List the key steps necessary to select a ribozyme with ligase activity through Systematic Evolution of Ligands by Exponential Enrichment (SELEX). Assume that you will complete multiple rounds of selection and that you want to diversify the pool each round. You add an RNA molecule that includes a sequence tag to the mixture of your potential ribozymes. If a ribozyme ligates the tag to its own 5' end, the ligating ribozyme can be purified out from the mix.

Sometimes the physical situation aids in the selection of substitution variables. For example, in an LRC circuit, the current is the derivative of the charge. In the LRC circuit governed by the equation $L Q^{\prime \prime}+R Q^{\prime}+\frac{1}{C} Q=E(t)$, show that the substitution I=Q' transforms the equation into the planar system

$$\begin{aligned} Q^{\prime} &=I, \\ I^{\prime} &=-\frac{R}{L} I-\frac{1}{L C} Q+\frac{1}{L} E(t) \end{aligned}$$

.

In each situation, explain whether the selection is made with replacement or without replacement.

a. The three digits in the lottery in the given example.

b. The three students selected to answer questions in the given case study.

c. The five people selected for extra security screening while boarding a particular flight.

d. The two football teams selected to play in the Rose Bowl in a given year.

e. The cars stopped by the police for speeding in five consecutive mornings on the same stretch of highway.

Calculate all first-order partial derivatives and the directional derivative $f^{\prime}(\mathbf{x} ; \mathbf{u})$ for each of the real-valued functions defined on $R^n$ as follows: (a) $f(\mathbf{x})=\mathbf{a} \cdot \mathbf{x}$, where a is a fixed vector in $R^n$. (b) $f(\mathbf{x})=\|\mathbf{x}\|^{4}$. (c) $f(\mathbf{x})=\mathbf{x} \cdot \mathbf{L}(\mathbf{x})$, where $\mathbf{L} : \mathbf{R}^{n} \rightarrow \mathbf{R}^{n}$ is a linear function. (d) $f(\mathbf{x})=\sum_{i=1}^{n} \sum_{j=1}^{n} a_{i j} x_{i} x_{j}$, where $a_{i j}=a_{j i}$.

Look at the following advice from the reading about walking in EL Yunque. After you read the selection, place a (checkmark) next to the sentence that is true.

Consejos para el caminante

• Nunca camine solo. Siempre vaya acompañado.
• Traiga agua y algo para comer.
• Use repelente para insectos.
• No abandone las veredas para no perderse (to get lost).
• No toque (touch) las plantas del bosque.

Never walk alone in the park._________

A regional infrastructure building and maintenance contractor must decide to buy a new compact horizontal directional drilling (HDD) machine now, or wait and buy it 2 years from now when a large pipeline contract will require the new equipment. The HDD machine will include an innovative pipeloader design and a maneuverable undercarriage system. The cost of the system is $68,000 if purchased now or an estimated$81,000 if purchased 2 years from now. At i = 10% per year and f = 5% per year, determine if the contractor should buy now or later (a) without any adjustment for inflation, and (b) with inflation considered.

The number of carbohydrates (in grams) per 8-ounce serving of yogurt for each of a random selection of brands is listed below. Calculate the true population variance and standard deviation for the number of carbohydrates per 8-ounce serving of yogurt with $95 \%$ confidence. Assume the variable is normally distributed.

$$\begin{array}{lllllllll} 17 & 42 & 41 & 20 & 39 & 41 & 35 & 15 & 43 \\ 25 & 38 & 33 & 42 & 23 & 17 & 25 & 34 & \end{array}$$

Implement the following Boolean function with a $4 \times 1$ multiplexer and external gates. (a) $F_{1}(A, B, C, D)=\Sigma(1,3,4,11,12,13,14,15)$ (b) $F_{2}(A, B, C, D)=\Sigma(1,2,5,7,8,10,11,13,15)$ Connect inputs A and B to the selection lines. The input requirements for the four data lines will be a function of variables C and D. These values are obtained by expressing F as a function of C and D for each of the four cases when AB = 00, 01,10, and 11. These functions may have to be implemented with external gates.

A campus club consists of five officers: president (P), vice president (V), secretary (S), treasurer (T), and activity coordinator (A) . The club can select two officers to travel to New Orleans for a conference; for fairness, they decide to make the selection at random. In essence, they are choosing a simple random sample of size n = 2.

a. What are the possible samples of two officers?

b. What is the chance that a particular sample of size 2 will be drawn?

c. What is the chance that the activity coordinator will be chosen?

Which of the following passages contributes most to the mood of this selection?

A "My great-grandfather on my mother's side ran a drugstore in a small town in central Illinois."

B "Some people drove all the way in from Decatur just to taste it."

C "As far as the family knew, the person who bought the recipe never used it."

D "The spin of my mother on the high stool with her whole life in front of her, something fine and fragrant still to happen."

A New York Times CBS News Poll typically involves the selection of random digits to be used for telephone numbers. The New York Times states that “within each (telephone) exchange, random digits were added to form a complete telephone number, thus permitting access to listed and unlisted numbers.” When such digits are randomly generated, what is the distribution of those digits? Given such randomly generated digits, what is a test for “goodness-of-fit”?

$\partial f / a n$ and $\partial g / \partial n$ denote directional derivatives of scalar fields f and g in the direction of the unit outer normal n to a closed surface S which bounds a solid V of the type described in the divergence theorem. That is, $\partial f / \partial \boldsymbol{n}=\mathbf{V} \mathbf{f} \cdot \boldsymbol{n}$ and $\partial g / \partial \boldsymbol{n}=\nabla g \cdot \boldsymbol{n}$. In each of these exercises prove the given statement. You may assume continuity of all derivatives involved. $\iint_{S} f \frac{\partial g}{\partial n} d S=\iiint_{v} f \nabla^{2} g d x d y d z + \iiint_{V} \nabla f \cdot \nabla g d x d y d z$.

Refer to another algorithm, called bubble sort, which sorts an n-element list. Bubble sort makes multiple passes through the list from front to back, each time exchanging pairs of entries that are out of order. Here is a pseudocode version: 1. Get values for n and the n list items. 2. Set the marker U for the unsorted section at the end of the list. 3. While the unsorted section has more than one element, do Steps 4 through 8. 4. Set the current element marker C at the second element of the list. 5. While C has not passed U, do Steps 6 and 7. 6. If the item at position C is less than the item to its left, then exchange these two items. 7. Move C to the right one position. 8. Move U left one position. 9. Stop. Suppose selection sort and bubble sort are both performed on a list that is already sorted. Does bubble sort do fewer exchanges than selection sort? Explain.

A researcher receives 100 containers of oxygen. Of those containers, 20 have oxygen that is not ionized, and the rest are ionized. Two samples are randomly selected, without replacement, from the lot. a. What is the probability that the first one selected is not ionized? b. What is the probability that the second one selected is not ionized given that the first one was ionized? c. What is the probability that both are ionized? d. How does the answer in part (b) change if samples selected were replaced prior to the next selection?