In an effort to determine the location of an operator site for a negatively regulated gene, you have made a series of deletions within the regulatory region. The extent of each deletion is shown by the line underneath the sequence, and the resulting expression from the operon (i = inducible; c = constitutive; - = no expression is also indicated).

. . . G G A T C T T A G C C G G C T A A C A T G A T A A A T A T A A . .

. . C C T A G A A T C G G C C G A T T G T A C T A T T T A T A T T . .

a. What can you conclude from these data about the location of the operator site?

b. Why do you think deletions 2 and 4 show no expression?

a. How many ribosomes are required (at a minimum) for the translation of $\operatorname{trp} E$ and $\operatorname{trp} C$ from a single transcript of the $\operatorname{trp}$ operon?

b. How would you expect deletion of the two tryptophan codons in the RNA leader to affect the expression of the $\operatorname{trp} E$ and $\operatorname{trp} C$ genes?

Translate each part into a Python statement using appropriate string methods:

(a) Assign to variable message the string 'The secret of this message is that it is secret.'

(b) Assign to variable length the length of string message, using operator len().

(c) Assign to variable count the number of times the substring 'secret' appears in string message, using string method count().

(d) Assign to variable censored a copy of string message with every occurrence of substring 'secret' replaced by 'xxxxxx', using string method replace().

Write a container class called PriorityQueue. The class should support methods:

insert(): Takes a number as input and adds it to the container

min(): Returns the smallest number in the container

removeMin(): Removes the smallest number in the container

isEmpty(): Returns True if container is empty, False otherwise

The overloaded operator len() should also be supported.

>>> pq = PriorityQueue()
>>> pq.insert(3)
>>> pq.insert(1)
>>> pq.insert(5)
>>> pq.insert(2)
>>> pq.min()
1
>>> pq.removeMin()
>>> pq.min()
2
>>> len(pq)
3
>>> pq.isEmpty()
False

Lou Goble, an internal auditor for a large manufacturing enterprise, received an anonymous note from an assembly-line operator who has worked at the company’s West Coast factory for the past 15 years. The note indicated that there are some fictitious employees on the payroll as well as some employees who have left the company. He offers no proof or names. What CAAT could Lou use to substantiate or refute the employee’s claims? (CIA Examination, adapted)

Let T be the linear operator on $R^{2}$, the matrix of which in the standard ordered basis is

$$\left[\begin{array}{ll}{2} & {1} \\ {0} & {2}\end{array}\right]$$

. Let WI be the subspace of $R^{2}$ spanned by the vector $\epsilon_{1}=(1,0)$. (a) Prove that $W_{1}$ is invariant under T. (b) Prove that there is no subspace $W_{2}$ which is invariant under T and which is complementary to $W_{1}$: $R^{2}=W_{1} \oplus W_{2}$.

Show that $R=l x-\left(1-x^{2}\right) D$ and $L=l x+\left(1-x^{2}\right) D$, where $D = d/dx$, are raising and lowering operators for Legendre polynomials. More precisely, show that $R P_{l-1}(x)=l P_{l}(x)$ and $L P_{l}(x)=l P_{l-1}(x)$. Note that, unlike the raising and lowering operators for Hermite functions, here R and L depend on l as well as x, so you must be careful about indices. The L operator operates on $P_l$, but the R operator as given operates on $P_{l−1}$ to produce $l P_{l}$. [If you prefer, you could replace l by l +1 to rewrite R as $(l+1) x-\left(1-x^{2}\right) D$; then it operates on $P_l$ to produce $(l+1) P_{l+1}$.] Assuming that all $P_{l}(1)=1$, solve $L P_{0}(x)=0$ to find $P_{0}(x)=1$, and then use raising operators to find $P_{1}(x)$ and $P_{2}(x)$.

Find the derivative. $f^{\prime}(3) \quad$ if $\quad f(x)=\frac{x^{3}}{9}-7 x^{2}$

Let T be a linear operator on a finite-dimensional vector space with minimal polynomial f(t)=\left(\phi_{1}(t)\right)^{m_{1}}\left(\phi_{2}(t)\right)^{m_{2}} \cdots\left(\phi_{k}(t)\right)^{m_{k}} where the $\phi_{i}(t)$ 's are distinct irreducible monic factors of $f(t) .$ Prove that for each $i, m_{i}$ is the number of entries in the first column of the dot diagram for $\phi_{i}(t) .$

The following is a sequence of the leader region of the his operon mRNA in Salmonella typhimurium. What bases in this sequence could cause a ribosome to pause when histidine is limiting (that is, when there is very little of it) in the medium?

5' AUGACACGCGUUCAAUUUAAACACCACCAUCAUCACCAUCACCUGACUAGUCUUUCAGGC $3^{\prime}$

A linear operator $T$ on $R^n$ and its characteristic polynomial are given. Find, if possible, a basis for $R^n$ consisting of eigenvectors of $T$. If no such basis exists, explain why.

$$\begin{equation*}T\begin{pmatrix}\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}\end{pmatrix}= \begin{bmatrix}3x_1-5x_3\\3x_2-5x_3\\-2x_3\\5x_3+3x_4\end{bmatrix}, (t+2)(t-3)^3\end{equation*}$$

E & T Excavation Company is planning an investment of $\$ 245,000$ for a bulldozer. The bulldozer is expected to operate for $1,500$ hours per year for five years. Customers will be charged $\$ 130$ per hour for bulldozer work. The bulldozer operator costs $\$ 32$ per hour in wages and benefits. The bulldozer is expected to require annual maintenance costing $\$ 15,000$. The bulldozer uses fuel that is expected to cost $\$ 42$ per hour of bulldozer operation.

Should E & T invest in the bulldozer, based on this analysis?

Work with a partner to create a role-play about purchasing for business. Consider presenting topics such as employee their, customer shoplifting disagreements about EOM/BOM totals, or whether to use centralized or decentralized buying. Submit a script for your role-play.

A firm is accustomed to training operators who do certain tasks on a production line. Those operators who attend the training course are known to be able to meet their production quotas 90% of the time. New operators who do not take the training course only meet their quotas 65% of the time. Fifty percent of new operators attend the course. Given that a new operator meets her production quota, what is the probability that she attended the program?