Discuss the following problem. The laboratory you joined is studying the life cycle of an animal virus that uses circular, double-strand DNA as its genome. Your project is to define the location of the origin(s) of replication and to determine whether replication proceeds in one or both directions away from an origin (unidirectional or bidirectional replication). To accomplish your goal, you broke open cells infected with the virus, isolated replicating viral genomes, cleaved them with a restriction nuclease that cuts the genome at only one site to produce a linear molecule from the circle, and examined the resulting molecules in the electron microscope. Some of the molecules you observed are illustrated schematically in Figure Q5–1. (Note that it is impossible to distinguish the orientation of one DNA molecule relative to another in the electron microscope.) You must present your conclusions to the rest of the lab tomorrow. How will you answer the two questions your advisor posed for you? Is there a single, unique origin of replication or several origins? Is replication unidirectional or bidirectional?

Fix a monomial ordering on $R=F\left[x_{1}, \ldots, x_{n}\right] .$ (a) Prove that $\left\{g_{1}, \ldots, g_{m}\right\}$ is a minimal Gröbner basis for the ideal $I$ in $R$ if and only if $\left\{L T\left(g_{1}\right), \ldots, L T\left(g_{m}\right)\right\}$ is a minimal generating set for $L T(I) .$ (b) Prove that the leading terms of a minimal Gröbner basis for $I$ are uniquely determined and the number of elements in any two minimal Gröbner bases for $I$ is the same. [Use (a) and the previous exercise.]

Poppycrock, Inc., manufactures large crates of microwaveable popcorn that are typically sold to distributors. Its main factory has the capacity to manufacture and sell 35,000 crates per month. The following information is available for the factory:

Boys and Girls of Canada is a not-for-profit organization that raises funds each year by selling popcorn door-to-door. It offers to pay Poppycrock $\$ 22$ per crate for a special-order batch of 5,000 crates. The special-order popcorn would include a unique label with information about the Boys and Girls of Canada. The additional cost of the label is estimated at $\$ 1.00$ per crate. In addition, the variable overhead for these special-order crates would decrease by $\$ .50$ because there would be no distribution costs.

a. What is the incremental cost of creating a normal crate of popcorn? A special-order crate of popcorn?

Consider the system

$$\begin{array}{ll}{y_{j}^{\prime}=y_{J+1}} & {(j=1, \ldots, k-1)} \\ {y_{k}^{\prime}=f(x)-\sum_{j=1}^{k} g_{j}(x) y_{j}}\end{array}$$

where $f, g_{1}, \ldots, g_{k}$ are continuous real functions on $[a, b],$ and derive a uniqueness theorem for solutions of the equation

$$y^{(k)}+g_{k}(x) y^{(k-1)}+\cdots+g_{2}(x) y^{\prime}+g_{1}(x) y=f(x)$$

subject to initial conditions $y(a)=c_{1}, \quad y^{\prime}(a)=c_{2}, \quad \ldots, \quad y^{(k-1)}(a)=c_{k}$.

Label the following statements as true or false. Assume that all vector spaces are finite-dimensional. $\\$ (a) Every linear operator T has a polynomial$p(t)$of largest degree for which$p($\mathbf{T}$)=$\mathbf{T}$_{0}$.$$(b) Every linear operator has a unique minimal polynomial.$$(c) The characteristic polynomial of a linear operator divides the minimal polynomial of that operator.$$(d) The minimal and the characteristic polynomials of any diagonalizable operator are equal.$$(e) Let T be a linear operator on an$n$-dimensional vector space$V, p(t)$be the minimal polynomial of$T,$and$f(t)$be the characteristic polynomial of T. Suppose that$f(t)$splits. Then$f(t)$divides$$(f) The minimal polynomial of a linear operator always has the same degree as the characteristic polynomial of the operator.$$(g) A linear operator is diagonalizable if its minimal polynomial splits.$$(h) Let$T$be a linear operator on a vector space$V$such that$V$is a T-cyclic subspace of itself. Then the degree of the minimal polynomial of T equals dim(V).$$(i) Let$T$be a linear operator on a vector space$V$such that$T$has$n$distinct eigenvalues, where$n=$\operatorname{dim}$(V) .$Then the degree of the minimal polynomial of T equals$n .$

Consider an entity type SECTION in a UNIVERSITY database, which describes the section offerings of courses. The attributes of SECTION are Section_number, Semester, Year. Course_number, Instructor, Room_no (where section is taught), Building (where section is taught), Weekdays (domain is the possible combinations of weekdays in which a section can be offered {‘MWF’, ‘MW’, ‘TT’, and so on}), and Hours (domain is all possible time periods during which sections are offered {‘9–9:50 a.m.’, ‘10–10:50 a.m.’, …, ‘3:30–4:50 p.m.’, ‘5:30–6:20 p.m.’, and so on}). Assume that Section_number is unique for each course within a particular semester/year combination (that is, if a course is offered multiple times during a particular semester, its section offerings are numbered 1, 2, 3, and so on). There are several composite keys for section, and some attributes are components of more than one key. Identify three composite keys, and show how they can be represented in an ER schema diagram.

