Eric Lassard is the highly-respected (albeit dreamy) Commandant, and Academy newcomer The Professor is also on hand, and also the cadets new friends, the K-9 Corps, a group of cops dogs, and stocking our crime-fighting heroes with an endless supply of wacky gadgets as they combat a motley crew of Kingpin and other recurring villains such as Numbskull, The Claw, Mr. Sleaze, Lockjaw and Amazona.
"Police Academy" was released through chronological volumes on VHS. At least six volumes were released, each including two episodes. In 2009 on DVD with the complete series.
Tottenham Hale station
Tottenham Hale , is a National Rail and London Underground Victoria Line station in Tottenham, north London. It is on Hale Road near the Tottenham Hale one-way system. The station is in Travelcard Zone 3.
The station opened on 15 September 1840 as Tottenham. It was on the Northern & Eastern Railway (N&ER) line from Stratford in east London to Broxbourne in Hertfordshire.
It was renamed Tottenham Hale on 1 September 1968 when it became an interchange station with London Underground on the opening of the first stage of the Victoria line.
Clapton railway station
Clapton railway station is on a suburban line operated by London Overground. It is located on Upper Clapton Road.
It links Upper Clapton to Liverpool Street in central London and Chingford.
Clapton is on a branch of the Lea Valley Lines. The branch separates into Tottenham Hale and St James Street. The line joins up with the Seven Sisters branch at Hackney Downs. Trains which call at Clapton do not serve the Tottenham Hale branch.
Clapton is in both Travelcard Zone 2 and Travelcard Zone 3.
Liverpool Street and
Chingford
List of Police Academy episodes
List of episodes of the animated series of .
P versus NP
P versus NP is the following question of interest to people working with computers and in mathematics: Can every solved problem whose answer can be checked quickly by a computer also be quickly solved by a computer? P and NP are the two types of maths problems referred to: P problems are fast for computers to solve, and so are considered "easy". NP problems are fast (and so "easy") for a computer to check, but are not necessarily easy to solve.
In 1956, Kurt Gödel wrote a letter to John von Neumann. In this letter, Gödel asked whether a certain NP complete problem could be solved in quadratic or linear time. In 1971, Stephen Cook introduced the precise statement of the P versus NP problem in his article "The complexity of theorem proving procedures".
Today, many people consider this problem to be the most important open problem in computer science. It is one of the seven selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution.
For instance, if you have a problem, and someone says "The answer to your problem is the set of numbers 1, 2, 3, 4, 5", a computer may be able, quickly, to figure out if the answer is right or wrong, but it may take a very long time for the computer to actually come up with "1, 2, 3, 4, 5" on its own. Another example includes finding prime numbers. It is easy to check if a number is prime, but it is very difficult to find those numbers in the first place. For some interesting, practical questions of this kind, we lack any way to find an answer quickly, but if we are provided an answer, it is possible to check—that is, to verify—the answer quickly. In this way, NP problems may be thought of as being like riddles: it may be hard to come up with the answer to a riddle, but once one hears the answer, the answer seems obvious. In this comparison (analogy), the basic question is: are riddles really as hard as we think they are, or are we missing something?
Because these kinds of "P versus NP" questions are so practically important, many mathematicians, scientists, and computer programmers want to prove the general proposition, that every quickly-checked problem can also be solved quickly. This question is important enough that the Clay Mathematical Institute will give $1,000,000 to anyone who successfully provides a proof or a valid explanation that disproves it.
Digging a little deeper, we see that all P problems are NP problems: it is easy to check that a solution is correct by solving the problem and comparing the two solutions. However, people want to know about the opposite: Are there any NP problems other than P problems, or are all NP problems just P problems? If NP problems are really not the same as P problems (P ≠ NP), it would mean that no general, fast ways to solve those NP problems can exist, no matter how hard we look. However, if all NP problems are P problems (P = NP), it would mean that new, very fast problem-solving methods do exist. We just have not found them yet.
Since the best efforts of scientists and mathematicians have not found general, easy methods for solving NP problems yet, many people believe that there are NP problems other than P problems (that is, that P ≠ NP is true). Most mathematicians also believe this to be true, but currently no one has proven it by rigorous mathematical analysis. If it can be proven that NP and P are the same (P = NP is true), it would have a huge impact on many aspects of day-to-day life. For this reason, the question of "P versus NP" is an important and widely studied topic.
Suppose someone wants to build two towers, by stacking rocks of different mass. One wants to make sure that each of the towers has exactly the same mass. That means one will have to put the rocks into two piles that have the same mass. If one guesses a division of the rocks that one thinks will work, it would be easy for one to check if one was right. (To check the answer, one can divide the rocks into two piles, then use a balance to see if they have the same mass.) Because it is easy to check this problem, called 'Partition' by computer scientists—easier than to solve it outright, as we will see—it is not a P problem.
How hard is it to solve, outright? If one starts with just 100 rocks, there are 2^{100-1}-1 = 633,825,300,114,114,700,748,351,602,687, or about 6.3 x 10^{29} possible ways (combinations) to divide these rocks into two piles. If one could check one unique combination of rocks every day, it would take 1.3 x 10^{22} or 1,300,000,000,000,000,000,000 years of effort. For comparison, physicists believe that the universe is about 1.4 x 10^{10} years old (450,000,000,000,000,000 or about 4.5 x 10^{17} seconds, or about one trillionth as old as the time it would take for our rock piling effort. That means that if one takes all of the time that has passed since the beginning of the universe, one would need to check more than two trillion (2,000,000,000,000) different ways of dividing the rocks "every second", in order to check all of the different ways.
If one programmed a powerful computer, to test all of these ways to divide the rocks, one might be able to check formula_1 combinations per second using current systems. This means one would still need formula_2 very powerful computers, "working since the origin of the universe," to test all the ways of dividing the rocks.
However, it may be possible to find a method of dividing the rocks into two equal piles without checking all combinations. The question "Is P equal to NP?" is a shorthand for asking if any method like that can exist.
There are many important NP problems that people don't know how to solve in a way that is faster than testing every possible answer. Here are some examples:
In the example above, we see that with formula_3 rocks, there are formula_4 ways to partition the set of rocks. With formula_5 rocks, there are formula_6 combinations. The function formula_7 is an exponential function. It's important to NP because it models the worst-case number of computations that are needed to solve a problem and, thus, the worst-case amount of time required.