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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hundred-dollar, Hundred-digit Challenge problems ： ウィキペディア英語版 Hundred-dollar, Hundred-digit Challenge problems The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by . A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digits. The deadline for the contest was May 20, 2002. In the end, 20 teams solved all of the problems perfectly within the required precision, and an anonymous donor aided in producing the required prize monies. The challenge and its solutions were described in detail in the book . ==The problems== From : # $\backslash lim\_\backslash int\_\backslash varepsilon^1\; x^\; \backslash cos\backslash left(x^\; \backslash log\; x\backslash right)\backslash ,dx$ # A photon moving at speed 1 in the ''xy''-plane starts at ''t'' = 0 at (''x'', ''y'') = (0.5, 0.1) heading due east. Around every integer lattice point (''i'', ''j'') in the plane, a circular mirror of radius 1/3 has been erected. How far from the origin is the photon at ''t'' = 10? # The infinite matrix ''A'' with entries $a\_=1,\; a\_=1/2,\; a\_=1/3,\; a\_=1/4,\; a\_=1/5,\; a\_=1/6,\; \backslash dots$ is a bounded operator on $\backslash ell^2$. What is $||A||$? # What is the global minimum of the function $\backslash exp\backslash left(\backslash sin\backslash left(50x\backslash right)\backslash right)\; +\; \backslash sin\backslash left(60e^y\backslash right)\; +\; \backslash sin\backslash left(70\; \backslash sin\; x\backslash right)+\backslash sin\backslash left(\backslash sin\backslash left(80y\backslash right)\backslash right)\; -\; \backslash sin\backslash left(10\backslash left(x+y\backslash right)\backslash right)\; +\; 1/4\backslash left(x^2\; +\; y^2\backslash right)$ # Let $f(z)=1/\backslash Gamma(z)$, where $\backslash Gamma(z)$ is the gamma function, and let $p(z)$ be the cubic polynomial that best approximates $f(z)$ on the unit disk in the supremum norm $||.||\_\backslash infty$. What is $||f-p||\_\backslash infty$? # A flea starts at $(0,0)$ on the infinite 2D integer lattice and executes a biased random walk: At each step it hops north or south with probability $1/4$, east with probability $1/4+\backslash varepsilon$, and west with probability $1/4-\backslash varepsilon$. The probability that the flea returns to (0, 0) sometime during its wanderings is $1/2$. What is $\backslash varepsilon$? # Let A be the 20000×20000 matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, ..., 224737 along the main diagonal and the number 1 in all the positions $a\_$ with $|i-j|=1,\; 2,\; 4,\; 8,\; \backslash dots,\; 16384$. What is the (1, 1) entry of $A^$? # A square plate $()\backslash times\; ()$ is at temperature $u=0$. At time $t=0$, the temperature is increased to $u=5$ along one of the four sides while being held at $u=0$ along the other three sides, and heat then flows into the plate according to $u\_\; =\; \backslash Delta\; u$. When does the temperature reach $u=1$ at the center of the plate? # The integral $I(\backslash alpha)=\backslash int\_0^2\backslash left()x^\backslash alpha\; \backslash sin\backslash left(\backslash alpha/\backslash left(2-x\backslash right)\backslash right)\backslash ,dx$ depends on the parameter α. What is the value of α in () at which ''I''(α) achieves its maximum? # A particle at the center of a 10×1 rectangle undergoes Brownian motion (i.e., 2D random walk with infinitesimal step lengths) till it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides? 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hundred-dollar, Hundred-digit Challenge problems」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.