LA Self Storage wishes to construct a facility comprising storage units of four different sizes: $5 \mathrm{ft} \times 5 \mathrm{ft}, 5 \mathrm{ft} \times 10 \mathrm{ft}, 5 \mathrm{ft} \times 15 \mathrm{ft}$, and $10 \mathrm{ft} \times 10 \mathrm{ft}$. The company has $11,250 \mathrm{ft}^2$ of floor space available for the construction of $200$ units. Experience with other storage facilities and market research suggest that four times the number of $5-\mathrm{ft} \times 15-\mathrm{ft}$ units and nine times the number of $10-\mathrm{ft} \times 10-\mathrm{ft}$ units should exceed the number of $5-\mathrm{ft} \times 5 \mathrm{-ft}$ units and 5 -ft $\times 10$-ft units by exactly $300$ units.

a. How many units of each size can the company construct if all the available floor space is utilized?

b. Give two possible solutions. (Note: Your answers will not be unique.)

Each part of the exercise is an implication of the form If "statement 1" then "statement 2" (all matrices are square $n\times n$). Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer.

Part 1:

a) If the equation $Ax = 0$ has only the trivial solution, then $A$ is row equivalent to the $n \times n$ identity matrix.

b) If the columns of $A$ span $\mathbb{R}^n$, then the columns are linearly independent.

c) If $A$ is an $n \times n$ matrix, then the equation $Ax = b$ has at least one solution for each $b$ in $\mathbb{R}^n$

d) If the equation $Ax = 0$ has a nontrivial solution, then $A$ has fewer than $n$ pivot positions.

e) If $A^T$ is not invertible, then $A$ is not invertible.

Part 2:

a) If there is an $n \times n$ matrix $D$ such that $AD = I$ then there is also an $n \times n$ matrix $C$ such that $CA = I$.

b) If the columns of $A$ are linearly independent, then the columns of $A$ span $\mathbb{R}^n$.

c) If the equation $Ax = b$ has at least one solution for each $b$ in $\mathbb{R}^n$, then the solution is unique for each $b$.

Choose the best way to write each underlined section and mark the letter of your answer. If the underlined section needs no change, mark the choice "Correct as is."

$\overset{(1)}{\text{\underline{{Fingerprints have provided a practical and effective method}}}}$ of identification for over a hundred years. $\overset{(2)}{\text{\underline{{Each person's pattern of loops and swirls are unique}}}}$. $\overset{(3)}{\text{\underline{{No one, not even identical twins, have exactly the same pattern}}}}$ as anyone else. $\overset{(4)}{\text{\underline{{Juan Vucetich and Edward R. Henry was the first to develop efficient methods}}}}$ of classifying fingerprints. Today, the police compare a suspect's fingerprints with thousands of prints stored in computer data banks. A computer can find a match in only a few minutes. $\overset{(5)}{\text{\underline{{From this work comes the facts that prosecutors need for their cases.}}}}$

A. Fingerprints has provided a practical and effective method

B. Fingerprints has been providing a practical and effective method

C. Fingerprints was providing a practical and effective method

D. Correct as is

Superpose a uniform stream of velocity $V_{\infty}$ and a line source of strength $\dot{V} / L$ at the origin. This generates potential flow over a two-dimensional half-body called the Rankine half-body. One unique streamline is the dividing streamline that forms a dividing line between free-stream fluid coming from the left and fluid coming from the source. (a) Obtain an equation for the dividing stream function $\psi_{\text {dividing }}$ as a function of $\dot{V} / L$. (Hint: The dividing streamline intersects the stagnation point at the nose of the body.) (b) Obtain an expression for half-height $b$ as a function of $V_{\infty}$ and $\dot{V} / L$. (Hint: Consider the flow far downstream.) $(c)$ Obtain an equation for the dividing stream function in the form of $r$ as a function of $\theta, V_{\infty}$, and $\dot{V} / L$. (d) Obtain an expression for stagnation point distance $a$ as a function of $V_{\infty}$ and $\dot{V} / L$ (e) Obtain an expression for $\left(V / V_{\infty}\right)^2$ (the squared nondimensional velocity magnitude) anywhere in the flow as a function of $a_f{ }^{\prime}$ canid $\theta_a$

With the help of a suitable explanation state whether the following statements are true or false.

