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In 1.2.1 Mathematical Induction section, Knuth presents mathematical induction as a two steps process to prove that P(n) is true for all positive integers n:
a) Give a proof that P(1) is true;
b) Give a proof that "if all P(1), P(2),..., P(n) are true, then P(n+1) is also true";
I have serious doubt about that. Indeed, I believe that point b) should be:
b) Give a proof that "if P(n) is true, then P(n+1) is also true". The major difference here is that you are only assuming that P(n) is true, not P(n-1), etc.
However, these books are old and have been read by many people (most of them being much more clever than I am^^).
So what is my confusion here?
The entire point here is that the choice of n is arbitrary. Since P(n) implies P(n+1) is the conerstone of induction, then all the intermediate values between 1 and n will also hold under the assumption of P(n). You are supposed to show that if P(0) implies P(1) and P(n) implies P(n+1) then all conditions hold by the nature of n being arbitrary.
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I have the following problem in dynamic programming.
A person has time machine and he can move in time either 1 year or 2. At the beginning he is at year 0 and he wants to reach year 100. Every step he does (1 or 2 years) he is paying some fixed fees. There is an array with 100 integers represents the fee he needs to pay if he went threw the specific year.
I need to find the minimum amount the person can pay to go from year 0 to year 100 using dynamic programming.
From what i have done so far i think that there should be something like
minCost(i) = min{A[i-1], A[i-2]}
and the base cases are years 1 and 2 which costs A[1], A[2] respectively. But i think this approach has more of greedy algorithm rather than dynamic programming.
I saw the bin packing algorithm of dynamic programming and i understood it and the matrix that represents it.
How should the matrix of the shown problem above look like?
And how should i build the function and the pseudo code for this problem?
You are almost there.
Think about how will you reach the i th year from i-1 th year and i-2 th year. There is a fee which you are forgetting to take into consideration.
MinCostToReachYear(i) = min( MinCostToReachYear(i-1) + fee(i-1), MinCostToReachYear(i-2) + fee(i-2) )
You already know the base cases year 1 and year 2. Can you think of extrapolating with the use of a for loop or more easily with a recursive function which you already know as mentioned above? I leave it as an exercise for you.
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I need to create a column in SQL Server database. Entries for that column will contain messages from chat. Previously such messages has been stored as comments.
My main quetion is:
What is typical text length for chat message and comment?
By the way:
What would happen if I used varchar(max)? How would it impact database size and performance? Is better to use powers of 2 or powers of 10 (e.g. 128 instead of 100) while considering text lengths?
Using VARCHAR(MAX) has a disadvantage: you can not define an index over this column.
Generally, your application should impose a maximum length for a chat message. How big that limit is depends very much on what the application is used for. But anything more than 1000 byte is probably less a legitimate message but an attempt to disrupt your service.
If your maximum value is a power of 2, or a power of ten or any other value has no influence on the performance as long as the row fits in one (8KB) page.
Short answer - it doesn't matter.
From MSDN:
The storage size is the actual length of the data entered + 2 bytes.
So VARCHAR(10) and VARCHAR(10000) will consume the same amount of data if the values don't exceed 10 characters.
Definitely use N/VARCHAR(MAX), it can grow to be 2GB (if I remember correctly). It will grow as required though, so it is very efficient with regards to space unless you are only storing very small amounts of data.
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This is findind limit calculus one. This would be very easy but i never factor high degrees. can anyone show me how to factor it.
lim((x^5-32)/(x-2),x=2)
lim((x^5-2^5)/(x-2),x=2)
As John suggested in the comments above: when x-->2 we handle a limit of type 0/0. In order to calculate it we use the derivative of the numerator and a derivative of the denominator:
f'(X^5-2^5) = 5x^4
---------- ----
f'(x-2) = 1
if we'll substitute x with 2 we'll get:
5*2^4
----- = 80
1
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I'm taking a crypto course, and we're going over substitution ciphers and their Key space.
per the instructor, the key space is 26! (approx 2^88) for the English alphabet. there is no reference to key length, probably because a subst cipher's length would be a function of the length of the alphabet, just as the number of options would.
per wikipedia the keyspace is the set of all possible keys of a certian length, and is calculated in the same way brute force try counts would be options^length or in this case 26^26.
so what am I not getting here?
That's a bit misleading, both your instructor and Wikipedia are correct.
Generally, key of 26 english letters defines a key space sized 2626.
For substitution ciphers over english alphabet 26! is the correct number representing the key space. That's because for substitution cipher the key is defined as a unique replacement of each letter with another one, e.g. A -> D, B -> M, C -> Y, etc. 26 letters --> key can be any permutation of 26-letter set --> 26!. Due to the uniqueness required for substitution, the key space is effectively smaller than the maximal 2626, because some (most) of the keys aren't possible - e.g., you can't map both A and B to D.
If your key is a set of digits, options^length is correct. Every digit may occur several times.
If your key is an alphabet, Factorial N is correct. Say, you want to place the A first. You have 26 options. After that, you have only 25 options for the B because A already occupies one. 24 For the C and so on.
26*25*24*...*1 = 26!
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I learn cryptography at my university, and I need to make an application for testing numbers as prime numbers.
I must to do it using Lehmann Test, but I don't know anything about it. Please, describe me the algorithm or give me an example (Java, C#, C++, etc). Thank you for helping.
Let's call PP your Potential Prime.
(1) Choose a random number a less than PP.
(2) Calculate a^(p-1)/2 mod PP.
(3) If a^(PP-1)/2 /= 1 or -1 (mod PP), then PP is not prime.
(4) If a^(PP-1)/2 = 1 or -1 (mod PP), then the probability that PP is not prime is less than 50%.