|学校||West Shore Junior/Senior High School|
|国家/州||FL，United States of America|
A Fine Classification of Second Minimal Odd Orbits
This paper presents the solution to the open problem of the fine classification of second minimal odd periodic orbits of continuous endomorphisms on the real line. It is rigorously proven that second minimal odd orbits (of any length 2k+1, where k>2), either have a Stefan structure, like that of a minimal odd orbit, or have one of 4k-3 types, each characterized with its own cyclic permutation and directed graph of transitions, unique up to an inverse. The result is applied to the overarching problem of deriving a universal law of distribution for periodic windows within the chaotic regime of the bifurcation diagram of the one-parameter family of unimodal maps. This paper adds to the growing evidence that a universal pattern of distribution can be revealed by fixing the maximum number of appearances for periodic windows, and comparing the topological structures of iterative orbits to the topological structure of the P-Linearization of Cyclic Permutations.
By proving a fine classification of second minimal odd orbits, similar to the fine classification of minimal odd orbits by Stefan in 1977, this paper has a tremendous impact in the field of Chaos Theory, offering a deeper understanding of the universal nature of periodic orbits that appear in the iterative functions used in applied mathematical fields such as Population Dynamics, Weather Prediction, and Fluid Turbulence.
高中生科研 英特尔 Intel ISEF
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高中生科研竞赛 英特尔 Intel ISEF 简介
英特尔国际科学与工程大奖赛，简称 "ISEF"，由美国 Society for Science and the Public（科学和公共服务协会）主办，英特尔公司冠名赞助，是全球规模最大、等级最高的中学生的科研科创赛事。ISEF 的竞赛学科包括了所有数学、自然科学、工程的全部领域和部分社会科学。ISEF 素有全球青少年科学竞赛的“世界杯”之美誉，旨在鼓励学生团队协作，开拓创新，长期专一深入地研究自己感兴趣的课题。
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The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. The deductive study of numbers, geometry, and various abstract constructs, or structures.
Algebra (ALB): The study of algebraic operations and/or relations and the structures which arise from them. An example is given by (systems of) equations which involve polynomial functions of one or more variables.
Analysis (ANL): The study of infinitesimal processes in mathematics, typically involving the concept of a limit. This begins with differential and integral calculus, for functions of one or several variables, and includes differential equations.
Combinatorics, Graph Theory, and Game Theory (CGG): The study of combinatorial structures in mathematics, such as finite sets, graphs, and games, often with a view toward classification and/or enumeration.
Geometry and Topology (GEO): The study of the shape, size, and other properties of figures and spaces. Includes such subjects as Euclidean geometry, non-Euclidean geometries (spherical, hyperbolic, Riemannian, Lorentzian), and knot theory (classification of knots in 3-space).
Number Theory (NUM): The study of the arithmetic properties of integers and related topics such as cryptography.
Probability and Statistics (PRO): Mathematical study of random phenomena and the study of statistical tools used to analyze and interpret data.
Other (OTH): Studies that cannot be assigned to one of the above subcategories. If the project involves multiple subcategories, the principal subcategory should be chosen instead of Other.