Cis regulаtоrs оf gene expressiоn will аlwаys be located on the same chromosome (or DNA molecule) as the gene they regulate
Instructiоns: On а sepаrаte sheet оf paper, answer each оf the exam problems shown below. Write your answers clearly. Unless otherwise stated, you will need to justify your answers to get the full credit. Problem 1. (15 pts) Given a quadratic form, f = 1 2 x ⊤ Q x − x ⊤ b + c , {"version":"1.1","math":"[ f =frac{1}{2}x^top Qx - x^top b +c, ]"}where f = 1 2 x ⊤ [ 2 0 0 1 ] x − x ⊤ [ 1 − 1 ] + 33 {"version":"1.1","math":"[ f=frac{1}{2}x^topbegin{bmatrix}2 & 0\0 &1end{bmatrix}x-x^top begin{bmatrix}1\-1end{bmatrix}+33 ]"} and d ( 0 ) = [ 1 − 1 ] ⊤ {"version":"1.1","math":"( d^{(0)}=left[begin{array}{cc} 1 & -1 end{array}right]^{top} )"}. Determine d 2 {"version":"1.1","math":"(d_2)"} so that d ( 0 ) {"version":"1.1","math":"(d^{(0)})"} and d ( 1 ) = [ − 1 / 5 d 2 ] ⊤ {"version":"1.1","math":"( d^{(1)}=begin{bmatrix}-1/5 & d_2 end{bmatrix} ^top)"}are Q {"version":"1.1","math":"(Q)"}-conjugate directions. Problem 2. (15 pts) Consider a series RLC{"version":"1.1","math":"( RLC )"} circuit consisting of a resistance R{"version":"1.1","math":"( R )"}, an inductance L{"version":"1.1","math":"( L )"}, and a capacitance C{"version":"1.1","math":"( C )"} driven by the voltage source Vin{"version":"1.1","math":"( V_{in} )"}. The current in the circuit is i{"version":"1.1","math":"( i )"}. The voltage across the capacitor is Vout{"version":"1.1","math":"( V_{out} )"}. Applying Kirchhoff's voltage law, we obtain the following differential equation modeling the circuit, Ldidt+Ri+Vout=Vin{"version":"1.1","math":"Lfrac{d i}{dt}+ Ri + V_{out} = V_{in}"} Our objective is to estimate L{"version":"1.1","math":"( L )"} and R{"version":"1.1","math":"( R )"} using available measurements of i{"version":"1.1","math":"( i )"}, didt{"version":"1.1","math":"( frac{d i}{d t} )"}, the output voltage Vout{"version":"1.1","math":"( V_{out} )"}, and the input voltage Vin{"version":"1.1","math":"( V_{in} )"} at three different instances of the circuit operation. The results of measurement experiments are given in the table below. Experiment No. didt{"version":"1.1","math":"( frac{di}{dt} )"} i{"version":"1.1","math":"( i )"} Vout{"version":"1.1","math":"( V_{out} )"} Vin{"version":"1.1","math":"( V_{in})"} 1 1 0 -1 0 2 0 1 0 2 3 0 -1 0 1 Obtain the least squares estimate of the circuit parameters L{"version":"1.1","math":"( L )"} and R{"version":"1.1","math":"( R )"}. Problem 3. (15 pts) Let A 0 = [ 1 0 1 1 ] , {"version":"1.1","math":"A_0=left[begin{array}{cc} 1 & 0\ 1 & 1 end{array}right],"} b ( 0 ) = [ 1 1 ] , {"version":"1.1","math":"quad b^{(0)}=left[begin{array}{c} 1\ 1end{array}right], "} and a 1 ⊤ = [ 0 1 ] , b ( 1 ) = 1. {"version":"1.1","math":"a_1^top=left[begin{array}{cc} 0 & 1 end{array}right], quad b^{(1)}=1. "} (5 pts) Use the recursive least squares to solve the combined system of equations. (10 pts) Compute the pseudoinverse of A = [ A 0 a 1 ⊤ ] . {"version":"1.1","math":"A=left[begin{array}{c} A_0\ a_1^top end{array}right]. "} Problem 4. (15 pts) minimize ‖ x ‖ 2 subject to [ b a b a ] x = [ 1 1 ] , {"version":"1.1","math":"begin{eqnarray*} mbox{minimize}&{}& |x|_2\ mbox{subject to}&{}&{}\ {}&{}& left[begin{array}{cc} b & a\ b & aend{array}right]x=left[begin{array}{c} 1\ 1 end{array}right], end{eqnarray*}"} where a{"version":"1.1","math":"( a )"} and b{"version":"1.1","math":"( b )"} are non-zero real parameters. Problem 5. (15 pts) Consider a particle whose current position and velocity vectors are: x c u r r e n t = [ 3 1 ] and v c u r r e n t = [ 1.5 2.5 ] . {"version":"1.1","math":"x_{current}=left[begin{array}{c} 3\ 1 end{array}right]quadmbox{and}quad v_{current}=left[begin{array}{c} 1.5\ 2.5 end{array}right]. "} Find the particle's next position, xnext{"version":"1.1","math":"( x_{next} )"}, using the PSO gbest algorithm, where the inertial constant ω=1{"version":"1.1","math":"( omega=1 )"}, the cognitive coefficient c1=2{"version":"1.1","math":"( c_1=2 )"}, and the social coefficient c2=2{"version":"1.1","math":"( c_2=2)"}. The pbest{"version":"1.1","math":"( pbest )"} is p=[0.51.5]{"version":"1.1","math":"p=left[begin{array}{c} 0.5\ 1.5 end{array}right] "} and the gbest{"version":"1.1","math":"( gbest ) "} is g=[56].{"version":"1.1","math":"g=left[begin{array}{c} 5\ 6 end{array}right]. "} Assume that the random vectors r{"version":"1.1","math":"( r )"} and s{"version":"1.1","math":"( s )"} {"version":"1.1","math":"( s )"} have the form, r=12[11]{"version":"1.1","math":"r=frac{1}{2}left[begin{array}{c} 1\ 1 end{array}right] "} and s=14[11].{"version":"1.1","math":"s=frac{1}{4}left[begin{array}{c} 1\ 1 end{array}right]. "} Problem 6. (10 pts) (5 pts) Consider the one-point crossover of a chromosome in the schema H = ∗ 1 ∗ 0 1 0 ∗ {"version":"1.1","math":"$$ H= begin{array}{ccccccc} ast & 1 & ast & 0 & 1 & 0 & ast end{array} $$"}where the probability that a chromosome is chosen for crossover is p c = 0.5 {"version":"1.1","math":"( p_c=0.5)"}. Find a lower bound on the probability that the schema H {"version":"1.1","math":"( H )"}survives the one-point crossover operation. (5 pts) Consider a chromosome in the schema H = ∗ 1 ∗ 0 ∗ ∗ ∗ {"version":"1.1","math":"$$ H= begin{array}{ccccccc} ast & 1 & ast & 0 & ast & ast & ast end{array} $$"}Find the probability that the mutation operation destroys the schema, where the probability of random change of each symbol of the chromosome is p m = 0.2 {"version":"1.1","math":"( p_m=0.2)"}. Problem 7. (15 pts) One can check that x = [ 1 3 1 2 ] ⊤ {"version":"1.1","math":"( x=begin{bmatrix}1 & 3 & 1 & 2 end{bmatrix}^top )"} is a feasible non-basic solution to the system of equations A x = b {"version":"1.1","math":"( Ax=b)"}, where A = [ a 1 a 2 a 3 a 4 ] = [ − 1 1 1 0 − 2 1 0 1 ] {"version":"1.1","math":"[ A=begin{bmatrix} a_1 & a_2 & a_3 & a_4 end{bmatrix}=begin{bmatrix}-1 & 1 & 1 & 0\-2 & 1 & 0 & 1 end{bmatrix} ]"} and b = [ 3 3 ] . {"version":"1.1","math":"[ b=begin{bmatrix} 3\3end{bmatrix}. ]"}We also have 2 a 1 + 3 a 2 + ( − 1 ) a 3 + a 4 = 0 {"version":"1.1","math":"[ 2a_1+3a_2+(-1)a_3+a_4=0 ]"}and a 2 + ( − 1 ) a 3 + ( − 1 ) a 4 = 0. {"version":"1.1","math":"[ a_2+(-1)a_3+(-1)a_4=0. ]"}Use the above data and the method of the proof of the Fundamental Theorem of Linear Programming to find a basic feasible solution (BFS). Congratulations, you are almost done with Midterm Exam 2. DO NOT end the Honorlock session until you have submitted your work to Gradescope. 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