# Mathematics

## Autonomous System

Miran Anwar, mth 235 ss21 1: Hw19-5.4-SDE-2x2NS-PP-NP. Due: 04/26/2021 at 11:00pm EDT. See in LN, § 5.4, See Subsection 5.4.3. 1. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systemx? =? 1 2x + xyy? = 3 2y? 1 2xy(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ x1 y1] .Find these components.x1 =y1 =Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1See in LN, § 5.4, See Subsection 5.4.3. 2. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systemx? =? 1 4x + 1 2xyy? = y? 1 2xy(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ x1 y1] .Find these components.x1 =y1 =Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X01Part 4: The Jacobian Matrix at X1See in LN, § 5.4, See Subsection 5.4.4. 3. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systemx? =? 1 4x + 1 2xyy? = y? 1 2y2 ? 1 2xy(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ 0 y1] , x2 =[ x2 y2] .Find these components.y1 =x2 =y2 = Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2See in LN, § 5.4, See Subsection 5.4.4. 4. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systex? =?x + xyy? = 9 8y? y2 ? 1 2xy(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ 0 y1] , x2 =[ x2 y2] .Find these components.y1 =x2 =y2 = 2Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2See in LN, § 5.4, See Example 5.4.5. 5. (10 points)

## The Characteristic Polynomial

Consider the BVP for the function y given byy??+ 25 y = 0, y(0) = 4, y (?5) = 5.(a) Find r1, r2, roots of the characteristic polynomial of the equation above.r1,r2 =(b) Find a set of real-valued fundamental solutions to the differential equation above.y1(x) =y2(x) =(c) Find all solutions y of the boundary value problem.y(x) =Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.See in LN, § 6.1, See Examples 6.1.2-6.1.3. 2. (10 points)Consider the BVP for the function y given byy??+ 36 y = 0, y(0) = 1, y(?) = 1.(a) Find r1, r2, roots of the characteristic polynomial of the equation above.r1,r2 =(b) Find a set of real-valued fundamental solutions to the differential equation above.y1(x) =y2(x) =(c) Find all solutions y of the boundary value problem.y(x) = 1Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.

## Differential Equations

Draw on paper the solution x(t) = v1 e?1t + v2 e?2t on the x1x2-plane and then answer the questions below.(a) Based on your graph, select the correct solution curve from the interactive graph below. Select One Curve 1 Curve 2 Curve 3 Curve 4 None(b) Use the graph below to find lim t?+??x(t)?, where ?x(t)? is the length of the solution vector x(t). Select One Zero Infinity None(c) Introduce the unit vectors u1 = v1/?v1? and u2 = v2/?v2?. Use the graph below to find thelim t?+?u(t), where u(t) = x(t) ?x(t)?, is a unit vector in the direction of the solution vector x(t).[Select One/U1/-U1/U2/-U2/None](d) Use the graph below to find lim t????x(t)?. Select One Zero Infinity None(e) Use the graph below to find the lim t???u(t), where u(t) = x(t) ?x(t)?[Select One/U1/-U1/U2/-U2/None]9(f) Characterize the zero solution, x0 = 0. Select One Source Node Source Spiral Sink Node Sink Spiral Saddle Center NoneComments on the graph below: The graph is interactive. You can click on the boxes to turn on and off possible solution curves. For each possible solution we display: the possible solution curve, the possible solution vectorx(t), and the associated unit vector u(t) = x(t)/?x(t)?. You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)change in time. You can move the eigenvectors v1 and v2 by dragging them from the endpoint, and then see howthe curves would change.

