Problem 1: Universal Approximation Theorem for Deep Neural Networks with ReLU Activations Statement: Prove that a feedforward neural network with a single hidden layer using ReLU (Rectified Linear Unit) activation functions is a universal approximator. Specifically, show that for any continuous function f : R" → R and for any € > 0, there exists a neural network N with one hidden layer such that: sup f(x) N(x)| < € xЄK where K is a compact subset of Rn. Key Points for the Proof: • • Utilize the properties of ReLU functions to construct piecewise linear approximations of f. Show that the linear combinations of ReLU activations can approximate any continuous function on compact subsets. Address the density of the set of functions representable by the neural network in the space of continuous functions.

Part ii)

a. The general continuous-time random walk is defined by g_(ij)={[mu_(i)",",j=i-1","],[-(lambda_(i)+mu_(i))",",j=i","],[lambda_(i)",",j=i+1","],[0","," otherwise "]:} Write out the forward and backward equations. b. The continuous-time queue with infinite buffer can be obtained by modifying the general random walk in the preceding problem to include a barrier at the origin. Put g_(0j)={[-lambda_(0)",",j=0","],[lambda_(0)",",j=1","],[0","," otherwise ".]:} Find the stationary distribution assuming sum_(j=1)^(oo)((lambda_(0)cdotslambda_(j-1))/(mu_(1)cdotsmu_(j))) < oo. If lambda_(i)=lambda and mu_(i)=mu for all i, simplify the above condition to one involving only the relative values of lambda and mu. c. Repeat a and b assuming there is a boundary at j=N. Comments.need explicit calculation process, step by step

A certain experiment produces the data (0, 1),(−1, 2),(1, 0.5),(2, −0.5). Find values for a, b, and c which describe the model that produces a least squares fit of the points by a function of the form f(x) = ax2 + bx + c.