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

first matrix A

[ 2 -1 3 ]

[-4 0 -2 ]

[2 -5 12 ]

[4 0 4 ]

amd b

[2]

[-2]

[5]

[0]

solve for Ax=b

using tan LU factorization of A

Let y′=y(4−ty) and y(0)=0.85.

Use Euler's method to find approximate values of the solution of the given initial value problem at t=0.5,1,1.5,2,2.5, and 3 with h=0.05.

Carry out all calculations exactly and round the final answers to six decimal places.

[ 1 -1 3 -3 5 2 ]

A=[ 1 -1 4 -1 9 -4 ]

[ -1 1 -3 3 -4 8  ]

[7]

b=[5]

[4]

use the row reduction algorithm to solve the following

Describe the solution set of Ax=b in parametric vector form

describe the solution set of Ax=0 as Span[ V1,V2,....,Vp]

Show that at least four of any 37 days must fall in the same month of the year

A study was conducted to determine whether the final grade of a student in an introductory psychology course is linearly related to his or her performance on the verbal ability test administered before college entrance. The verbal scores and final grades for 1010 students are shown in the table below.

Find the least squares line.

y=___ +___ x

III. Use Table to generate a list of ordered pairs (x,sin x) for x=0, \[Pi]/12, 2\[Pi]/12, 3\[Pi]/12, ..., 2\[Pi]. You should end up with a list of lists. Then do it again but without using Table.

Mathematica assignment, any ideas?

Fix a group G. We say that elements g₁, g₂∈G are conjugate if there exists h∈G such that

hg₁h⁻¹ = g₂.

1. Prove that conjugacy is an equivalence relation.
2. Prove that if g∈Z(G), the center of G, then its conjugacy classes has cardinality one.
3. Let G = S_n. Prove that h(i₁i₂ ... i_t)h⁻¹  = (h(i₁) h(i₂) ... h(i_t)), where i_j∈{1, 2, ... , n }.
4. Prove that the partition of S₃ into conjugacy classes is {{e} , {(1 2), (2 3), (1 3)} , {(1 2 3), (1 3 2)}} .That is, there are three distinct conjugacy classes: the set consisting of the 1-cycle e is one class, the set of 2-cycles is another class, and the set of 3-cycles forms the last conjugacy class.
5. Describe (with justification) the partition of S₄  into conjugacy classes explicitly. Be sure to be clear as to exactly how many conjugacy classes there are, give a representative element of each, and tell us how to determine which conjugacy class a given element of S₄ belongs. [Hint: You might want to invent a concept of "cycle type" to describe your answer.]
6. Are the elements

   1	2	3
   0	2	-7
   0	0	5

   and

   1	0	0
   0	5	π
   -1	0	2

   conjugate in the group GL₃(R)? Justify.

A recent 10-year study conducted by a research team at the Medical School was conducted to assess how age, blood pressure, and smoking relate to the risk of strokes. Assume that the following data are from a portion of this study. Risk is interpreted as the probability (times 100) that the patient will have a stroke over the next 10-year period. For the smoking variable, define a dummy variable with 1 indicating a smoker and 0 indicating a nonsmoker.

What action might the physician recommend for this patient?

Consider the mixing process shown in the figure. A mixing chamber initially contains 2 liters of a clear liquid. Clear liquid flows into the chamber at a rate of 10 liters per minute. A dye solution having a concentration of 0.75 kilograms per liter is injected into the mixing chamber at a constant rate of r liters per minute. When the mixing process is started, the well-stirred mixture is pumped from the chamber at a rate of 10+r liters per minute.

Part A and B provided. Please solve part C...

(a) Develop a mathematical model for the mixing process. Let Q represent the amount of dye in kilograms in the mixture.
dQ/dt = _______ kg / min

     3/4*r-Q/2(10+r)

(b) The objective is to obtain a dye concentration in the outflow mixture of 0.1 kilograms per liter. What injection rate r is required to achieve this equilibrium solution?
r =______ L / min

     20/13

Would this equilibrium value of r be different if the fluid in the chamber at time t=0 contained some dye?

(c) Assume the mixing chamber contains 2 liters of clear liquid at time t=0. How many minutes will it take for the outflow concentration to rise to within 1% of the desired concentration of 0.1 kilograms per liter?
t = ___________ min

prove that sign p=sign ^-p (if p is a permutation).