(1) Consider the adjacency matrix A of a directed network of size N =4 given by 1 A = 1 0 0 1 0 we will indicate with 1 the column vector with elements N and we will indicate with Ithe identity matrix. In the following 1 for i = 1,2, - (a) Draw the network (b) Calculate the eigenvector centrality using its definition. (c) Calculate the Kate centrality. %3D %3D ... (1) lk gontrolity
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- 2. Show the following cordinate frame matrix for cordinate space that has the following transform applied to it [order given, in global/world space]: rotation around the X axis of 30 degrees translation along (X=20,Y=10,Z=50) rotation around the Y axis of 135 degrees scaling factor of 6 along the Z axis translation along (X=50,Y=0,Z=50)3.9 Consider an autoassociative net with the bipolar step function as the activation func- tion and weights set by the Hebb rule (outer products), with the main diagonal of the weight matrix set to zero. a. Find the weight matrix to store the vector Vi=(,1,1,1,1, 1. b. Test the net, using V, as input. ¢. Test the net, using T =(,1,11,-1,-1. d. Find the weight matrix to store the vector Ya= (il 1 ~1,~1, ~1), Test the net, using V; as input. f. Test the net, using Lh=(,11,-10,0). Find the weight matrix to store both V, and V,. . Test the neton V,, V5, Ty, T>. FwBu = ƒ and Cu = f might be solvable even though B and C are singular. Show that every vector f = Bu has ƒ1 + ƒ2+ ……. +fn = 0. Physical meaning: the external forces balance. Linear algebra meaning: Bu = ƒ is solvable when ƒ is perpendicular to the all – ones column vector e = (1, 1, 1, 1…) = ones (n, 1).
- For the two clusters data points [(3,1),(3,-1), (6,1),(6,-1)] and [(1,0),(0,1), (0,-1), (-1,0)] mark the support vectors and find the equation for linear support vector machine for this linearly separable data points.3.6 a. Use the hybrid (binary/bipolar) form of Hebb rule learning as described in Example 3.6 to find the weight matrix for the associative memory network based on the following binary input-output vector pairs: s=0100 0 (=01 0 =01 001 @@= 0 s3) =0 1.0 0 t3)=0 1 s@=0 110 H4) =@ 1) b. Using the unit step function (with threshold 0) as the output units’ activation function, test the response of your network on each of the input patterns. Describe the results obtained. c. Test the response of your network on various combinations of input patterns with “‘mistakes” or ‘‘missing-data’ (as in Example 3.8). Discuss the results you ob- serve.1. Develop a system of linear equations for the network by writing an equation for each router (A, B, C, D, and E). Make sure to write your final answer as Ax=b where A is the 5x5 coefficient matrix, x is the 5x1 vector of unknowns, and b is a 5x1 vector of constants. 2. construct the augmented matrix [A b] and then perform row reduction using the rref() function. Write out your reduced matrix and identify the free and basic variables of the system. 3. Use MATLAB to compute the LU decomposition of A, i.e., find A = LU. For this decomposition, find the transformed set of equations Ly = b, where y = Ux. Solve the system of equations Ly = b for the unknown vector y. 4. Compute the inverse of U using the inv() function. 5. Compute the solution to the original system of equations by transforming y into x, i.e., compute x = inv(U)y 6. Check your answer for x1 using Cramer’s Rule. Use MATLAB to compute the required determinants using the det() function.
- ind the general solution of the following linear systems. Sketch the phase portraits. Pick several points on the phase plane to plot the vector field and show your work. Plot also few trajectories. Please be neat in showing your solutions. Use Matlab to plot the vector field (on one figure) and to plot the solution to the system (on another figure). a)*?′+2?+3?=0 *?′ + 3? + 2? = 0 b)?′ =4?−12? ?′ = 5? + ? c) ?′ = −? + 3 ? ? =−6?−2? d)?′=5?−5? ?′ =5?−3?Let f ∈ C+ 2π with a zero of order 2p at z. Let r>p and m = n/r. Then there exists a constant c > 0 independent of n such that for all nsufficiently large, all eigenvalues of the preconditioned matrix C−1 n (Km,2r ∗ f)Tn(f) are larger than c.Show the residual graph for the network flow given in answer to part (a) Show the final flow that the Ford-Fulkerson Algorithm finds for this network, given that it proceeds to completion from the flow rates you have given in your answer to part (a), and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?). Identify a cut of the network that has a cut capacity equal to the maximum flow of the network.
- solve it with Julia programming 1: Modify your models and coding of Chebyshev center and building a house to get the right anwsers. # Given matrix A and vector bA = [2 -1 2; -1 2 4; 1 2 -2; -1 0 0; 0 -1 0; 0 0 -1]b = [2; 16; 8; 0; 0; 0] answer wil be:x is: 0.75y is: 3.25z is: 0.7500000000000001r is: 0.7500000000000001objective value is: 0.7500000000000001Given the complement of a graph G is a graph G' which contains all the vertices of G, but for each unweighted edge that exists in G, it is not in G', and for each possible edge not in G, it is in G'. What logical operation and operand(s) can be applied to the adjacency matrix of G to produce G'? AND G's adjacency matrix with 0 to produce G' XOR G's adjacency matrix with 0 to produce G' XOR G's adjacency matrix with 1 to produce G' AND G's adjacency matrix with 1 to produce G'Consider the following directed network with flows written as the first number and edge capacity as the second on each edge: Part 1 Draw the residual network obtained from this flow. Part 2 Perform two steps of the Ford Fulkerson algorithm on this network, each using the residual graph of the cumulative flow, and the augmenting paths and flow amounts specified below. After each augment, draw two graphs, preferably side by side; these are graphs of: a) The flow values on the edges b) Residual network The augmenting paths and flow amounts are: i) s → b→d c→t with flow amount 7 Units. ii) s → b→ c→ t with 4 units. Note for continuity your second graph should be coming from the one in (i) NOT from the initial graph. Part 3 Exhibit a maximum flow with flow values on the edges, state its value, and exhibit a cut (specified as a set of vertices) with the same value.