[分享]誰能行行好 幫我英翻中(7大難題)

[分享]誰能行行好 幫我英翻中(7大難題)

Herbie 於 星期四 三月 06, 2003 1:40 pm

1.The P versus NP Problem home / millennium prize problems / the P versus NP problem

It is Saturday evening and you arrive at a big party. Feeling shy, you wonder how many people you already know in the room? Your host proposes that you must certainly know Rose, the lady in the corner next to the dessert tray. In a fraction of a second you are able to cast a glance and verify that your host is correct. However, in the absence of such a suggestion, you are obliged to make a tour of the whole room, checking out each person one by one, to see if there is anyone you recognize. This is an example of the general phenomenon that generating a solution to a problem often takes far longer than verifying a given solution is correct. Similarly, if someone tells you that the number 13,717,421 can be written as the product of two smaller numbers, you might not know whether to believe him, but if he tells you that it can be factored as 3607 times 3803 then you can easily check that it is true using a hand calculator. The problem of deciding whether the answer can be quickly checked can really take much longer to solve, no matter how clever a program we write, is considered one of the outstanding problems in logic and computer science. It was formulated by Stephen Cook in 1971.
2.The Hodge Conjecture home / millennium prize problems / the hodge conjecture

In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.
3.The Poincare Conjecture home / millennium prize problems / the poincare conjecture

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the the surface of the apple is 'simply connected,' but that the surface of the doughnut is not. Poincare, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out be be extraordinarily difficult, and mathematicians have been struggling with it ever since.
4.The Riemann Hypothesis home / millennium prize problems / the riemann hypothesis

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function "z(s)" called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation z(s) = 0 lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.
5.Yang-Mills Theory home / millennium prize problems / yang-mills theory

The equations of quantum physics describe the world of elementary particles. Almost fifty years ago, the physicists Yang and Mills discovered a remarkable relationship between geometry and particle physics, embodied in these equations. In so doing, they paved the way to later combination of the laws for electro-magnetic forces with those for strong and weak ones. The predictions culled from these equations describe particles observed at laboratories around the world, including Brookhaven, Stanford, and CERN. However, the gauge theories of Yang and Mills are not known to have solutions compatible with quantum mechanics, nor to describe the particles observed in nature. Despite this, the "mass gap" hypothesis concerning supposed solutions to the equations is taken for granted by most physicists and provides an explanation of why we do not observe "quarks." Solving this mathematical problem requires establishing a mathematical proof of this phenomenon.
6.Navier-Stokes Equations home / millennium prize problems / navier-stokes equations

Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. These and other fluid phenomena are described by the mathematical equations known by the names of the mathematicians Navier and Stokes. Unlike many problems in quantitative science, the solutions to these equations are not known, nor is it even known how to solve these equations. The solution to this problem entails showing the existence and smoothness of solutions to the Navier-Stokes equations.
7.The Birch and Swinnerton-Dyer Conjecture home / millennium prize problems / the birch and swinnerton-dyer conjecture

Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like x^2 + y^2 = z^2 . Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function z(s) near the point s=1. In particular this amazing conjecture asserts that if z(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if z(1) is not equal to 0, then there is only a finite number of such points.


