Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
UAF DEPARTMENT OF MATHEMATICS AND STATISTICS MATH 103X CONCEPTS AND CONTEMPORARY APPLICATIONS OF MATHEMATICS COURSE SYLLABUS GUIDELINES FALL 2014-FALL 2015 Syllabi must clearly satisfy university requirements, available at : http://www.uaf.edu/uafgov/faculty-senate/curriculum/course-degree-procedures-/uafsyllabus-requirements/ Course Description: Applications of mathematics in modern society. Topics may include voting systems, probability and statistics and applications of graph theory in management science; uses of probability and statistics in industry, government and science; and applications of geometry to engineering and astronomy. Problem solving emphasized. Effective From Fall 2014 until Fall 2015 Prerequisites: Placement into Math 103x determined by the UAF Math Placement Policy as stated in the UAF Catalog. Enforcement of prerequisites is expected, though instructors have permission to override this requirement when appropriate. Textbook: Excursions in Modern Mathematics 8th ed by Tannenbaum ISBN: ??? Students should have the option of purchasing the e-book for this course through MyMathLab. Software: MyMathLab (optional) For MyMathLab Help (click here) If you do not plan to use MyMathLab but would like to have the resources available to your students who have access (click here for access information). Drop Date: Please note that the University Drop Date Sept 19 2014 deadline will be strictly enforced. Withdrawal Date: Please note that the University Withdrawal Date Oct 31, 2014 deadline will be strictly enforced. Instructor Availability: All Instructors will have regular, posted office hours (or online meetings) during the week. Grading Policy: The final grade in this course will be determined as follows: Written Assessed Work At most 30% Online Assessed Work At most 15% At least 2 Midterm Exams At least 40% total Comprehensive Final Exam At least 20% Instructors should provide written feedback to students approximately weekly throughout the semester. This can be through humanly-graded assignments or email correspondence. Students must have a mechanism for estimating their current grade in the course. To avoid confusion for the student, the syllabus should state: You need to earn a grade of C- or better in this course to receive core credit. The syllabus must include a grading scale in some form. One example is below. Plus/minus grading is at the discretion of the instructor, but must be stated explicitly. A 90-100 B 80-89 C 70-79 D 60-69 F 0-59 Exams: All exams for this course are closed book. All exams should be paper-pencil exams, written and graded by a faculty member. Midterm and final exams from previous semesters should not be reused by an instructor. Limited reuse of edited problems is acceptable. An exam can have no more than 10% of points from multiple choice questions. The final exam will be a comprehensive exam. Students will not be allowed to take the exam early or late unless there is written verifiable proof of the reason for missing the exam (e.g., a doctor’s note, police report, court notice, etc., clearly stating the date AND time of the conflicting circumstances). In the event the final exam is not taken, under rare circumstances where the student has a legitimate reason for missing the exam, a makeup exam will be administered. The comprehensive final exam should contain problems of the types listed below. (IS THIS LAST SENTENCE CORRECT – that is: do you really mean there should be one of these type problems on the final??) Finding the winner of an election using the four methods Identify winning coalitions of a weighted voting system Calculate the Banzhaf power index Calculate the Shapley-Shubik power index Determine basics of a graph (vertices, edges, connected, path, circuit, etc.) Find Euler paths and/ or circuits Find a minimum-cost Hamiltonian Circuit Understand the Brute Force, Nearest Neighbor and Cheapest-Link Algorithms Find a minimal spanning tree of a graph Find mean, median, and quartiles of a data set Create a box plot of a data set Find the standard deviation of a data set Compute the probability of an event Find the probability space for a random phenomena Find probability using permutations and/or combinations Calculate the mean and standard deviation of a given statistic Find the mean and standard deviation of a normal curve Use the 68-95-99.7 rule to calculate normal probabilities Sample Course Schedule: Week 1: Introduction and begin Chapter 1(this is always a short week Thursday and Friday) Week 2: Chapter 1 Week 3: Chapter 1 Week 4: Chapter 2 Week 5: Chapter 2 Week 6: Midterm Week 7: Chapter 5 Week 8: Chapter 6 Week 9: Chapter 6 Week 10: Chapter 7 Week 11: Midterm Week 12: Chapter 15 Week 13: Chapter 16 Week 14: Chapter 16 Week 15: Chapter 17 Week 16: Final Exam Course Content and approximate timing: Mathematics of Social Choice (Chapters 1&2 – 9 hours) The Mathematics of Voting Analyze and interpret a preference schedule. Rearrange a preference schedule to accommodate the elimination of one or more alternatives. Explain the difference between majority rule and the plurality method. List and discuss the four voting methods. Determine the winner from a preference schedule using each of the 4 voting methods. Explain what is meant by a Condorcet Candidate. Describe the process of insincere voting. List three factors that can affect the outcome of an election. List and discuss the four fairness criteria. Understand how a method “violates” a fairness criterion. Discuss “Arrow’s Impossibility Theorem.” Weighted Voting Systems Interpret the symbolic notation