Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the definition of Frechet differentiation, show that ∇f(x) = x for all x ̸= 0. Furthermore, show that f(x) is not Frechet differentiable at x = 0.

V Company is expecting to have cash on hand on September 1 of $23,400. V Company is expecting to have sales of $92,750 in August and $76,500 in September. Of these amounts, 85 percent are expected to be cash sales and 15 percent are expected to be charge sales (i.e. credit sales). Based on company experience, the company expects 60 percent of credit sales to be collected in the month of sale, 38 percent of credit sales to be collected in the month following the month of sale, and 2 percent of credit sales not to be collected due to bad debts.

V Company is a retailer and also expects to purchase $62,000 of inventory in August and $59,000 of inventory in September, all on account. V Company usually pays 70 percent of charged purchases in the month of purchase and the other 30 percent of charged purchases in the month following purchase. Other costs for wages, utilities, rent, insurance, etc. are expected to be $26,200 per month.   

V Company normally likes to have at least a $20,000 cash balance on hand at all times.

REQUIRED: Prepare a Cash Budget for V Company for September only. Then indicate in a short paragraph after the budget whether V Company will have to seek financing to meet its desired minimum balance goal, and if so, provide a general figure of how much financing will be needed.

Please use Excel versus paper for answer.

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

a. How many amplifiers do you need to cascade to get at least a 1000 voltage gain with a load resistance of 1.0 kΩ?
b. What is the effective voltage gain Av obtained? (Respond with a whole number - rounded -)
c. Find the Open Circuit Voltage Gain for Cascading (Answer with a rounded integer)
d. If you have an input rms voltage of 1.5mV, what is the efficiency of the amplifier?
e. What is the transconductance of the circuit?

Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈ V, and nonnegative edge capacities {ce}. Give a polynomial-time algorithm to decide whether G has a unique minimum s-t cut (i.e., an s-t of capacity strictly less than that of all other s-t cuts).

Do you agree with the statement that “V” visualization is 6^th V for Big Data Analytics? If yes, provide examples to support your answer.

Proof:

Let S ⊆ V be a subset of a vector space V over F. We have that S is linearly dependent if and only if there exist vectors v1, v2, . . . , vn ∈ S such that vi is a linear combination of v1, v2, . . . , vi−1, vi+1, . . . , vn for some 1 ≤ i ≤ n.