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Category: Mathematics

1.Making cocktails with the Z-transform:The very famous Fibonacci sequence goes like this:0,1,1,2,3,5,8,13,21,34,55,89The magical and mystical ‘golden ratio’φ= 1.618033988749895 is the limiting value of the ratiobetween two terms in the sequence eg.,5534= 1.61765,8955= 1.6181818The Fibonacci sequence is generated by the following difference-equation:y(k+ 2) =y(k+ 1) +y(k)k= 0,1,2,…(1)withy(0) = 0,y(1) = 1 as initial conditions. Here each term in the sequence is the sum of the twoprevious. Thusy(2) =y(1) +y(0), y(3) =y(2) +y(1), y(4) =y(3) +y(2)etc.Given that you know the difference equation that generates the Fibonacci sequence, use Z-transformsto calculatey(k) andφ, as set out in the following steps.(a) Using the difference-equation above, show that the Z-transformY(z) of the Fibonacci sequencey(k) is:Y(z) =zz2−z−1(b) Show thatY(z) can be equivalently written as:Y(z) =z(z−α)(z−β)whereα=φ=1+√52= 1.618033988749895,known as the golden ratio,andβ= 1−φ.[Hint: Factorizez2−z−1 by hypothetically lettingz2−z−1 = 0 and then finding theαandβyou need so thatz2−z−1 = (z−α)(z−β) = 0. That is you use the ’formula’ to find rootsofz2−z−1 = 0.](c) Find the partial fraction decomposition for1zY(z) =1(z−α)(z−β)=Az−α+Bz−βThat is, show thatA=φ/√5, and find B.(d) Use the inverse Z-transform to find an exact expression fory(k), namelyy(k) =φk√5−(1−φ)k√5.(e) Find the first five terms in the sequence from the solution you found above fory(k) in (d) andshow that it conforms to the solution expected of the difference equation Eqn.1. (f) Showy(k)≈φk√5for largek.(g) Hence showy(k+ 1)y(k)≈φ(h) Why is the golden ratio considered magical or mystical? Relate this to physical quantities suchas the size of the pyramids, the Mona Lisa, pine cones, door frames, cocktails (see attached),and other pleasing structural ratios etc.(i) Why might it be incorrect to refer to the numbers 0,1,1,2,3,5,8.13… as the Fibonacci series?(a=3, b=2, c=3, d=3, e=2, f=2, g=2, h=2, i=1. Total =20 marks)2.Fourier TransformA beaker of water of temperaturey(t) at timetis cooling to room temperature. To keep thingssimple and without loss of generality, we will suppose room temperatute is 0 degrees centigrade.The temperature of the watery(t) satisfies Newton’s Law of cooling which states:dy(t)dt=−ky(t) +x(t)Herex(t) are external temperature fluctuations that impact the cooling process of the beaker. Onewonders how these fluctuations influence the cooling process. We want to study the frequencyresponse of this system to the forcing fluctuationsx(t).(a) DenoteY(ω) andX(ω) as the Fourier transforms ofy(t) andx(t) respectively.By taking Fourier transforms of the above equation, and giving working, show thatY(ω) is ofthe form:Y(ω) =ab+jcX(ω)Find a,b, and c.(b) For a general input, the frequency response of the system is given byH(ω) =Y(ω)X(ω). FindH(ω).(c) How does one obtainH(ω) in practice? [This is not an assessment question!] One has toexamine the system for all different frequency inputsω. It might be possible to test onefrequencyω0after another and thus sweepωto obtain a picture ofH(ω) for allω.A neater solution requires only pulsing the system with a large impulse that can be modelledby the delta functionx(t) =δ(t). This will give us the so called impulse responseH(ω) (orfrequency response) to a delta signal.Why do this? Recall that the Fourier Transform ofx(t) =δ(t) isF(δ(t)) =X(ω) = 1. Thismeans that the impulsex(t) =δ(t) has energy for any and all frequency componentsω. So asingle pulsex(t) =δ(t) will automatically yield the spectrumY(ω) =H(ω)X(ω) =H(ω) forallω.Now that you have calculatedH(ω) for the beaker system, separate it into its real and imag-inary components i.e., in the formH(ω) =a+jb.Find alegebraic expressions for themagnitude|H(ω)|and phaseφ(ω)for this system.(d) Sketch a graph of|H(ω)|as a function ofω. (e) Provide a careful interpretation of this graph. What does it mean in terms of the inputfluctuationsx(t) (eg., anx(t) that is dominated by high frequencies as compared to anx(t)that is just a low frequency oscillation).(f) Suppose the inputx(t) =e−tu(t). FindY(ω).(g) Take the inverse Fourier transform ofY(ω) and findy(t). Sketchy(t).(h) What is the solution of the above differential equation for cooling whenx(t) = 0. Sketchthis solution on the same axes as above and compare with your last plot. Explain what youobserve.(a=2, b=1, c=4, d=1, e=2, f=4, g=4, h=2. Total =20 marks)3. Without using tables (i.e., from first principles) find the Fourier transform of:y(t) =etu(t+ 1) +e3tu(t−3).(7 marks)4. Find the inverse Fourier transform ofY(ω) =δ(ω−3). [Hint: Use a shift theorem]Can you give some intuitive interpret to your solutiony(k)?(7 marks)5. Find the inverse Z-transform ofY(z) =zz2−5z+ 6.(6 marks