for a weighted voting system by identifying the quota and the weight of each voter. Identify the winning coalitions in a given weighted voting system. Determine whether a weighted voting system has a dictator, any dummies, or voters with veto power. Calculate the Banzhaf power index for a given weighted voting system. List the possible permutations (sequential coalitions) for a three- or four-voter weighted voting system. Calculate the Shapley-Shubik index for a three voter weighted voting system. Management Science (Chapters 5,6,&7 - 12 hours) Euler Circuits Determine by observation if a graph is connected, and determine the degree of each vertex. Construct graphs that model real world situations. Define an Euler circuit, and determine whether a graph contains an Euler circuit. Find an Euler circuit and identity the circuit by numbering the edges. If a graph does not contain an Euler circuit, “eulerize” the graph by duplicating a minimum number of edges. Identify the types of problems whose solutions involve Euler circuits. Hamilton Circuits Give the definition of a Hamilton circuit. Explain the difference between an Euler circuit and a Hamilton circuit. Identify a given application as being an Euler circuit problem or a Hamilton circuit problem. Explain what is meant by a complete graph on N vertices. Calculate N! for a given value of N. Calculate the number of Hamilton circuits and the number of edges in a complete graph with a given number of vertices. Define the term algorithm. Explain the brute force method for finding the minimum-cost Hamilton circuit. Find an approximate solution to the TSP by applying the nearest-neighbor and cheapest-link algorithms. The Mathematics of Networks Explain the difference between a graph and a tree. Explain what is required for a graph to be a tree. Identify which types of applications are solved by using Euler circuits, Hamilton circuits, or minimum spanning trees. Find a minimum-cost spanning tree by applying Kruskal’s algorithm. Statistics (Chapters 15,16,&17 - 12 hours) Descriptive Statistics Calculate the median, mean, and 1st and 3rd quartiles of a set of data. Calculate the range and inter-quartile range of a given data set List the five-number summary for a given data set and construct a box plot. Find the standard deviation for a small data set. Explain in your own words the meaning of standard deviation. Explain the difference between a bar graph and a histogram. Construct a bar graph or histogram for a small data set. Find the mean, median, and quartiles of data represented by a bar graph or frequency table. Analyze a pie chart. Chances and Probability Describe the sample space for a given random phenomena. Explain what is meant by the probability of an outcome. List the two laws of probability. Apply the laws of probability to determine the validity of a probability space. Identify which probability law is not satisfied for a given illegitimate probability space. Compute the probability of an event when the probability space of the experiment is given. Write the probability space for a given random phenomena. Identify the different situations in which permutations or combinations are used. Normal Distributions Define statistical inference. Explain the difference between a parameter and a statistic. Identify both the parameter and the statistic in a simple inferential setting. Explain what a random variable is. Explain the difference between the honest and dishonest-coin principles. Using an appropriate formula, calculate the mean and standard deviation of a given statistic. Discuss the effect of an increased sample size on the statistic's sampling error. Explain the difference between the population mean and the sample mean. Describe a normal curve. Locate the mean and standard deviation from a graph of a normal curve. Explain the 68-95-99.7 rule and apply it to compute normal probabilities. Give the mean and standard deviation of a normally distributed data set, and compute the percent of the population that falls within a given interval. 1 Section from the following – 5 hours Mathematics of Sharing (Ch3) Mathematics of Apportionment (Ch4) Mathematics of Scheduling (Ch8) Growth in Nature (Ch9) Financial Mathematics (Ch10) Censuses, Surveys, Polls and Studies (Ch14) Topics for Assessment: For the final exam students should be able to: Differentiate between the methods for finding a winner of an election and be able to use the method to determine a winner.(Plurality, Borda Count, Plurality with elimination and pairwise comparisons) Understand the mechanics behind a weighted voting system Find the power distribution of players using both the Banzhaf and Shapely-Shubik Power indices Find edges and vertices of a graph Find paths, circuits, Euler paths and Euler circuits Find Hamiltonian paths or circuits using different algorithms (Brute forces, nearest neighbor, repetitive nearest neighbor, or cheapest link) Find a spanning tree of a graph and be able to apply Kruskal’s Algorithm Find quartiles, range, interquartile range, mean, median and standard deviation for a data set Describe a sample space for a random phenomena List outcomes and probabilities for an event Find probabilities using permutations or combinations Find the mean and standard deviation for a normal curve Use the 68-95-99.7 rule to calculate normal probabilities Criteria for Assessment: Students are able to master problem solving skills Students learn to manipulate abstract symbols Students learn a broad spectrum of mathematical applications Basic statistics Graph theory and its applications Probability Social choice and voting systems