Homework Answers
Prove that {a i b j | i ≤ 2j and j ≤ 2i} is nonregular...
Prove that {a i b j | i ≤ 2j and j ≤ 2i} is nonregular using the Myhill-Nerode theorem. That is, exhibit infinitely many pairwise distinguishable strings.
Prove that aw і < 2] and J < 2i} is nonregular using the Myhill-Nerode theorem....
Prove that aw і < 2] and J < 2i} is nonregular using the Myhill-Nerode theorem. That is, exhibit infinitely many pairwise distinguishable strings
Imprecise Counting - Long Runs in Binary Strings Let n=2^k for some positive integer k and...
Imprecise Counting - Long Runs in Binary Strings Let n=2^k for some positive integer k and consider the set Sn of all n-bit binary strings. Let c be an integer in {0,…,n−k}. Consider any j∈{1,…,n−k−c+1}. How many strings b1,…,bn∈Sn have bj,bj+1,…,bj+k+c−1=00…0? In other words, how many strings in Sn have k+c consecutive zeros beginning at position j? For each j∈{1,…,n−k+c+1}, let Xj be the subset of Sn consisting only of the strings counted in the previous question. Show that (n−k−c+1)∑(j=1)...
VI |JI Make a sketch showing the vectors J , K and J-K . Explain why...
VI |JI Make a sketch showing the vectors J , K and J-K . Explain why -HN and use this relationship to determine J +K 8 , 0 3 g | K 1250.001-303 M 90.0 180g
Prove that if an integer n is not divisible by 3, then n^2=3k+1 for some integer...
Prove that if an integer n is not divisible by 3, then n^2=3k+1 for some integer k. Note: “not divisible by 3” means either “n=3m+1 for some integer m” or “n=3m+2 for some integer m”.
16. Prove that, for each integer k, 24k = 1, 24k+1 = 1, 24k+2 = -1,...
16. Prove that, for each integer k, 24k = 1, 24k+1 = 1, 24k+2 = -1, 24k+3 = -i. Show how this result gives a formula for in for all n by writing n = 4k+],0 j <3.
Prove that the given language is Not Context Free L2 = { w ∈ {0,1,2}∗ | w follows 0^(i)1^(j)2^(k)...
prove that the given language is Not Context Free L2 = { w ∈ {0,1,2}∗ | w follows 0^(i)1^(j)2^(k) pattern, where i < j and i < k and i, j, k >= 0 }
Prove that, for large integer k 〉 0, the 2-norm of an arbitrary matrix Ak behaves...
Prove that, for large integer k 〉 0, the 2-norm of an arbitrary matrix Ak behaves asymptotically like ー2+1 where j is the largest order of all diagonal submatrices J of the Jordan form with o(%)-ρ(A) and v is a positive constant. (Hint: refer to Greenbaum for an expression of the kth power of a j-by-j Jordan block)
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k,...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
Consider a random vector Y () y(2). y(k) where the elements y(i) are made yi)wi), j-1,...
Consider a random vector Y () y(2). y(k) where the elements y(i) are made yi)wi), j-1, ...k where w(j) are independent, identically distributed, Gaussian, zero-mean, and with the variance σ2 i.e., N(0, σ2). 1. Find the Maximum Likelihood (ML) estimator for xr, i.e., ML 2. Find the Mean Square Error (MSE) of ML estimator, i.e., MSE(XML) Ξ Var@sL) 3. Is this estimator consistent? Prove your answer 4. Is this estimator efficient? Prove your answer
Prove that aw і < 2] and J < 2i} is nonregular using the Myhill-Nerode theorem. That is, exhibit infinitely many pairwise distinguishable strings
VI |JI Make a sketch showing the vectors J , K and J-K . Explain why -HN and use this relationship to determine J +K 8 , 0 3 g | K 1250.001-303 M 90.0 180g
16. Prove that, for each integer k, 24k = 1, 24k+1 = 1, 24k+2 = -1, 24k+3 = -i. Show how this result gives a formula for in for all n by writing n = 4k+],0 j <3.
Prove that, for large integer k 〉 0, the 2-norm of an arbitrary matrix Ak behaves asymptotically like ー2+1 where j is the largest order of all diagonal submatrices J of the Jordan form with o(%)-ρ(A) and v is a positive constant. (Hint: refer to Greenbaum for an expression of the kth power of a j-by-j Jordan block)
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
Consider a random vector Y () y(2). y(k) where the elements y(i) are made yi)wi), j-1, ...k where w(j) are independent, identically distributed, Gaussian, zero-mean, and with the variance σ2 i.e., N(0, σ2). 1. Find the Maximum Likelihood (ML) estimator for xr, i.e., ML 2. Find the Mean Square Error (MSE) of ML estimator, i.e., MSE(XML) Ξ Var@sL) 3. Is this estimator consistent? Prove your answer 4. Is this estimator efficient? Prove your answer
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