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1.Making cocktails with the Z-transform:The very famous Fibonacci sequence goes like this:0,1,1,2,3,5,8,13,21,34,55,89The magical and mystical ‘golden ratio’φ= 1.618033988749895 is the limiting value of the ratiobetween two terms in the sequence eg.,5534= 1.61765,8955= 1.6181818The Fibonacci sequence is generated by the following difference-equation:y(k+ 2) =y(k+ 1) +y(k)k= 0,1,2,…(1)withy(0) = 0,y(1) = 1 as initial conditions. Here each […]

Explain the measuring scale associated with the question and if the data collected is cross-sectional or time series.

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Your first should be a question associated with a categorical (quantitative) variable. Explain the measuring scale associated with the question and if the data collected is cross-sectional or time series. What might you be able to infer about the data you would collect? Your second should be a question associated with a quantitative variable. Explain […]

Use the Shell Sort algorithm to rearrange the numbers 5, 9, 1, 7, 4, 6 into ascending order. Explain the method, demonstrate all passes and state the total number of comparisons and swaps.

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Use the Shell Sort algorithm to rearrange the numbers 5, 9, 1, 7, 4, 6 into ascending order. Explain the method, demonstrate all passes and state the total number of comparisons and swaps. (8 marks) Using the list of numbers, 5, 9, 1, 7, 4, 6 given in question 1, Use the Quicksort Algorithm to […]

Explain why a person would be wrong to claim, “My correlation coefficient is positive, therefore the two variables are highly correlated.”

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1.A student is analyzing how two variables are correlated: size of committee and “efficiency” of committee. He reports to his friends that he had a correlation coefficient (r) of -0.85. What does the negative sign on the “r” tell us? (Hint: Don’t just say that the relationship is negative – explain what that means!!) 2. […]

Do you think you would also collect some descriptive data, such as gender, age, or shift? Why do you think it makes sense to collect descriptive data?

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Clearly define the problem or issue you are addressing. Provide a brief background of any research you have found that might affect your research hypothesis. Create a research hypothesis based on the information provided in each scenario. You have been given a data set (Microsoft® Excel® document) with two sets of interval data (just the […]

How much noisier are streets where cars travel faster? A department of transportation worker wants to fit a simple linear regression model /41„ = + fl rx where x is the speed of the car in mph and y is the noise level in decibels. State the null and alternate hypothesis whether the speed of the car is significantly linearly associated with the noise level.

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‘ssfl ,onsider the table you used in question 22. Can you conclude that the mpg is less for tread B in for tread A? Use the a = 0.01 level and interpret in the context of the problem. 6 of 6 a random sample of 340 cars driven at low altitudes, 46 of them exceeded […]

What are the pros and cons of the significance level you selected (i.e., use the terms Type I and Type II error in your response)?Discuss

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Part A A researcher is conducting a study to determine if a newly developed drug, X, is an effective treatment for treating Basal cell carcinoma (i.e., a common kind of skin cancer). The researcher collects a sample of 2,000 participants from the population of people suffering from Basal cell carcinoma. As part of this study, […]

A polymer material, subjected to a constant strain of 0.5, undergoes stress relaxation from 3.5 MPa to 0.5 MPa in 30 seconds. What is its modulus of elasticity at 10 second?Calculate

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1.Calculate the weight average molecular weight for a thermoplastic that has the following weight fractions for the molecular weight ranges listed.Hint: consult pages17and 18 of lecture file 210426.Molecular Weight Range(g/mol)Weight FractionMolecular Weight Range(g/mol)Weight Fraction0 –5,0000.0120,000 –25,0000.195,000 –10,0000.0425,000 –30,0000.2110,000 –15,0000.1630,000 –35,0000.1515,000 –20,0000.1735,000 –40,0000.072.The average molecular weight of a random poly(isobutylene-isoprene) copolymer is known to be 77,630 […]

In your own words, name two effects of the Digital Millennium Copyright Act (DMCA) that are mentioned in the article.Discuss

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Question #1: Give your own single-sentence synopsis of the article. Question #2: Do you think that government has a role in overseeing cryptographic research? Question #3: Do you think that private citizens should be allowed to practice cryptography? Question #4: In your own words, name two effects of the Digital Millennium Copyright Act (DMCA) that […]

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