE，Tiling理论
结论摘要：

英文主题词fractal; multifractal analysis; substitution dynamical systems; PDEs on fractals; tiling theory

成果综合统计
成果类型
数量
  • 期刊论文
  • 会议论文
  • 专利
  • 获奖
  • 著作
  • 84
  • 0
  • 0
  • 0
  • 0
期刊论文
ON THE DIMENSIONS OF SECTIONS FOR THE GRAPH-DIRECTED SETS
On single-matrix graph-directed iterated function systems
Dimensions of weighted Besicovitch sets
IFS ON BOUNDARIES OF HOMOGENEOUS TREES: WQB CONDITION AND MULTIFRACTAL ANALYSIS
Schrodinger equations and heat kernel upper bounds on metric spaces
ON HAUSDORFF DIMENSION AND TOPOLOGICAL ENTROPY
Integral self-affine tiles of Bandt's model
The pointwise dimensions of Moran measures
Comparison inequalities for heat semigroups and heat kernels on metric measure spaces
Maximal pattern complexity of higher dimensional words
DOUBLING MEASURES WITH DOUBLING CONTINUOUS PART
Dynamics of Mandelbrot cascades
The self-affine property of (U, r)-Carlitz sequences of polynomials deciphered in terms of graph dir
The geometry of Whitney's critical sets
Hausdorff and packing dimensions of subsets of Moran fractals with prescribed mixed group frequency
Super-stationary set, subword problem and the complexity
On the relative growth rate of degrees of partial quotients in continued fractions over the field of
Generalized Bessel and Riesz Potentials on Metric Measure Spaces
Hypergeometric functions for function fields and transcendence
Hausdorff Measure of Circular Moran Sets
The L-q-spectrum of a class of graph directed self-affine measures
Continued fractions with sequences of partial quotients over the field of Laurent series
Fractional-moment CAPM with loss aversion
Some fractals associated with Cantor expansions
DIVERGENCE POINTS WITH FAST GROWTH ORDERS OF THE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS
Egoroff's theorem and maximal run length
Hausdorff dimensions of bounded-type continued fraction sets of Laurent series
On substitution invariant sturmian words: An application of rauzy fractals
The self-similarity structure on infinite intervals
Topology and separation of self-similar fractals in the plane
Some properties of the Tribonacci sequence
A multifractal decomposition according to rate of returns
Heat kernel estimates on effective resistance metric spaces
SELF-SIMILARITY OF SATISFIABLE BOOLEAN EXPRESSIONS DECIPHERED IN TERMS OF GRAPH DIRECTED ITERATED FU
Gauges for the self-similar sets
DOUBLING MEASURES AND NONQUASISYMMETRIC MAPS ON WHITNEY MODIFICATION SETS IN EUCLIDEAN SPACES
On the triplex substitution - Rauzy fractal
Dimensions of level sets related to tangential dimensions
Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces
An analytical approach to heat kernel estimates on strongly recurrent metric spaces
Metric properties and exceptional sets of beta-expansions over formal Laurent series
Beta-expansion and continued fraction expansion over formal Laurent series
A remark on continued fractions with sequences of partial quotients
Cantor sets determined by partial quotients of continued fractions of Laurent series
A problem of Hirst on continued fractions with sequences of partial quotients
Packing dimensions of homogeneous perfect sets
Gap sequence, Lipschitz equivalence and box dimension of fractal sets
On the triplex substitution - combinatorial properties
Beta-expansion and continued fraction expansion
Self-similar structure on intersections of triadic Cantor sets
Hausdorff dimension of certain sets arising in continued fraction expansions
Obtaining upper bounds of heat kernels from lower bounds
The entropies and multifractal spectrum of some compact systems
A Billingsley type theorem for Bowen entropy
Doubling measures on Whitney modification sets
A note on the limiting weak-type behavior for maximal operators
The multifractal dimension functions of homogeneous Moran measure
Approximation ratio of the digits in Oppenheim series expansion
An iterated logarithm law related to decimal and continued fraction expansions
一类广义Sierpinski地毯的Huasdorff维数
一类广义齐次Moran集的Hausdorff维数
一般集合上的两个函数和的图像的盒维数估计
一类弱自相似集的强正则性
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