Let Z denote the set of all integers. Give an explicit bijection f : Z → N

Can someone please explain to me the relationship between Stoke's theorem, Green's theorem, and the Gradient theorem. and give one example each for 0,1 and 2 forms. As a Calc 3 student, I am very confused about what this theorem is, I know it is essential to what linear algebra. Please help.

Use Warshall’s algorithm to find the transitive closure of the relation R={(1,1),(1,3),(1,4),(2,4),(2,5),(3,1),(4,4),(5,3)}

prove that positive operators have unique positive square root

use Simpson's 3/8th Rule on the first 3 segments, and multiple application of Simpson's 1/3rd Rule on the rest of the segments.

?(?)=400?⁵−900?⁴+675?³−200?²+25?+0.2

a = 0.12
b = 1.56
n = 7

(a) Seek power series solutions of the given differential equation about the given point x0;
find the recurrence relation.
(b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates
sooner).
(c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set
of solutions.
(d) If possible, find the general term in each solution.

1. y''-y=0, x0=0

2. y''-xy'-y=0, x0=0

3. (4-x^2)y''+2y=0, x0=0

4. 2y''+(x+1)y'+3y=0, x0=2

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).