Due October 25. Let R denote the set of complex numbers of the form a + b √ 3i, with a, b ∈ Z. Define N : R → Z≥0, by N(a + b √ 3i) = a 2 + 3b 2 . Prove: (i) R is closed under addition and multiplication. Conclude R is a ring and also an integral domain. (ii) Prove N(xy) = N(x)N(y), for all x, y ∈ R. (ii) Prove that 1, −1 are the only units in R.

The reported net incomes for the first 2 years of Sandhill Co. were as follows: 2017, $169,000; 2018, $225,000. Early in 2019, the following errors were discovered. 1. Depreciation of equipment for 2017 was overstated by $7,900. 2. Depreciation of equipment for 2018 was understated by $15,400. 3. December 31, 2017 ending inventory was overstated by $20,300. 4. December 31, 2018 inventory was overstated by $28,300.

Prepare the correcting entry necessary when these errors are discovered. Assume that the books are closed.

A channel of distribution is the part of the supply chain that focuses on getting the product to the ultimate consumers. Over the past 10 years, retailers have seen a shift from brick and mortar to online buying of goods by consumers. As a result, more than 7000 retailers have closed up physical shop this year. Please are some advantages and disadvantage of online versus on ground shopping. If you are hired to slow and increase the number of on ground stores, what would be your strategy from a marketing and distribution strategy?

1.A 0.230 H inductor is wired in series with a LaTeX: 9.20\:\mu F9.20 μ F capacitor. The capacitor is fully charged when the circuit is closed at time t = 0. Calculate the rate of change in current through the inductor at t = 0. Assume that the initial charge (t = 0) on the capacitor is LaTeX: 5.30\mu C5.30 μ C

2.On the previous question at what time will the charge on the capacitor fall to half its initial value? State the first time that this occurs

Consider a permanent adverse productivity shock (for example, a permanent increase in the price of oil) which lowers the current and future productivity of capital and the current and expected future real incomes of households. Assuming a closed economy, first analyze the general equilibrium effects of this shock on output, employment, the real wage, the price level, and the real interest rate in both the classical and the Keynesian versions of macroeconomic adjustment. Second, explain how the adjustment effects would differ in the case of a temporary adverse productive shock.

Consider a hypothetical cell whose plasma membrane is impermeant to everything. The membrane is studded with channel proteins for Ca2+, but these channels are initially closed. The concentration of Ca2+ outside the cell is 100 mM and the concentration of Ca2+ inside the cell is 1 mM. Initally, the membrane potential is 0 mV.

After a while, the net flow of Ca2+ ceases because dynamic equilibrium has been reached. What is the membrane potential of the cell when it reaches dynamic equilibrium? Show how you arrived at this answer.

Assume that in a small open economy where full employment always prevails, national saving is 250.

a. If domestic investment is given by I = 650 – 22 r, where r is the real interest rate, what would the equilibrium "r" be if the economy were closed?   

b. If the economy is open and the world interest rate is 11 percent, what will investment be?   

c. Using your answer from part b., what is the current account surplus/deficit ?

d. What is the net capital flow?

For the following, assume air is an ideal gas with constant specific heats at 300K. Find the work and heat transfer and state whether they are into or out of the air.

In an open system, air is heated at constant pressure from 27°C, 400 kPa to 500K.

w = 0 kJ/kg, q = 201 kJ/kg

In a closed system, air is heated at constant pressure from 27°C, 100 kPa to 500K.

w = 57.4 kJ/kg, q = 201 kJ/kg