D4 = {(1),(1, 2, 3, 4),(1, 3)(2, 4),(1, 4, 3, 2),(1, 2)(3, 4),(1, 4)(2, 3),(2, 4),(1, 3)}
M = {(1),(1, 4)(2, 3)}
N = {(1),(1, 4)(2, 3),(1, 3)(2, 4),(1, 2)(3, 4)}
Show that M is a subgroup N; N is a subgroup D4, but that M is not a subgroup of D4
In: Advanced Math
Let a be a positive constant number. Draw the graph of a catenary y=acosh(x/a). Calculate the arc length s from the point (0,a) to the point (x,acosh(x/a)), and find the expression of the curve in terms of the parameter s.
In: Advanced Math
Another of the assumptions of the basic SIR model is that the population does not lose the immunity once it has been acquired. But in real life that doesn't always happen. In the basic model, make the necessary modifications to include the case in which the population loses immunity and is again susceptible to illness. Perform a full model analysis.
In: Advanced Math
One of the assumptions in the basic SIR model was that the population was mixed homogeneously and therefore all susceptible elements were equal before the disease, however, in reality that does not always happen. Sometimes the geographical region or of different age groups. In the basic model, make the necessary modifications to include the case in which the population is not homogeneous, and perform a complete analysis of the model.
In: Advanced Math
Astronauts in training are required to practice a docking maneuver under manual control. As part of this maneuver, they are required to bring an orbiting spacecraft to rest relative to another orbiting craft. The hand controls provide for variable acceleration and deceleration, and there is a device on board that measures the rate of closing between the two vehicles. The following strategy has been proposed for bringing the craft to rest. First, look at the closing velocity. If it is zero, we are done. Otherwise, remember the closing velocity and look at the acceleration control. Move the acceleration control so that it is opposite to the closing velocity, and proportional in magnitude. After this time, look at the closing velocity again and repeat the procedure. Under what circumstances would this strategy be effective?
In the example from the textbook, there were no assumptions made with regards to any of the reaction times, but to make this a bit more concrete, we are going to assume that the astronaut’s reaction time is five seconds, that he or she waits 10 seconds before making the next observation of closing velocity, and that the constant of proportionality between the closing velocity and manual acceleration is 0.02.
Reconsider the docking problem of Example 4.3, and now assume that c=5 sec, w=10 sec, and k=0.02.
a. Assuming an initial closing velocity of 50 m/sec, calculate the sequence of velocity observations v0, v1, v2. . ., predicted by the model. Is the docking procedure successful?
b. An easier way to compute the solution in part (a) is to use the iteration function G(x)=x+F(x), with the property that x(n+1)=G(x(n)). Compute the iteration function for this problem, and use it to repeat the calculation in part (a).
c. Calculate the solution x(1), x(2), x(3), . . ., starting x(0)=(1,0). Repeat, starting at x(0)=(0,1). What happens as n→∞? What does this imply about the stability of the equilibrium (0,0)? [Hint: Every possible initial condition x(0)=(a,b) can be written as a linear combination of the vectors (1,0) and (0,1) and G(x) is a linear function of x].
d. Are there any states x for which G(x)=λx for some real λ? If so, what happens to the system if we start with this initial condition?
In: Advanced Math
Answer with clear explanation.
Find the irreducible factors of x6 − 1 over R.
b. Find all monic irreducible polynomials of degree ≤ 3 over Z3.
c. Find a polynomial q(x) such that (x2 + 2x + 1)q(x) ∼= 1(mod x3 + x 2 + 1) over Z3.
In: Advanced Math
How can you determine the standard matrix of a linear transformation?
What is the difference between a one-to-one linear transformation and an onto linear transformation?
In: Advanced Math
how would you recognize from the reconstruction deck of a graph whether it is bipartite
In: Advanced Math
Bob, Peter and Paul travel together. Peter and Paul are good
hikers; each walks p miles per hour. Bob has a bad foot and drives
a small car in which two people can ride, but not three; the car
covers c miles per hour. The threee friends adopted the following
scheme: They start together, Paul rides in the car with Bob, Peter
walks. After a while, Bob drops Paul who walks on; Bob returns to
pick up Peter, and then Bob and Peter ride in the car till they
overtake paul. At this point, they change: Paul rides and Peter
walks just as they started and the whole procedure is repeated as
often as necessary.
A. How much progress (how many miles) does the company make per
hour?
In: Advanced Math
USING MATLAB
Q2. Determine and plot the frequency response function H(e^jw)
(magnitude and phase normalized in w). Choose n = [-20, 20].
1. h(n) = (0.9)^│n│
2. h(n) = [ (0.5)^n + (0.4)^n] u(n)
USING MATLAB
In: Advanced Math
Use the Simplex method to solve the following LP
minimize −2x1 −3x2 + x3 + 12x4
subject to −2x1 −9x2 + x3+ 9x4 + x5 = 0
1/3x1 + x2 −1/3x3 −2x4 + x6 = 0
x1,x2,x3,x4,x5,x6
≥ 0
In: Advanced Math
In: Advanced Math
Consider n-dimensional finite affine space: F(n,p) over the field with prime p-elements.
a. Show tha if l and l' are two lines in F(n,p) containing the origin 0, then either l intersection l' ={0} or l=l'
b. how many points lie on each line through the origin in F(n,p)?
c. derive a formula for L(n,p), the number of lines through the
origin in F(n,p)
In: Advanced Math
Using field axioms and order axioms prove the following theorems (explain every step by referencing basic axioms)
(i) The sets R (real numbers), P (positive numbers) and [1, infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1 is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of natural numbers) then M = N
The following definitions are given:
A subset S of R is called inductive, if 1 is an element of S and if x + 1 is an element of S whenever x is an element of S.
The intersection of all inductive sets if called the set of natural numbers and is denoted by N
In: Advanced Math
a. Give an example of a finitely generated module over an integral domain which is not isomorphic to a direct sum of cyclic modules.
b. Let R be an integral domain and let M=<m_1,...,m_r> be a finitely generated module. Prove that rank of M is less than or equal to r.
In: Advanced Math