#p-np

[curiosamathematica]

Famous Fluid Equations Are Incomplete

A 115-year effort to mathematically bridge the particle and fluid descriptions of nature has led physicists to an unexpected answer.

The Navier-Stokes equations—despite being exceptionally useful for modeling the weather, ocean currents, pipes, cars, airplane wings and other hydrodynamic systems, and despite the million-dollar prize offered for their exact solutions—are incomplete. Truer equations of fluid dynamics can be found in a little-known, relatively unheralded theory developed by the Dutch mathematician and physicist Diederik Korteweg in the early 1900s. And yet, for some gases, even the Korteweg equations fall short, and there is no fluid picture at all.

[curiosamathematica]

Randomization can help in attempts to solve the traveling salesman problem, asking for the shortest possible route visiting a number of given cities.

[sevennet-blog]

How to: What's "P=NP?", and why is it such a famous question?

How to: What's "P=NP?", and why is it such a famous question?

What’s "P=NP?", and why is it such a famous question?

The question of whether P=NP is perhaps the most famous in all of Computer Science. What does it mean? And why is it so interesting?

Oh, and for extra credit, please post a proof of the statement’s truth or falsehood. :)

Answer: What’s "P=NP?", and why is it such a famous question?

http://en.wikipedia.org/wiki/Complexity_classes_P_and_NP

Answ…

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[amindamazed]

Saw this book at the university library today; link goes to the author's website.

The P-NP problem is the most important open problem in computer science, if not all of mathematics. Simply stated, it asks whether every problem whose solution can be quickly checked by computer can also be quickly solved by computer. The Golden Ticket provides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. In this informative and entertaining book, Lance Fortnow traces how the problem arose during the Cold War on both sides of the Iron Curtain, and gives examples of the problem from a variety of disciplines, including economics, physics, and biology. He explores problems that capture the full difficulty of the P-NP dilemma, from discovering the shortest route through all the rides at Disney World to finding large groups of friends on Facebook. But difficulty also has its advantages. Hard problems allow us to safely conduct electronic commerce and maintain privacy in our online lives.

[curiosamathematica]

Some common time complexities of algorithms.

The time complexity of an algorithm describes the amount time the algorithm takes to handle an input of a certain length (in worst case scenario). For instance, an algorithm checking whether a certain number is present in a given list, possibly needs to run over the whole list to find the answer, so its complexity f(n) is a lineair function of the length n of the list (f(n)=n). It has been shown that to sort a generic list, the fastest algorithms possible are of the complexity class n ln(n).

More difficult problems generally require more time to be solved. For example, solving the traveling salesman problem via brute-force search has time complexity n!, and there are even algorithms with complexity 2^(2^n) or worse, which plainly is not useful for practical applications. An algorithm is said to be efficient if its time complexity is bounded by a polynomial, because then its runtime remains "reasonable" for bigger input values. Notice the plot above is a log plot, so complexities like n! grow even faster by comparison to polynomials.

The Millennium Problem P versus NP is related to this concept, as it asks whether or not one can find a solution to a problem in polynomial time (deterministically) if one can check a solution in polynomial time.