Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes. Prove that the height of the rooted tree (T, r) is at most twice the height of the rooted tree (T, r′).

Assume that G is connected and let
                   (G) = min{deg(v) ∈ Z ≥0 | v ∈ V (G)}
denote the lowest vertex degree that occurs in a graph G and let
                    λ(G) = min{|K| ∈ Z ≥0 | K ⊆ E(G) is a cutset of G}
denote the edge connectivity of G.
Prove that λ(G) ≤ δ(G)

Voltage amplifiers are available with Avoc = 8 V / V, Rin = 1.8 kΩ, Ro = 850 Ω. With a 12V DC power source, each amplifier draws 1.5 mA average current.

a) How many amplifiers do you need to cascade to get at least a voltage gain of 1000 with a load resistance of 1.0 kΩ?

b) What is the voltage gain Av obtained? (Respond with a rounded whole number)

c) For the cascade connection, find the voltage gain in open circuit. (Respond with a rounded whole number)

d) If you have a 1.5 mV input, how efficient is the equivalent amplifier?

e) Find the transconductance of the entire circuit.

Give the finite-state automaton and the regular grammar for the following:

(a) (ab v ba)* v (ab)*

(b) (11*)*(110 v 01)

(c) All strings over {0,1} containing the string 010

(d) All strings over {0,1} which do not contain the string 010

An incompressible fluid oscillates harmonically (V = V0 sin(wt), where V is the velocity) with a certain frequency in a 19 cm diameter pipe. A 1/4 scale model is to be used to determine the pressure difference per unit length, ∆?? at any instant along the pipe. Assume that ∆?? = ?(?, ?0,?,?,?,?)

Where D is the pipe diameter, ω the frequency, t the time, μ the fluid viscosity, and ρ the fluid density.

a) Determine the similarity requirements for the model and the prediction equation for ∆??.

b) If the same fluid is used in the model and the prototype, and the model frequency is 250 rad/s, what is the prototype frequency?

In the figure below, the battery has potential difference V = 9.0 V, C2 = 3.0

Put the terms in the appropriate blanks.

What is the difference between a 10% (v/v) methanol (CH₃OH) solution and a 10% (m/m) methanol solution?

A 10% (v/v) methanol solution can be made by adding _________ of methanol to _________ of water, while a 10% (m/m) methanol solution can be made by adding _________ of methanol _________ of water.

a. 10 mL

b. 90 mL

c. 90 g

d. 100 g

e. 100 mL

f. 10 g

How do temperature and pressure affect the solubility of solids and gases in water?

An increase in temperature usually __________ the solubility of solids and ________ the solubility of gases in water. An increase in pressure of a gas above a liquid _________ the solubility of a gas within that solution and _________ the solubility of solids.

a. increases

b. decreases

c. has no effect on

Describe the formation of an aqueous LiBr solution, when solid LiBr dissolves in water.

The _________ at the surface of the solid are pulled into solution by the _________ water molecules, where the ________ process surrounds

The output voltage of an AC generator is given by ?v = 121 V sin (32?t). The generator is connected across a 0.007 H inductor. Find the following.

(a) frequency of the generator?=____ Hz

(b) rms voltage across the inductor?=____ V

(c) inductive reactance?=____ ohms

(d) rms current in the inductor?=____ A

(e) maximum current in the inductor?=____ A

(f) average power delivered to the inductor?=____ W

(g) Find an expression for the instantaneous current. (Use the following as necessary: t.)

" i=_________ "

(h) At what time after t = 0 does the instantaneous current first reach 1.00 A? (Use the inverse sine function.)?=_____ s

provide correct answers for five star rating

Suppose two batteries, with unequal emfs of 2.00 V and 3.00 V, are connected as shown in (Figure 1) If each internal resistance is r = 0.400? , and R = 4.00 ?, what is the voltage across the resistor R?

A proton moves from a location where V =75 V to a spot whereV = -20 V.

(a) What is the change in the proton's kinetic energy?

(b) If we replace a proton with an electron, what is the change in kinetic energy?