Problem 28. Prove that, for every IL FSC C = (Q,δ,ν,s) and every n ∈ N, there is an input string w ∈ {0,1}^n such that |C(w)|≥n−log2 |Q| .

For S = {1,{1,2},3} find the power set P(S)

Design an iterative function Collatz(n,k) that computes the k^th number in the hailstone sequence starting at n: Collatz(n,k)= { c(c(c(...c(n)...))) }<----- k iterations

Select all correct Statements - logn ∈ O(n) - n2 ∈ O(nlgn) - log3n ∈ o(n) - 5nlogn + 2n2 ∈ Θ(nlogn) - f(n) + g(n) ∈ O(max(f(n), g(n))) - f2(n) ∈ Ω(f(n))

Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) : V4? + 1? = /17 and dist(a, b) = v2? + 0² = 2. Because many pairs of points have identical distances (e.g. dist(h, c) V5), the above diagram has more than one minimum-weight spanning tree. dist(h, b) = dist(h, f) Determine the total number of minimum-weight spanning trees that exist in the above diagram. Clearly justify your answer.

3. 4. Given a directed acyclic graph G = (V, E) and two vertices s, te V, design an efficient algorithm that computes the number of different directed paths from s to t. Define the incidence matrix B of a directed graph with no self-loop to be an nxm matrix with rows indexed by vertices, column indexed by edges such that Bij = -1 1 0 if edge j leaves vertex i, if edge jenters vertex i, otherwise. Let BT be the transpose of matrix B. Find out what the entries of the n x n matrix BBT stand for.

Create an R function pwfun(), which computes values of the piece-wise (mathematical) function pwfun defined in the following way: x < -1 Its graph is as follows. e pwfun(c(-2, 0.5, 3)) 208 pwfun(x) = - x = c(-3, -2, -1, 0, 1, 2, 3.2) pwfun(x) 4 2 0 0 0 39.24 -2x-2 0 x²-1 18 Your R function pwfun() should also accept x as a vector of arbitrary length and return vector of the same length with entries defined as values of the mathematical function pwfun, evaluated at the corresponding entries of vector x. Here are some input vectors and what you should get as a corresponding output. pufun(-1) .-1≤x≤1 1

In this problem, we consider the class of n-dimensional axis-aligned boxes. Each function c in this class is specified by aset of 2n values ℓ1, . . . , ℓn and u1, . . . , un in [0, 1], which defines an axis-aligned n-dimensional box. Given an n-dimensionalinput vector x, c(x) is defined to be 1 (or ‘+’) if for every i ∈ {1, . . . , n} the i’th coordinate of x lies in [ℓi, ui]. Otherwise,c(x) is defined to be 0 (or ‘-’) .An example of a 2-dimensional axis-aligned box is shown in the figure below. (a) Let C be the class of n-dimensional axis-aligned boxes, and c ∈ C be the unknown target function. State an efficientalgorithm that takes as input an arbitrary set of points x1, x2, . . . , xm each a point in n-dimensional space, and theircorresponding labels c(x1), c(x2), . . . , c(xm), and outputs a function h ∈ C that is consistent with this data. Youralgorithm should run in polynomial time in m, n and facilitate answering part (b) below.   Let D be an arbitrary, unknown distribution, and…

Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with  uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2,   T = A · D = A3,   Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V   Tu,u

Let A = {(N1, N2) | N1 and N2 are NFAS and L(N1)N L(N2) = Ø}. Show that A is decidable.