how to derive wave equation?
(not second order, but first order)

An Introduction of the Theory of Groups - Fourth Edition (Joseph J. Rotman)

If H ≤ G, then show that G acts transitively on the set of all left cosets of H (Theorem 3.14) and G acts transitively on the set of all conjugates of H (Theorem 3.17).

Theorem 3.14 - f H ≤ G and [G: H] = n, then there is a homomorphism ρ: G -> Sn with ker ρ ≤ H.

Theorem 3.17 - Let H ≤ G and let X be the family of all the conjugates of H in G. There is a homomorphism ψ: G -> S_x with ker ψ ≤ N_G(H).

at t=0,a tank contains Q0 g of the salt dissolved in 100L of water, assume that water containing 1/4g of salt per L in entering the tank at a rate of r L/min. at the same time, the well-mixed mixture is draining from the tank at the same rate.

a) set up the differential equation and initial condition for the salt amount Q as a function of time.

b)find the amount of salt in the tank as a function of time t, Q(t), by solving the differential equation in the above problem.

c) when t->infinit , meaning after a long time, what is the limt amount Qlimit?

Give an example of an augmented matrix in echelon form corresponding to a system of 2 equations in three unknowns satisfying each of the following conditions or explain why it is not possible.

(a) No solutions

(b) One unique solution

(c) Infinitely many solutions.

Statistically analyzing the difference between two or more means is important for Psychological research

Statistical Analysis Assignment # 1

Upload the Excel Data Analysis Program onto your computer. For the data set below, please calculate:

• the mean and standard deviation for the number of hours studied and
• the mean and the standard deviation for test score.
• the correlation between the number of hours studied and the obtained test score.

What conclusion can you draw from these results?

Test Score          Hours Studied

1. 100                4

2.   90                4

3.   20                1

4.   52                2

5.   83                3

6.   76                2

7.   55                1

8.   98                4

9.   89                4

10. 70               2

I study mathematics-economics in the second year of a bachelor programme: I’m one month into the course “Analysis I”. I would like to get more familiar with certain subjects. For a reasonable answer for this Q&A, I’d like a formal definition, a simple example/proof and summary of the mentioned subject.

The subject is Fourier series.

I study mathematics-economics in the second year of a bachelor programme: I’m one month into the course “Analysis I”. I would like to get more familiar with certain subjects. For a reasonable answer for this Q&A, I’d like a formal definition, a simple example/proof and summary of the mentioned subject.

The subject is differentiability of one real function of one variable.

Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y be a continuous onto map with the property that f^-1[{y}] is connected for every y∈Y. Show that ifY is connected then so isX.

For this question, a block is a sequence of 20 characters, where each character is one of the 26 lowercase letters a-z. For example, these are blocks:

iwpiybhunrplsovrowyt
rpulxfsqrixjhrtjmcrr
fxfpwdhwgxtdaqtmxmlf

1. How many different blocks are there?
2. A block is squarefree if no character appears two times consecutively. The first and third example above are squarefree, but the second example is not because of the two consecutive occurrences of r. How many squarefree blocks are there?
3. A block is non-local if the number of characters between any two occurrences of the same character is at least 2. The first example above is non-local. The second example is not, because there are two occurrences of r with no characters between them. The third example is not because there are two occurrences of m with only one character between them. How many non-local blocks are there?
4. A block is k-non-local if the number of characters between any two occurrences of the same character is at least k. Write a formula for the number of k-non-local blocks that is valid for any k∈{0,…,20}.

Sanity check: The formula you get for 2.4 gives the answer to 2.2 when k=1 and gives the answer to 2.3 when k=2.

Let X be the space of all continuous functions from [0, 1] to [0, 1] equipped with the sup metric. Let Xi be the set of injective and Xs be the set of surjective elements of A and let Xis = Xi ∩ Xs. Prove or disprove: i) Xi is closed, ii) Xs is closed, iii) Xis is closed, iv) X is connected, v) X is compact.

Let φ : R −→ R be a continuous function and X a subset of R with closure X' such that φ(x) = 1 for any x ∈ X. Prove that φ(x) = 1 for all x ∈ X.'  

Show that for any k ≥ 2, if n + 1 distinct integers are chosen from the set [kn] = {1, 2, . . . , kn}, then there will be two integers which differ by at most k − 1. Please demonstrate the steps so that I can learn from it and solve other problems by following the reasoning!

Consider a damped forced mass-spring system with m = 1, γ = 2, and k = 26, under the influence of an external force F(t) = 82 cos(4t).

a) (8 points) Find the position u(t) of the mass at any time t, if u(0) = 6 and u 0 (0) = 0

. b) (4 points) Find the transient solution uc(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time?

c) (4 points) Find the amplitude R and the phase angle δ for this motion and express U(t) as a single trigonometric term: U(t) = R cos(ωt − δ).

d) (4 points) Justify the following description of what happens: “the transient motion uc(t) dies out with the passage of time, leaving only the steady state periodic motion U(t)