Write a program that prompts the user to enter the center of a rectangle, width, and height, and displays the rectangle.

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.

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

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

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

If H is a subgroup of G and A is an abelian group let $M_{G / H}(A)$ denote the abelian group of all maps from the left cosets gH of H in G to A. (a) Prove that

$$M_{1}^{G}(A) \cong M_{1}^{H}\left(M_{G / H}(A)\right)$$

as H-modules. [If $\left\{g_{i}\right\}_{i \in \mathcal{I}}$ is a choice of left coset representatives of H in G define the correspondence between $f \in M_{1}^{G}(A)$ and

$$F: H \rightarrow M_{G / H}(A)$$

by

$$F(h)\left(g_{i} H\right)=f\left(g_{i} h\right),$$

and check that this is an isomorphism of H-modules. (b) A G-module A such that $H^{n}(H, A)=0$ for all n $\geq$ 1 and all subgroups H of G is called cohomologically trivial. Prove that $M_{1}^{G}(A)$ is a cohomologically trivial for any abelian group A. (c) If G is finite, prove that $\mathbb{Z} G \otimes_{\mathbb{Z}} A$ is cohomologically trivial for all abelian groups A.

In 1975, an airplane carrying an atomic clock flew back and forth for $15 \mathrm{~h}$ at an average speed of $140 \mathrm{~m} / \mathrm{s}$ as part of a time-dilation experiment. The time on the clock was compared to the time on an atomic clock kept on the ground. How much time did the airborne clock "lose" with respect to the clock on the ground?

the distance east or west from the prime meridian

Write a program that prompts the user to enter the center x-, y-coordinates, width, and height of two rectangles and determines whether the second rectangle is inside the first or overlaps with the first.

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?

The Tethered Satellite discussed in this module is producing 5.00 kV, and a current of 10.0 A flows. (a) What magnetic drag force does this produce if the system is moving at 7.80 km/s? (b) How much kinetic energy is removed from the system in 1.00 h, neglecting any change in altitude or velocity during that time? (c) What is the change in velocity if the mass of the system is 100,000 kg? (d) Discuss the long term consequences (say, a week-long mission) on the space shuttle's orbit, noting what effect a decrease in velocity has and assessing the magnitude of the effect.

In a pulse power circuit, the voltage of a 10$\mu$F capacitor is zero for t<0 and $v=5(1-e^{-4000t})$ V for t>0. Determine the capacitor current and the energy stored in the capacitor at t=0ms and t=10ms.

Hot exhaust gases leaving a stationary diesel engine at $450^{\circ} \mathrm{C}$ enter a $15-\mathrm{cm}$-diameter pipe at an average velocity of $7.2 \mathrm{~m} / \mathrm{s}$. The surface temperature of the pipe is $180^{\circ} \mathrm{C}$. Find the pipe length if the exhaust gases are to exit the pipe at $250^{\circ} \mathrm{C}$ after transferring heat to water in a heat recovery unit. Employ properties of air for exhaust gases.

The distance-time graph for Rolf intersects the distance-time graph for Annette at the point (2.5, 7). (Each graph gives the distance in mile s and the time in hours.) What does the common point (2.5, 7) tell you?

Enzymes that are only produced when a substrate is present are termed _____.