384 patients with coronavirus have been hospitalised in the city hospital. 240 have been treated with anti-viral medication and 144 have not.  

Perform a hypothesis test to determine if there is a relationsip between patients being given anti-viral drugs and developing pneumonia. Use a 1% level of significance.

The observed values are presented below.

Anti-viral drug treatment:

1- fill the expected values in the table below

anti-viral drug treatment

2- fill in the two missing x² values

contribution to the x² test statistics

3- calculate test statistics (2dp)

4- reject H₀ if x² > ___ (3dp)

5- (reject/ do not reject) H₀, there is (sufficient/insufficient) evidence to indicate that there is a relationship between drug treatment and pneumonia

Suppose that X is a non-empty, complete, and countable metric space. Use Baire Category Theorem only to prove directly that X has an isolated point. Include a precise statement of the theorem in your proof and indicate how it is being applied.

QUESTION 1: Solve the linear programming model given below using the simplex method. Write the primal and dual results from the optimal table you obtained.

MAX Z = 10?₁ + 20?₂ + 5?₃

6?₁ + 7?₂ + 12?₃ ≥ 560

5?₁ − 3?₂ +   x₃ ≤ 100

2000?₁ + 1000?₂ + 1000?₃ ≤ 62298

?₁,?₂,?₃ ≥ 0

IMPORTANT REMINDER ABOUT THE QUESTION SOLUTION: NEW ORDER CALCULATIONS SHOULD BE WRITTEN DETAILED WHEN CREATING THE SYMPLEX TABLES. WHEN THE CALCULATIONS ARE SHOWED AND THE TABLES ARE CREATED, THE DECIMAL SECTION WILL STAY WITH ITS Rational Form.

Find the special fundamental matrix for the following system such that Φ(0) = I.

x' = row 1 ( 4 3 )

row 2 ( 3 -4 ) * x

1. Encode the message "TEST".
2. Find  NOTE: your matrix entries should be integers in the range
3. Use your answer above to decode the message "ETAJ".

Find all the complex numbers z for which the multiple-valued function (1+i)^z provides complex numbers which all have the same absolute value.

solve

y' +(x+2/x)y =(e^x)/x^2 ; y(1)=0

We define the Liouville function λ(n) by setting λ(1)=1. If n>1, we consider the prime power factorization n=p_1^(a_1 ) p_2^(a_2 )…p_m^(a_m ) and define λ(n)=(-1)^(a_1+a_2+⋯+a_m ) Prove that the summatory function of λ,Λ(n)=∑_d|n┤▒〖λ(d)〗 is multiplicative.

Is reflection across a line L through the origin an invertible transformation of R 2 ? If so, find the matrix representation of the inverse; if not, explain why not.

Take the smallest cellular structure forthe circle X = S1 consisting
of a single 0-cell and a single 1-cell. Let Y be the same circle with two 0-cells and two
1-cells with the ends attached to the two different 0-cells. Count (in particular, describe)
all possible (up to homotopy) cellular maps X → Y and Y → X

question related to Topological data analysis 2

Suppose that the rate of change of the price "x " of a good increases in the
time at a constant ratio "c" as a result of inflation
constant, at the same time falls proportionally to the difference
between supply "y" at time "t" and some equilibrium supply "y0", is
say x´ (t) = c - a (y - y0). It is also assumed that the exchange rate
of the offer is proportional to the difference between the price and some
equilibrium price "x0", that is y´ (t) = b (x - x0), where "a" and "b" are
positive proportionality constants. Assuming that in
t = 0, x = x0 and y = y0.

i) Do the price and / or the offer oscillate around x0 and y0 + c / a,
respectively?

ii) If at time t = T0 the price is maximum, at what time is the supply
maximum?

Solve the differential equation by Laplace transform y^(,,) (t)-2y^' (t)-3y(t)=sint   where y^' (0)=0 ,y=(0)=0

The temperature at some instant (t = 0) of a laterally insulated solid rod of 1 unit length is given by ?(?,?) = sin ?? for 0  x  1, where x is measured from left end of the rod to the other end on the right. Also, at time t = 0, the left end of the rod is subjected to 0oC and maintained over the time, while the right end is insulated when t > 0. Note that the term x in the expression of the temperature is in radian. The temperature variation in the rod,

?(?,?), satisfies the heat equation: ?? ?? = ?( 2 ?^2? /??^ 2 )

Solve the above heat equation numerically with del x = 0.2 and del t = 0.04 using explicit formula: ?(?? ,??+1) = ??(??−1,??) + (1 − 2?)?(?? ,??) + ??(??+1,??)

Given that, c ^2 = 0.1 and ? = ? ^2 Δ? (Δ?)2 .

Estimate the numerical values for ?(?,?) at t = 0.08, giving your answers at 4 decimal points

4 a) Find the Fourier Integral representation of ?(?) = 1 ?? |?| < 1

0 ?? |?| > 1

b) Find the Fourier Sine Transform of ?(?) = ? −|?| . Hence evaluate ∫ ?????? 1+?2 ??.