Exercises 2.4.4 and 2.5.4 establish the equivalence of the Axiom of Completeness and the Monotone Convergence Theorem. They also show the Nested Interval Property is equivalent to these other two in the presence of the Archimedean Property.

(a) Assume the Bolzano-Weierstrass Theorem is true and use it to construct a proof of the Monotone Convergence Theorem without making any appeal to the Archimedean Property. This shows that BW, AoC, and MCT are all equivalent.

(b) Use the Cauchy Criterion to prove the Bolzano-Weierstrass Theorem, and find the point in the argument where the Archimedean Property is implicitly required. This establishes the final link in the equivalence of the five characterizations of completeness discussed at the end of Section 2.6.

(c) How do we know it is impossible to prove the Axiom of Completeness starting from the Archimedean Property?

Can you provide me the definition of "Mixing Problem" in Differential Equation? Also, an example question and its solution for "Mixing Problem". Thanks a lot.

Write a program to solve the boundary value problem ? ′′ = ? ′ + 2? + cos ? for ? ? [0, ?/2] with ?( 0) = 0.3, ?( ?/ 2) = 0.1. Check your numerical solution with actual using necessary plot.(MATLAB)

Question 13 (1 point)

Every invertible matrix is diagonalizable.

Question 13 options: True False

Question 14 (1 point)

Every diagonalizable matrix is invertible.

Question 14 options: True False

Let Q1=y(1.1), Q2=y(1.2), Q3=y(1.3), where y=y(x) solves...

1) y'''+2y''-5y'- 6y=4x^2 where y(0)=1, y'(0)=2, y''(0)=3

2) y'''- 6y''+11y'- 6y=6e^(4x) where y(0)=4, y'(0)=10, y''(0)=30

3) y''- 6y'+9y=4e^(3x) ln(x) where y(1)=, y'(1)=2

Please show all steps and thank you!!!

solve for matrix B

Let I be Identity matrix

(I-2B)^-1=

Discrete math

Summarize all the theorem regarding to graph theory.

e.g.) A connected graph has a Euler circuit iif every vertex is of even degree.

Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if and only if ad=bc. (a) Show that R is an equivalence relation. (b) What is the equivalence class of (1,2)? List out at least five elements of the equivalence class. (c) Give an interpretation of the equivalence classes for R. [Here, an interpretation is a description of the equivalence classes that is more meaningful than a mere repetition of the definition of R. Hint: Look at the ratio a/b corresponding to (a, b).

Can ?⃑⃑ = ( 11 −5 9 ) be written as a linear combination of the vectors ?⃑ 1⃑ = ( 1 0 1 ), ?⃑⃑2⃑ = ( −2 3 −2 ), and ?⃑⃑3⃑ = ( −6 7 5 )? Does your answer allow you to conclude whether or not ? = {?⃑ 1⃑, ?⃑⃑2⃑, ?⃑⃑3⃑} is a basis for ? 3 ?

Create a vector ?⃑ such that the set {( 1 0 0 ) , ( 1 3 2 ) , ?⃑} is a basis for ? 3 . Explain.

ANSWER BOTH PARTS WILL RATE

Prove the following statements by using the definition of convergence for sequences of
real numbers.

a) If {cn} is a sequence of real numbers and {cn} converges to 1 then {1/(cn+1)} converges to 1/2

b) If {an} and {bn} are sequences of real numbers and {an} converges A and {bn} converges to B and B is not equal to 0 then {an/bn} converges to A/B

*NUMBER THEORY*

1.Find all the possible solutions for the following diphantine equations by using the euclidian algorithim. You must show all the process to get credit.

a.           3x + 5y = 7

b.           3x − 12y = 7

c.           1990x − 173y = 11

d.           21x + 48y = 6

e.           2x + 3y + 5z = 11

Winkler Furniture manufactures two different types of china cabinets: a French provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue.

Formulate as an LP problem and obtain the revenue