A MOST PROFOUND MATH PROBLEM

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On August 6, 2010, a computer scientist named Vinay Deolalikar published a paper with a name as concise as it was audacious: “P ≠ NP.” If Deolalikar was right, he had cut one of mathematics’ most tightly tied Gordian knots. In 2000, the P = NP problem was designated by the Clay Mathematics Institute as one of seven Millennium Problems—“important classic questions that have resisted solution for many years”—only one of which has been solved since. (The Poincaré Conjecture was vanquished in 2003 by the reclusive Russian mathematicianGrigory Perelman, who refused the attached million-dollar prize.)

 

A few of the Clay problems are long-standing head-scratchers. The Riemann hypothesis, for example, made its debut in 1859. By contrast, P versus NP is relatively young, having been introduced by the University of Toronto mathematical theorist Stephen Cook in 1971, in a paper titled “The complexity of theorem-proving procedures,” though it had been touched upon two decades earlier in a letter by Kurt Gödel, whom David Foster Wallace branded “modern math’s absolute Prince of Darkness.” The question inherent in those three letters is a devilish one: Does P (problems that we can easily solve) equal NP (problems that we can easily check)?

Take your e-mail password as an analogy. Its veracity is checked within a nanosecond of your hitting the return key. But for someone to solve your password would probably be a fruitless pursuit, involving a near-infinite number of letter-number permutations—a trial and error lasting centuries upon centuries. Deolalikar was saying, in essence, that there will always be some problems for which we can recognize an answer without being able to quickly find one—intractable problems that lie beyond the grasp of even our most powerful microprocessors, that consign us to a world that will never be quite as easy as some futurists would have us believe. There always will be problems unsolved, answers unknown.Click link to read full article;

http://www.newyorker.com/online/blogs/elements/2013/05/a-most-profound-math-problem.html

( Posted by F. Sheikh)

Different forms of mathematical thought

Posted by Noor Salik

Different forms of mathematical thought
One makes the distinction in mathematics between:
(i) Continuous thinking (for example real numbers and limits), and
(ii) Discrete thinking (for example natural numbers and number theory).
Experience shows that continuous problems are often easier to treat than discrete ones.
The great successes of the continuous way of thinking are based on the notion of limits
and the theories connected with this notion (calculus, differential equations, integral
equations and the calculus of variations) with diverse applications in physics and other
natural sciences.

In contrast, number theory is the prototype for the creation of effective mathematical
methods for treating discrete problems, arising in today’s world in computer science,
optimization of discrete systems and lattice models in theoretical physics for studying
elementary particles and strings.

The epochal discovery by Max Plank in 1900 that the energy of the harmonic oscillator
is not continuous but rather discrete (quantized), led to the important mathematical
problem of generating discrete structures from continuous ones by an appropriate,
non-trivial quantization process.

Oxford Users’ Guide to Mathematics

Interesting Simple Math Quiz on Birthday

How many people would be enough to make the odds of  birthday match at least 50-50?


Guess the answer and then read the following paragraph;

You have to stay with the explanation for a while to finally get it.

 

By an amazing coincidence my sister, Cathy, and my Aunt Vere have the same birthday: April 4 Actually, it’s not so amazing. In any extended family with enough siblings, aunts, uncles and cousins, you’d expect at least one such birthday coincidence. Certainly, if there are 366 people in the family — more relatives than days of the year — they can’t all have different birthdays, so a match is guaranteed in a family this big. (Or if you’re worried about leap year, make it 367.) But suppose we don’t insist on absolute certainty. A classic puzzle called the “birthday problem” asks: How many people would be enough to make the odds of a match at least 50-50? The answer, just 23 people, comes as a shock to most of us the first time we hear it. Partly that’s because it’s so much less than 366. But it’s also because we tend to mistake the question for one about ourselvesMy birthday.

To read explanation click on article below:

 

http://opinionator.blogs.nytimes.com/2012/10/01/its-my-birthday-too-yeah/?emc=eta1