Problem 2: Recall that pi(n) denotes the number of integer partitions of n with exactly k parts. Show that Pk(n)an- m11 n20 Problem 2: Recall that pi(n) denotes the number of integer partitions...
3) Let a -pip?.. .Pk and blp22.. .P%* where pi are distinct positive primes and ri, si are non-negative integers. Then show that where for each i, ni minfri, si) (Recall that min{c, d\ denotes the minimum of c and d.) 3) Let a -pip?.. .Pk and blp22.. .P%* where pi are distinct positive primes and ri, si are non-negative integers. Then show that where for each i, ni minfri, si) (Recall that min{c, d\ denotes the minimum of c...
6.2 Prove that for fixed positive integers k and n, the number of partitions of n is equal to the number of partitions of 2n + k into n + k parts.
Given: •pk denotes the probability that the number of claims equals k for k= 0,1,2,... •pn/pm=m!/n!, m≥0, n≥0 Using the corresponding zero-modified claim count distribution with P0M= 0.1, calculate P1M
Show that the number of partitions of n in which each part appears either 0, 2, 5, or 7 times is the same as the number of partitions of n in which each part is either 2 mod 4, or 5 mod 10.
Problem 4. Let w be a positive continuous function and let n be a nonnegative integer. Equip P.(R) with the inner product (p,q) = $' p(x)q(x)"(x) dx. You do not need to check that this is an inner product. (a) Prove that P.(R) has an orthonormal basis po..., Pr such that deg pk = k for each k. (b) Show that (Pk, pk) = 0 for each k, where the polynomials pį are from the preceding part. Here pé denotes...
2. Count the number N(k) of all finite automata with exactly k states. Prove that in the definition of the regularity we can't restrict the num- ber or states to be less than some fixed integer
2. Recall that Matnxn(F) denotes the vector space of n x n-matrices with entries in F, define T: Mn + Mn by T(A) = A -AT. Show that T is a linear transformation and find its kernel and image.
Problem A: For any integer k, 0 s k sS n, determine the number of vectors (x,xn), such that each x is or 1 and 2-1xi2k.
A fourth order, Type I, linear phase, FIR filter, h[n], is to be designed using the window method. The ideal impulse response of the filter is defined as:hd[n] = sin([pi/4]*[n - N/2]) / ([n - N/2]*pi) ,where N is the filter order and 'pi' denotes the mathematical (irrational) constant number 3.14159.... Given that a stopband attenuation of 50 dB is required,a) Find and sketch h[n]b) Determine the transfer function of the resulting digital filterc) Draw the filter block diagramd) Determine...
Exercise 7 (2 points) Recall the binomial coefficient for integer parameters 0 Sk< n. Prove that Exercise 8 (2 points) Prove the following: if z is an integer with at most three decimal digits aia2a3, then x is divisible by 3 if and only if aut a2 +a3 is divisible by 3. Exercise 9 (3 points) A square number is an integer that is the square of another integer. Let x and y be two integers, each of which can...