Consider a system with the input/output relationship y(t) = x(t)cos(15πt).
(a) Is this system (i) linear, (ii) causal, (iii) stable, (iv) memoryless, (v) time-invariant, and (vi) invertible. Justify each answer with a clear mathematical argument. (b) Find the Fourier Transform Y (f) of y(t) in terms of the transform X(f) of x(t).

Repeat problem (2) for the system with the input-output relationship y(t) =R1 τ=0(1−τ)2x(t−τ)dτ.

A fluid flow is defined by u = (0.4x2 + 2t) m/s and v = (0.8x + 2y) m/s, where x and y are in meters and t is in seconds.

Part A

Determine the magnitude of the velocity of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s.

V=

Part B

Determine the direction of the velocity of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s. Enter your answer as the angle θV, which the velocity makes with the x axis, measured counterclockwise from the positive x axis.

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Part C

Determine the magnitude of the acceleration of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s.

Part D

Determine the direction of the acceleration of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s. Enter your answer as the angle θa, which the acceleration makes with the x axis, measured counterclockwise from the positive x axis.

Set up a spreadsheet that implements the secant method and then solves each of the problems. Use the graph of each function to select an initial guess. Recall the iteration formula for the secant method: X^k+1=x^k-[f(x^k)/f(x^k)-f(x^k-1)](x^k-x^k-1)

Put the formula for the function under the heading f(xk-1) and f(xk). In the cell under xk+1, put the secant method iteration formula. In the second row, replace the previous xk-1 with xk and then xk with xk+1. Now copy the two formulas down one row. At this point, one iteration of the secant method is displayed. To see more iterations, just copy the second row down for as many iterations as desired. If too many iterations are copied and the function difference becomes exactly zero, a divide by zero error will appear.

a. f(x) x-x^1/3-2

b.f(x)=xtanx-1

c.f(x)=x^4-e^x+1

d.f(x)x^2e^x-1

excel problem

Find the roots of the functions given using the bisection method. Use the graph of each function to choose points that bracket the root of interest.

a. f(x) x-x^1/3-2

b.f(x)=xtanx-1

c.f(x)=x^4-e^x+1

d.f(x)x^2e^x-1

TAM and its value and should we expect the use of mathematical models to hopefully provide more data/science-based justification?

A bipartite graph is drawn on a channel if the vertices of one partite set are placed on one line in the plane (in some order) and the vertices of the other partite set are placed on a line parallel to it and the edges are drawn as straight-line segments between them. Prove that a connected graph G can be drawn on a channel without edge crossings if and only if G is a caterpillar. (***Please do on paper)

Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <= 4 to prove that every planar graph with than 12 vertices can be 4-colored.

CLIQUE

INPUT: Graph G, positive integer l

PROPERTY: G has a set of l manually adjacent nodes.

CLIQUE COVER

INPUT: graph G’, positive integer k

PROPERTY: N’ is the union of k or fewer cliques.

So, Question is : Show that CLIQUE and CLIQUE COVER is cycle base on the property that is given?

What does it mean no computer!!

Let a logistic curve be given by

dP/dt = 0.02*P*(50-P)

with the initial condition

P(0)=5

This is an IVP (“initial value problem”) for the population P versus time t.

Report the following numbers in the given order (use two digits):

1. The growth coefficient k=?
2. The carrying capacity M=?
3. The population level when the growth just starts to slow down (“the inflection point” population level) P_IP=?
4. The time when the population just starts to slow down t_IP=?
5. The population level when t=5: P(5)=?
6. The long-term approximation for the population level limit P(t) as t-> infinity.

Let G be a group. The center of G is the set Z(G) = {g∈G |gh = hg ∀h∈G}. For a∈G, the centralizer of a is the set C(a) ={g∈G |ga =ag }

(a)Prove that Z(G) is an abelian subgroup of G.

(b)Compute the center of D₄.

(c)Compute the center of the group G of the shuffles of three objects x₁,x₂,x₃.

○n: no shuffling occurred

○s₁₂: swap the first and second items

○s₁₃: swap the first and third items

○s₂₃: swap the second and third items

○m₁: move the last item to the front

○m₂: move the front item to the end

(d)Compute the center of GL₂(R).

(e)Prove that Z(G) = ∩_a∈GC(a).

please explain every subquestion

3. Solve the following differential equations by using LaPlace transformation:

2x'' + 7x' + 3x = 0; x(0) = 3, x'(0) = 0

x' + 2x = ?(t); x(0-) = 0

where ?(t) is a unit impulse input given in the LaPlace transform table.

Solve the following initial value: y ^''+ 4y = 2 cos 2t, y(0) = −2 and y 0 (0) = 0