Prove that $G^{i}$ is a characteristic subgroup of G for all i.

Let W be the set of all vectors of the form shown, where a, b and c represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is not a vector space.

$$\left[\begin{array}{c} 1 \\ 3 a-5 b \\ 3 b+2 a \end{array}\right]$$

Markov chains have been used to study music. The relative frequency with which one note is followed by another in a particular song can be recorded in a transition matrix, which can then be used to generate random sequences of notes with the same frequency pattern. If the rhythms are also incorporated, new songs that are in some way similar to the original song can be generated. (a) Consider the simple melody "Merrily We Roll Along," with the following sequence of notes in the key of G:

$$\begin{array}{l} BAGABBB \\ AAABBB \\ BAGABBB \\ AABAG \end{array}$$

Suppose this song is played in a loop, so that the final G is followed by the initial B. Construct a transition matrix for this melody. For example, of the 12 occurrences of B, 5 are followed by A, so the entry in row B, column A, is 5/12. (b) Determine an equilibrium vector for the matrix in part (a). Then explain why, in retrospect, this result is obvious by looking at the frequency of notes in the original melody.

Let

$$G=A_{1} \times A_{2} \times \cdots \times A_{n}$$

and for each i let $B_{i}$ be a normal subgroup of $A_{i}$. Prove that

$$B_{1} \times B_{2} \times \cdots \times B_{n} \unlhd G$$

and that

$$\left(A_{1} \times A_{2} \times \cdots \times A_{n}\right) /\left(B_{1} \times B_{2} \times \cdots \times B_{n}\right) \cong\left(A_{1} / B_{1}\right) \times\left(A_{2} / B_{2}\right) \times \cdots \times\left(A_{n} / B_{n}\right)$$

Identify the two events described in the study. Do the results indicate that the events are independent or dependent? Explain your reasoning. A study found that people who suffer from obstructive sleep apnea are at increased risk of having heart disease.

A micrometer is 0.000001 meter. Use your calculator to determine how many micrometers are in each of the following. Write your answer in both calculator and scientific notation. 5,000 meters

Determine whether the data set is a population or a sample. Explain your reasoning. The annual revenue of each store in a shopping mall

An access code consists of three digits. Each digit can be any number from 0 through 9, and each digit can be repeated.

What is the probability of randomly selecting the correct access code on the first try?

Factor each expression. If the expression cannot be factored, write cannot be factored. Use algebra tiles if needed. 3x + 9

Data have been collected on the likelihood that a teacher, or a student with a declared interest in teaching, will continue on that career path the following year. We have simplified the classification of the original data to four groups: high school and college students, new teachers, continuing teachers, and those who have quit the profession. The transition probabilities are given in the following matrix.

$$\begin{matrix} \text{ } & \text{Student} & \text{New} & \text{Continuing} & \text{Quit}\\ \text{Student} & \text{0.70} & \text{0.11} & \text{0} & \text{0.19}\\ \text{New} & \text{0} & \text{0} & \text{0.86} & \text{0.14}\\ \text{Continuing} & \text{0} & \text{0} & \text{0.88} & \text{0.12}\\ \text{Quit} & \text{0} & \text{0} & \text{0} & \text{1}\\ \end{matrix}$$

(a) Find the expected number of years that a student with an interest in teaching will spend as a continuing teacher. (b) Find the expected number of years that a new teacher will spend as a continuing teacher. (c) Find the expected number of additional years that a continuing teacher will spend as a continuing teacher. (d) Notice that the answer to part (b) is larger than the answer to part (a), and the answer to part (c) is even larger. Explain why this is to be expected. (e) What other states might be added to this model to make it more realistic? Discuss how this would affect the transition matrix.

What is the relationship between the intermolecular forces in a solid and its melting temperature?

Write a program that outputs the conjunctive normal form of a Boolean expression p(x, y, z).

Find the given inverse Laplace transform by finding the Laplace transform of the indicated function. $\mathscr{L}^{-1}\left\{\frac{1}{s^{2}\left(s^{2}+a^{2}\right)}\right\} ; f(t)=a t-\sin a t$

Solve the equation first by completing the square and then by factoring. x$^{2}$ - 10x + 21 = 0