SEM 2 Flashcards

1
Q

Can the hcf be negative

A

Yes

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2
Q

What is the prime number formula (Works up to n = 41)?

A

P(n) = n^2 - n + 41

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3
Q

What are the steps of mathematical induction?

A

Prove P(N) is true

Prove P(k+1) is true given P(k) is true

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4
Q

Strong induction?

A

Like mathematical except

P(k + 1) is true when P(N), P(N + 1) … is true

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5
Q

What is fermats little theorem?

A

a^p = a mod p

a^p-1 = 1 mod p (if p is not factor of a)

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6
Q

Key generation: n = ?

A

pq

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7
Q

Key Gen: e = ?

A

e is between 1 and (p-1)(q-1) and

e and (p-1)(q-1) are coprime

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8
Q

Key gen: d = ?

A

d > 0 such that

de == 1 mod (p-1)(q-1)

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9
Q

What is the public key?

A

(n,e)

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10
Q

What is private key?

A

d

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11
Q

How to encrypt?

A

For each value M

c = M^e mod n

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12
Q

How to decrypt?

A

M = c^d mod n

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13
Q

How to find inverse of 2x2 matrix?

A

1/ad - bc (d, -b, -c, a)

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14
Q

How to find inverse of any matrix (not 2x2)

A
  • Write identity matrix next to it
  • Turn left side into identity matrix and right side is inverse
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15
Q

What is minor?

A

Determinant (without working out result)

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16
Q

What is cofactor?

A

Multiply minor by -1

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17
Q

When does a system of linear equations have a unique solution?

A

When determinant != 0

18
Q

When is a matrix invertible?

A

When the determinant != 0

19
Q

If a matrix is diagonal what are its eigenvalues?

A

The diagonal elements

20
Q

What makes a matrix similar?

A

B = P^-1 A P

21
Q

Do similar matrices have the same characteristic polynomials and eigenvalues?

A

Yes

22
Q

What makes A diagonalisable?

A

If there is a diagonal matrix which is similar

23
Q

Cartesian product?

A

Set of ordered pairs

24
Q

Permutation? P(n,k)

A

n! / (n-k)!

  • ordered collection
25
Q

Combinations? C(n,k)

A

n! / k! (n-k)!

26
Q

Probability space consists of?

A

Sample space and probability function

27
Q

Conditional probability? A given B

A

P(A n B) / P(B)

28
Q

Bayes theorem? (A given B)

A

P(B | A)P(A) / P(B)

29
Q

Law of total probability?

A

P(A) = P(A|B1)P(B1) + … + P(A|Bn)P(Bn)

30
Q

Expectation value?

A

Sum of:

x * p(x)

31
Q

Var?

A

E(X^2) - E(X)^2

32
Q

sd?

A

Root varience

33
Q

Bernoulli distribution?

A
  • Range of possible values 0 -1
  • P(x = 1) = p
  • P(X = 0) = 1- p
  • E(X) = p
  • var(X) = p(1-p)
34
Q

Binomial distribution

A
  • Range up to n
  • P(X = k) = C(n,k)p^k(1-p)^n-k for each k
  • E(X) = np
  • var(X) = np (1-p)
35
Q

Poisson distribution?

A
  • Unlimited range
  • P(X = k) = (lambda)^k/ k! * e^ -(lambda)
  • E(X) and var(X) = lambda
36
Q

Difference eqution?

A

vk+1 = Avk

37
Q

Solve difference equation?

A

vk = A^k v0

38
Q

Markov process?

A
  • Calculate prop for event a and event b using given info
  • Place in Mx = b (each value matrix)
  • For given time distribution is given by M^k x
39
Q

Transition matrix?

A

Probability of moving from state to state as a vector

40
Q

Markov process (Long term behavior)?

A

vk = v

  • v is (lambda)-eigenvector of M