PUMaC 2024 · 加试 · 第 11 题
PUMaC 2024 — Power Round — Problem 11
题目详情
- There are two places where you may ask questions about the test. The first is Piazza. Please ask your coach for instructions to access our Piazza forum. On Piazza, you may ask any question so long as it does not give away any part of your solution to any problem . If you ask a question on Piazza, all other teams will be able to see it. If such a question reveals all or part of your solution to a power round question, your team’s power round score will be penalized severely. For any questions you have that might reveal part of your solution, or if you are not sure if your question is appropriate for Piazza, please email us at pumac@math.princeton.edu. We will email coaches with important clarifications that are posted on Piazza. Introduction and Advice In this power round, we formally investigate fractals, exploring how to make the idea of self-similarity and non-integer dimensions rigorous. We hope that this introduction pro- vides not only an interesting perspective on something often seen as pop-math, but also an introduction to measure theory, a fascinating branch of mathematics. A large part of the difficulty in this power round will arise from the many different perspectives that one needs to understand the material and tackle the problems. For ex- ample, understanding the geometry of fractals is essential to proving facts about them, but grasping set theory and topology is essential to making many of these intuitions formal. Here is some further advice with regard to the Power Round: • Read the text of every problem! Many important ideas are included in problems and may be referenced later on. In addition, some of the theorems you are asked to prove are useful or even necessary for later problems. • Make sure you understand the definitions . A lot of the definitions are not easy to grasp; don’t worry if it takes you a while to fully understand them. If you don’t, then you will not be able to do the problems. Feel free to ask clarifying questions about the definitions on Piazza (or email us). • Don’t make stuff up : on problems that ask for proofs, you will receive more points if you demonstrate legitimate and correct intuition than if you fabricate something that looks rigorous just for the sake of having “rigor.” • Check Piazza often! Clarifications will be posted there, and if you have a question it is possible that it has already been asked and answered in a Piazza thread (and if not, you can ask it, assuming it does not reveal any part of your solution to a question). If in doubt about whether a question is appropriate for Piazza, please email us at pumac@math.princeton.edu. • Don’t cheat : as stated in Rules and Reminders, you may NOT use any references such as books or electronic resources. If you do cheat, you will be disqualified and banned from PUMaC, your school may be disqualified, and relevant external institu- tions may be notified of any misconduct. Good luck, and have fun! – Colby Riley We would like to acknowledge and thank many individuals and organizations for their support; without their help, this Power Round (and the entire competition) could not exist. Please refer to the solutions of the power round for full acknowledgments and references. Contents n 1 Topology in R 6 2 Measures 8 2.1 Hausdorff measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Interlude: some fractal constructions 12 3.1 Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Sierpinski Carpet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Minkowski Sausage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Koch Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Iterated Function Systems 14 4.1 Dimension of IFS’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Mass Distribution Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5 Sierpinski Triangle 21 Notation and Basic Concepts • { x ∈ S : C ( x ) } : the set of all x in the set S satisfying the condition C ( x ). Ex.: √ { n ∈ N : n ∈ N } is the set of perfect squares. • A × B : cartesian product. It is the set consisting of ordered pairs { ( a, b ) : a ∈ A, b ∈ n B } . Ex.: R = R × ... × R • A ⊂ B : subset. Ex.: { 1 , 2 } ⊂ { 1 , 2 , 3 } , and { 1 , 2 } ⊂ { 1 , 2 } • f ( C ): for a function f : A → B and subset C ⊆ A, the set of elements of the form f ( c ) , for c ∈ C. • N : the natural numbers, { 1 , 2 , 3 , . . . } . • Z : the integers. • Q : the rational numbers. • R : the real numbers. • for a < b , [ a, b ] = { x : a ≤ x ≤ b } , ( a, b ) = { x : a < x < b } , and ( a, b ] , [ a, b ) are defined similarly. • sup X : the supremum. For X ⊂ R , The supremum is the smallest value s such that for all x ∈ X , x ≤ s . Always exists, but may be ±∞ . Ex.: sup { 1 , 2 , 3 } = 3 . sup N = ∞ . • sup f ( x ): the supremum applied to the set f ( X ). x ∈ X • inf X : the infimum, the largest value s such that for all x ∈ X , x ≥ s . Always exists, but may be ±∞ . • for a decreasing sequence a , lim a is defined to be inf a . n n →∞ n n ∈ N n • for an increasing function f : (0 , a ) → R defined near 0, lim f ( δ ) is defined to be δ → 0 2 inf f ( δ ). Ex.: lim ( δ + 5) = 5 . For a decreasing function, use sup. δ> 0 δ → 0 • log with no base refers to the natural logarithm. • a function f : A → B is injective if for all x, y ∈ A , f ( x ) = f ( y ) = ⇒ x = y . It is surjective if for all b ∈ B , there is some a ∈ A such that f ( a ) = b . It is bijective if it is injective and surjective. • a set is countable if there exists a bijection from it to N or a finite set. Notes on the use of calculus in the Power Round: we here at PuMaC do not expect anyone in this competition to necessarily know calculus. Thus, we have tried to minimize the use of calc in this power round. Accordingly, for proofs which involve limits, infimums, and supremums, we do not expect formal, ϵ − δ type proofs, nor do we want you to be citing continuity results willy nilly. Thus, the limit parts of your proofs will be graded to a lesser standard than other parts of your proofs, as a little treat. Still try to be as rigorous as possible, and we will keep it in mind during grading. Thank you! n 1 Topology in R This section is a crash course in the parts of topology relevant for this power round. There are no problems, but the concepts introduced here are important technical details that we need for fractals. Without them, fractals would be too hard to study and understand. n 1 n R is the set of n -tuples ( x , ..., x ) with that every x ∈ R . Two points of R can be 1 n i added term-wise and multiplied by a scalar c ∈ R : ( x , ..., x ) + ( y , ..., y ) = ( x + y , ..., x + y ) 1 n 1 n 1 1 n n cx = c ( x , ..., x ) = ( cx , ...cx ) 1 n 1 n n Denote the special element (0 , ..., 0) as simply 0. Given an element x = ( x , ..., x ) ∈ R , 1 n we define the Euclidean norm to be v u n X u 2 t | x | = x i i =0 2 For example, in R , this is the distance formula q 2 2 | x | = x + x 1 2 The Euclidean metric is how we measure distance. We define dist( x, y ) = | x − y | . This function satisfies 3 very important properties:
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