文章: 841
註冊時間: 2003-01-16
來自: 港都-打狗

yll 於 星期四 三月 06, 2003 7:43 pm

先看些故事 ㄏㄏㄏ


德國數學家David Hilbert於1900年在巴黎舉行的第二屆國際數學家協會中公布了他的23個數學難題,百年來,已經解出了20個問題,而這些結果間接促成了文明史上醫學、科技、與安全問題的重大突破。
在「克萊數學院」宣佈7大難題懸賞舉行的新聞發佈會上,身為「克萊數學院」委員,並在1995年因修正了「費瑪最後定理(Fermat's Last Theorem)」的邏輯漏洞而名噪一時的懷爾斯(Andrew Wiles)說:「這些是二十世紀最難解的七大數學問題了。希望透過獎金獎勵,可以吸引並發掘新一代的數學家。」他自己對於興趣在一個數學家成長過程中的作用有著深刻的體會。懷爾斯回憶說,他10歲時在一本連環畫上首次知道了什麼是『費爾馬大定理』,這成為他不斷探索問題解答的起點。「克萊數學院」揮金如土的另一個原因,是因為此次懸賞求解的7大難題是20世紀中仍未被數學家解決的數學題。過去100年來,最優秀的數學家面對它們都無計可施。而這幾道難題的破解,極有可能為密碼學等研究帶來革命。例如,有關專家指出,7大難題中最有名的『黎曼假設』一旦獲得解答,將有助於研製出提高網路上資訊傳輸的安全性,客戶的信用卡賬號資訊、醫療和金融資料等將獲得到更高的保障。而其餘的"普安卡雷猜想"、"霍奇猜想"、"戴爾猜想"、"斯托克斯方程"、"米爾斯理論"以及"P對NP問題"等6大難題,解決後可能給航太等領域帶來突破性進展,並開展出空前的數學研究領域。  

1.黎曼假設 The Riemann Hypothesis
2.普安卡雷猜想 The Poincare Conjecture
3.霍奇猜想 The Hodge Conjecture
4.戴爾猜想The Birch and Swinnerton-Dyer Conjecture
5.斯托克斯方程(流體力學的N-S方程式)Navier-Stokes Existence and Smoothness
6.米爾斯理論「The Yang-Mills Theory」(楊密規範場論)Yang-Mills Existence and Mass Gap
7.P對NP問題 P versus NP

文章: 4367
註冊時間: 2002-08-28
來自: 天父的小花園~

yll 於 星期四 三月 06, 2003 7:44 pm

視窗Window「踩地雷」遊戲藏有數學大秘密  耍酷

微軟視窗Window電腦遊戲「踩地雷」,可能是破解複雜的網路安全密碼的重要關鍵。英國伯明罕大學數學教授Richard Kaye認為,如果有人在可以在一個大型複雜的「踩地雷」遊戲中,解出決定所有地雷組合的運算法,這個人就有可能解出「The P vs NP Problem」。美國麻薩諸塞州的「克萊數學學院」(Clay Mathematics Institute),已經為這個能解決問題的人,準備了100萬美元的獎金。
微軟視窗系統中的「踩地雷」遊戲,是一個老少咸宜,簡單易玩的遊戲。玩法是,玩家在格子圖中嘗試著找出哪些小格子內藏有電腦預設的地雷。每個格子內的數字,表示附近區域埋有多少地雷。Richard Kaye玩了幾個星期的「踩地雷」後,突發奇想,如果在一個更大型的格子圖上玩,而不只限於電腦的話,「踩地雷」有著和其他被認為無解的問題一樣的數學特徵。他說:「我一向對帶有數學內涵的遊戲感興趣。數學和遊戲是十分契合的。我只是突然理解到,這遊戲隱藏著一個很棒的數學原理,只是,我仍不清楚我能找的是什麼。」這個發現為解決數學界的大難題,即所謂「P versus NP Problem」,提供了有利線索,「P versus NP Problem」這道難題試圖判定,一些看似無法在短期內解決的問題,事實上是否有可能存在著一個相當簡單的解決方法,只是尚未被人發現而已。這個發現對全世界有著更重大的影響。Richard Kaye說:「如果能找到一個有效率的方法來玩踩地雷,那麼也會找到一個破解密碼的有效途徑。」

美國美國麻薩諸塞州的「克萊數學學院」(Clay Mathematics Institute)的院長Arthur Jaffe說,在知道Richard Kaye的研究前,他自己也是個「踩地雷」迷。他經常在失眠的時候,玩這遊戲。現在,有了Richard Kaye研究發現,他從此不介意他的工作人員或是小孩,花時間玩「踩地雷」。Arthur Jaffe表示,「當我告訴了14歲的女兒關於Richard Kaye的研究時,她也很驚訝這個遊戲竟然會有教育意義。」

文章: 4367
註冊時間: 2002-08-28
來自: 天父的小花園~