## The Jacobian Matrix

Similar problem in LN, § 5.3, Example 5.3.6, 5.3.7. 1. (10 points)Find the critical points (also called equilibrium solutions) of the predator-prey systemx? =?3 x + 2 x y y? = 7 y?8 x yCritical PointsNote: A point is an ordered pair (x,y), and your answer must be a comma separated list of points.Similar problem in LN, § 5.3, Example 5.3.6, 5.3.7. 2. (10 points)Find the critical points (also called equilibrium solutions) of the competing species systemx? = 3 x?x2 ?2 x y y? = 2 y?y2 ?x yEquilibrium Points:Note: A point is an ordered pair (x,y), and your answer must be a comma separated list of points.See in LN, § 5.3, See Examples 5.3.8-5.3.10. 3. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systemx? =?9 x + x3y? =?2 y(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ x1 0] , x2 =[ x2 0] .where x1 > 0 > x2. Find these components.x1 = x2 =Part 2: The Jacobian Matrix 1Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 See in LN, § 5.3, See Examples 5.3.8-5.3.10. 4. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systemx? =?16 x + x3y? = 3 y(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ x1 0] , x2 =[ x2 0] .where x1 > 0 > x2. Find these components.x1 = x2 =Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 See in LN, § 5.3, See Examples 5.3.8-5.3.10. 5. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systex? = 4 y?y3y? =?9 x?y2

## Sampling Variability

Number of Alcoholic Drinks Consumed per Week: 5, 6, 5, 1, 7, 4, 3, 0, 3, 5, 6, 4, 2, 0, 1, 3, 8, 7, 4, 4.1. (2 points). The above distribution captures the number of alcoholic drinks, on average, twenty college age students consume in a week. Using the data, calculate the following and show your work:a. Mean:b. Median:c. Mode:d. Range:e. Variance:f. Standard deviation (note: for this and all future standard deviation calculations, please assume that we are working with a population):2. (2 points). Calculate the Z-scores for the following values randomly selected from the population: 0, 4, 5, 83. (3 points). Using the Z-scores you calculated, identify the below percentages for the area under the curve using Appendix A Area Under Normal Curve and the Calculating Z-Score Probabilities Handout.a. What percentage of the distribution would we expect to fall between the Z-score for the value of 0 from the original distribution and a Z-score of 0?b. What percentage of the distribution would we expect to fall between the Z-score for the value of 4 from the original distribution and a Z-score of 0?c. What percentage of the distribution would we expect to fall between the Z-score for the value of 5 and the rest of the distribution to the right of that value?d. What percentage of the distribution would we expect to fall between the Z-score for the value of 0 from the original distribution and the Z-score for the value of 8 from the original distribution?

## Cummulative Frequency

Identify the data set’s level of measurement. 1) hair color of women on a high school tennis teamA) nominal B) ordinal C) ratio D) interval 1)2) number of milligrams of tar in 85 cigarettes A) interval B) ordinal C) nominal D) ratio2)3) the ratings of a movie ranging from “poor” to “good” to “excellent” A) ratio B) nominal C) ordinal D) interval3)For questions 4 & 5: The heights (in inches) of 30 adult males are listed below.70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 69 71 68 67 73 74 70 71 69 68Class with = ______________________________________4) Construct a frequency distribution, a relative frequency distribution, and a cumulative frequency distribution using five classes.Class Frequency MidpointRelativefrequency Cummulative frequency1Use the frequency distribution table from question #4 to answer the following question. 5) Construct a frequency polygon using five classes. Use the Midpoint for the Horizontal axis.Provide an appropriate response. 6) The table lists the smoking habits of a group of college students.Sex Non-smoker Regular Smoker Heavy Smoker Total Man 135 34 5 174 Woman 187 21 10 218 Total 322 55 15 392 If a student is chosen at random, find the probability of getting someone who is a man or a non-smoker. Round your answer to three decimal places.7) Find the probability of getting four consecutive aces when four cards are drawn without replacement from a standard deck of 52 playing cards.2Decide if the situation involves permutations, combinations, or neither. Explain your reasoning. 8) The number of 5-digit pin codes if no digit can be repeatedProvide an appropriate response. 9) How many different permutations of the letters in the word PROBABILITY are there?10) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three boys in ten births.311)Find the area of the indicated region under the standard normal curve.Provide an appropriate response. Use the Standard Normal Table to find the probability. 12) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standarddeviation of 2.5 inches. The U.S. Army requires that the heights of women be between 58 and 80 inches. If 200 women want to enlist in the U.S. Army, how many would you expect to meet the height requirements?

