the distance east or west from the prime meridian

Find the distance between the points. $(5,2),(-1,4)$

Let $\varphi$ : G $\rightarrow$ GL(V) be a representation of G. The centralizer of $\varphi$ is defined to be the set of all linear transformations, A, from V to itself such that

$$A \varphi(g)=\varphi(g) A$$

for all g $\in$ G (i.e., the linear transformations of V which commute with all $\varphi(g)^{\prime}s$). (a) Prove that a linear transformation A from V to V is in the centralizer of $\varphi$ if and only if it is an FG-module homomorphism from V to itself (so the centralizer of $\varphi$ is the same as the $\left.\operatorname{ring} \operatorname{Hom}_{F G}(V, V)\right)$. (b) Show that if z is in the center of G then $\varphi(z)$ is in the centralizer of $\varphi$. (c) Assume $\varphi$ is an irreducible representation (so V is a simple FG-module). Prove that if H is any finite abelian subgroup of GL(V) such that A$\varphi$(g) = $\varphi$(g)A for all A $\in$ H then H is cyclic (in other words, any finite abelian subgroup of the multiplicative group of units in the ring Hom$_{F G}$(V, V) is cyclic). Hom$_{F G}$(V, V) is a division ring, so this reduces to proving that a finite abelian subgroup of the multiplicative group of nonzero elements in a division ring is cyclic. Show that the division subring generated by an abelian subgroup of any division ring is a field and use Proposition 18. (d) Show that if $\varphi$ is a faithful irreducible representation then the center of G is cyclic. (e) Deduce from (d) that if G is abelian and $\varphi$ is any irreducible representation then G/ker $\varphi$ is cyclic.

Write a program that prompts the user to enter the radius and draws four circles in the center of the screen.

Find the distance from $(-1,7)$ to the line $x=3$.

Write the correct word in the space next to each definition.

random $\rule{2cm}{0.15mm}$

Phoenix Industries has pulled off a miraculous recovery. Four years ago it was near bankruptcy. Today, it announced a $1 per share dividend to be paid a year from now, the first dividend since the crisis. Analysts expect dividends to increase by$1 a year for another 2 years. After the third year (in which dividends to increase by $1 a year for another 2 years. After the third year (in which dividends are$3 per share) dividend growth is expected to settle down to a more moderate long-term growth rate of 6%. If the firm's investors expect to earn a return of 14% on this stock, what must be its price?

What different perspectives of Progressive Era environmentalism do these two historians emphasize?

Dr. Detail is a mobile car wash. The price for a standard wash is $35 and takes half an hour. Each worker is paid$20/hr, uses $5 of materials and$0.50 per mile to use their own vehicle to travel from job to job. The average job is 5 miles.

Ernest Kuhn’s son got sick in the car, and Ernest Kuhn has asked Dr. Detail to detail his car instead of doing a simple wash and vacuum.

D. What other issues would you need to consider?

Dr. Detail is a mobile car wash. The price for a standard wash is $35 and takes half an hour. Each worker is paid$20/hr, uses $5 of materials and$0.50 per mile to use their own vehicle to travel from job to job. The average job is 5 miles.

Ernest Kuhn’s son got sick in the car, and Ernest Kuhn has asked Dr. Detail to detail his car instead of doing a simple wash and vacuum.

B. If the price is raised to $150, what is the differential income change?

Set U: integers from 1 to 20, Set P: prime numbers, Set T: multiples of 3, Set F: multiples of 5. Create a Venn diagram of these sets.

Write each set in roster form and in set-builder notation. P is the set of natural numbers that are factors of 25.

Differentiate the functions in problem with respect to x. $y=\frac{\sin x}{e^{x}-x}$

Scientists think that the global population of tigers is falling exponentially. Estimates suggest that in 1970 there were 37 000 tigers but by 1980 the number had dropped to 22 000. (a) The number T of tigers n years after 1970 can be modelled by

$$T = k a ^ { n }$$

. (i) Write down the value of k. (ii) Show that a = 0.949 to three significant figures. (b) What does the model predict that the population will be in 2020? (c) When the population reaches 1000, the tiger population will be described as ‘near extinction’. In which year will this happen? In the year 2000 a worldwide ban on the sale of tiger products was implemented, and it is believed that by 2010 the population of tigers had recovered to 10 000. (d) If the recovery has been exponential, find a model of the form

$$T = k a ^ { m }$$

connecting the number of tigers (T) with the number of years after 2000 (m). (e) If each year since 2000 the rate of growth has been the same, find the percentage increase in the tiger population each year.