Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove that if (X, dX ) is compact, then f is a homeomorphism. (Hint: it might be convenient to use that a function is continuous if and only if the inverse image of every open set is open, if and only if the inverse image of every closed set is closed).

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V a subspace of R3?

Find the modulus of:

(a) (3 – j4) (-5 + j12)

(b) (2+?)/(4?+(1+?)^2)

Express (6 – ?8)^−3 in the standard form ? + ??. Find its conjugate.

If ? + ?? = (?+?)/ (?−?) ,where a, b and c are real, prove that ?^2 + ?^2 = 1 and ?/?= (2?)/ (?^2−1)

Please show all your steps so I can understand

Thank you

Present two cases of the Cardano’s formula

The legislature in a state has 59 seats. Apportion these seats to the five counties below using Jefferson's method. County Population Seats received

Adams 131,000

Grant 350,000

Colton 300,000

Davis 347,000

Hayes 236,000

6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic
to the subfield of C consisting of complex numbers with rational real and
imaginary parts.

A tank contains 60 kg of salt and 1000 L of water. Pure water enters a tank at the rate 12 L/min. The solution is mixed and drains from the tank at the rate 6 L/min.

(a) What is the amount of salt in the tank initially?
amount =  (kg)

(b) Find the amount of salt in the tank after 3 hours.
amount =   (kg)

(c) Find the concentration of salt in the solution in the tank as time approaches infinity. (Assume your tank is large enough to hold all the solution.)
concentration =   (kg/L)

Let R be a commutative ring with identity with the property that every ideal in R is principal. Prove that every homomorphic image of R has the same property.

An unknown radioactive element decays into non-radioactive substances. In 340 days the radioactivity of a sample decreases by 37 percent. (a) What is the half-life of the element? (b) How long will it take for a sample of 100 mg to decay to 74 mg? time needed: (days)

Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 205 degrees Fahrenheit when freshly poured, and 1 minutes later has cooled to 190 degrees in a room at 64 degrees, determine when the coffee reaches a temperature of 150 degrees.
The coffee will reach a temperature of 150 degrees in how many minutes?

A round-robin tournament involving n plays is modeled with digraph D where, for every two distinct vertices (players) u and v, either (u,v) is an edge (player u defeats player v) or (v,u) is an edge (player v defeats play u). Prove that if D is acyclic, i.e., no directed cycles, then there always exists a player who has defeated everyone (out-degree is n – 1) and a player who has lost to everyone (in-degree is n – 1).

Use the dual simplex method to solve the following linear programming problems. Clearly indicate all the steps, the entering and departing rows and columns and rows, the pivot and the row operations used. Use the simplex method to solve the following linear programming problems. Clearly indicate all the steps, the entering and departing rows and columns and rows, the pivot and the row operations used. 2.2.1 An electronics manufacturing company has three production plants, each of which produces three different models of a particular MP3 player. The daily capacities (in thousands of units) of the three plants are shown in the table. Basic model Gold model Platinum model Plant 1 8 4 8 Plant 2 6 6 3 Plant 3 12 4 8 The total demands are 300,000 units of the Basic model, 172,000 units of the Gold model, and 249,500 units of the Platinum model. The daily operating costs are $55,000 for plant 1, $60,000 for plant 2, and $60,000 for plant 3. How many days should each plant be operated in order to fill the total demand while keeping the operating cost at a minimum? What is the minimum cost? Use the method of the Dual.

Jack has two children. What is the probability that both are boys. In addition, what is the probability the oldest is a boy, at least one is a boy, at least oen boy is born on a monday.

Thanks for the help!