## The Square Matrix

Mathematics II (Applied) BESS Assignment 3Exercise 1 Consider the square matrixA =? ? 1 2 10 1 0 1 0 1? ?(a) Calculate the eigenvalues of A and decide whether the matrix is diagonalizable. Justify your answer. (b) Find the associated eigenspaces of A. (c) Write the eigendecomposition of the power matrix A3 and of the exponential matrix eA. (d) Consider the matrixB = A ·eAProve B is diagonalizable and find the diagonal matrix D similar to B.Exercise 2 Let b : R ? R. Consider the dynamical systemx? = x2b(t)(a) Provide a suffi cient condition for the function b : R ? R such that for every initial point (x0, t0) ? R2 there exists a unique solution of the IVP. Justify your answer. (b) Puttingb(t) = etcompute the general solution ?(t;c) of the ODE. (c) Find the particular solution ?? of the IVP with starting point (x0, t0) = (0,1) and specify its domain I ? R.

## Data Analytics

1. Nulls in Database?When designing a database are nulls ok? In a Customer table, what do we do about middle initial and suffix? What are the pros and cons of adding a column or just creating another table? Please explain.2. Agile vs Waterfall?List 3 types of IT Projects that you would use an SDLC (Waterfall) approach and 3 types of IT Projects you would use an Agile approach. Please explain why.3. Cloud?What are some of the considerations for deciding whether to purchase your own physical servers or putting your organization’s data on the cloud? Please discuss the merits and risks of both.4. IaaS vs PaaS vs SaaSPlease describe the difference between IaaS vs PaaS vs SaaS, each of their purpose, anf their benefits.5. Microservices and ContainersBriefly describe Microservices and Containers and their purpose.6. Scrum and KanbanPlease briefly describe the difference between Scrum and Kanban and what environment would be suitable for each.

## Descriptive Statistics

Analysis of Data with Descriptive StatisticsOn the questionnaire, question 5 utilizes a 4-point Itemized Rating scale (illustrated below). This scale is balanced and can be assumed to yield interval scale / metric data. Given the preceding, invoke SPSS to calculate the mean and standard deviation for all of the variables in 5 (Q5a – Q5i).1. Using only the mean for each of the variables, which of the movie theater items was considered most important?Answer:2. Using only the standard deviation for each of the variables, for which question was there the greatest amount of agreement? Hint: Least amount of dispersion regarding the response to the movie item.Answer:3. Questions 4 and 6 utilize multiple-choice questions that yield nonmetric data but that are ordinal scale. The appropriate measure of central tendency for nonmetric data are the median and the mode.a. What is the median response for question 4, concerning the amount a person spends on food / drink items at a movie?Answer:b. Concerning question 6, the distance a person would drive to see a movie on a big screen, what is the mode of that distribution of responses?Answer:4. In this question, the objective will be to compare the results of median and mean responses for Q3.Answer:a. Mean response:b. Median response:c. Standard deviation:d. Minimum response:e. Maximum response:5. When the responses to a question contain extreme values, the mean response can lie in the upper or lower quartile of the response distribution. In such a case, the median value would be a better indicator of an average response than the mean value. Given the information you obtained from answering #4 above, is the mean or median a better representative of the average response to Q3?Answer:Chapter 15 SPSS Exercise 4 (20 points)

## The Nutrients Folic Acid

A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce:Type A Type B Type CFolic acid (mg) 3 1 3Choline (mg) 4 2 4Inositol (mg) 3 2 4Suppose food type C has been improperly labeled, and it actually contains 4 mg of folic acid, 6 mg of choline, and 5 mg of inositol per ounce. Would it still be possible to use matrix inversion to find the combination of foods giving the required supply.2.5 Points

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