(a) If the rank of $A$ is $n$, then $A$ cannot have more than $n$ rows.

(b) If matrices $A$ and $B$ are of the same size and $k$ is a scalar, then $\operatorname{rank}(A)+\operatorname{rank}(B)=\operatorname{rank}(A+B)$ and $\operatorname{rank}(k A)=k \operatorname{rank}(A)$

(c) Consider a system of $n$ equations in $n$ variables. If the matrix of coefficients has rank $n$, the solution is unique.

(d) If the vectors $\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n$ are linearly dependent, then the system of linear equations $\left[\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\right] \mathbf{x}=\mathbf{b}$ has many solutions.

(e) If the system of linear equations $\left[\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\right] \mathbf{x}=\mathbf{b}$ has no solution, then $\operatorname{rank}\left(\left[\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\right]\right)<\operatorname{rank}$ $\left(\left[\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n \mathbf{b}\right]\right)$.

Let $R$ and $S$ be rings with 1 and let $M$ and $N$ be left $R$ -modules. Assume also that $M$ is an $(R, S)$ -bimodule. (a) For $s \in S$ and for $\varphi \in \operatorname{Hom}_{R}(M, N)$ define $(s \varphi) : M \rightarrow N$ by $(s \varphi)(m)=\varphi(m s)$ Prove that $s \varphi$ is a homomorphism of left $R$ -modules, and that this action of $S$ on Hom $_{R}(M, N)$ makes it into a left $S$ -module. (b) Let $S=R$ and let $M=R$ (considered as an $(R, R)$ -bimodule by left and right ring multiplication on itself). For each $n \in N$ define $\varphi_{n} : R \rightarrow N$ by $\varphi_{n}(r)=r n$ i.e., $\varphi_{n}$ is the unique $R$ -module homomorphism mapping $1_{R}$ to $n .$ Show that $\varphi_{n} \in$ Hom $_{R}(R, N) .$ Use part (a) to show that the map $n \mapsto \varphi_{n}$ is an isomorphism of left $R$ -modules: $N \cong \operatorname{Hom}_{R}(R, N) .$ (c) Deduce that if $N$ is a free (respectively, projective, injective, flat) left $R$ -module, then Hom $_{R}(R, N)$ is also a free (respectively, projective, in jective, flat) left $R$ -module.

One of the unique proteins found in the meiotic cohesion complex is Rec 8. This protein, expressed only during meiosis, is not cleaved during meiosis I but is cleaved during meiosis II to finally allow sister chromatids to segregate. Scientists hypothesized that a protein protects the Rec 8 protein from cleavage and degradation during meiosis I. To identify such a protein, researchers first produced the Rec 8 protein in mitotically dividing yeast cells. In these cells, Rec 8 was cleaved during mitosis and the cells suffered no harmful effects. To find a protein that protects Rec 8 from cleavage, researchers then expressed other proteins in the cell expressing Rec 8 mitotically and were able to identify Shugoshin that protects Rec 8 from degradation. What effect do you think expressing Shugoshin had on the mitotically dividing cells expressing Rec 8? What phenotype would the cells show?

Trader Joe's is a privately held chain of specialty grocery stores in the United States. Starting out as a small chain of convenience stores, it has expanded to over 340 stores as of June 2010 (www.traderjoes.com). It has developed a reputation as a unique grocery store selling products such as gourmet foods, beer and wine, bread, nuts, cereal, and coffee. One of their best-selling nuts is Raw California Almonds, which are priced at $\$ 4.49$ for 16 ounces. Since it is impossible to pack exactly 16 ounces in each packet, a researcher has determined that the weight of almonds in each packet is normally distributed with a mean and a standard deviation equal to $16.01$ ounces and $0.08$ ounces, respectively.

a. Discuss the sampling distribution of the sample mean based on any given sample size.

b. Find the probability that a random sample of 20 bags of almonds will average less than 16 ounces.

c. Suppose your cereal recipe calls for no less than 48 ounces of almonds. What is the probability that three packets of almonds will meet your requirement?