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New York City Col­lege of Tech­nol­ogy Math­e­mat­ics De­part­ment Fall 2012 Prof. Urmi Ghosh-Dasti­dar Course: In­tro­duc­tion to Lin­ear Al­ge­bra MAT2580 Sec­tion (6643) (3 cred­its) Top­ics in­clude sys­tems of equa­tions, ma­tri­ces, de­ter­mi­nants, eigen­val­ues and eigen­vec­tors, inner prod­ucts, vec­tor spaces, and sub­spaces. Pre­req: MAT1575 (Cal­cu­lus II) Meet­ing Time – Tues/Thurs: 2:30 – 3:45 p.m. N723 Text: Lin­ear Al­ge­bra and its Ap­pli­ca­tions,4th Edi­tion by David C. Lay. Pub­lisher: Ad­di­son Wes­ley. In­struc­tor’s name: Urmi Ghosh-Dasti­dar Of­fice Hours: Tues/Thurs: 3:50 – 4:50 p.m. (Namm 726) Of­fice: N726 ; Ph: (718)260-5349 Of­fice: Pearl 616 (by ap­point­ment only) If you want to meet me other than the of­fice hours please make an ap­point­ment. e-mail: ughosh-dasti­dar@​citytech.​cuny.​edu Note: All exams will take place in-class un­less stated oth­er­wise. The final exam date and time is fixed. You have to make your­self avail­able for all in-class exams and final exam Tech­nol­ogy pre­req­ui­sites: A graph­ing cal­cu­la­tor is re­quired: We rec­om­mend a cal­cu­la­tor which can com­pute eigen­val­ues. E-mail: All stu­dent must use City Tech e-mail ad­dress while tak­ing this course. Read­ing e-mail on a reg­u­lar basis is nec­es­sary. I may need to con­tact you via e-mail if sit­u­a­tion arises. City Tech has pro­vided all stu­dents with a City Tech email ad­dress. Your email ad­dress is the first let­ter of your first name, fol­lowed by your last name, fol­lowed by @​campus.​citytech.​cuny.​edu. You can ac­cess your email by going to the fol­low­ing web site: http://​campus.​citytech.​cuny.​edu/. For help with ac­cess­ing email, you can also send an email to helpdesk@​campus.​citytech.​cuny.​edu. In case of emer­gency, you can call 718-254-8565 or email: epak@​citytech.​cuny.​edu or rhoque@​citytech.​cuny.​edu for tech­ni­cal help. Theme: Bio­di­ver­sity: Eco-Math link through Lin­ear Al­ge­bra A Brief In­tro­duc­tion Bio­di­ver­sity and the Hud­son River Flow­ing from the Lake Tear of the Clouds, North the Hud­son River jour­neys 315 miles and drops 4,322 feet in el­e­va­tion be­fore emp­ty­ing it­self into New York Har­bor. The Hud­son River is home to di­verse pop­u­la­tions of fish, birds, and mam­mals that co­habit and com­pete among them­selves for re­sources. Re­cently the Amer­i­can shad, At­lantic stur­geon, river her­ring (blue back her­ring and alewife), Amer­i­can eel, and large­mouth bass are in de­cline. In­tense eco­nomic har­vest­ing pres­sure and over­ex­ploita­tion cause coastal and ma­rine species to de­cline. There­fore, har­vest­ing and fish­ing should be man­aged prop­erly and care­fully to avoid de­cline of cur­rent pop­u­la­tion. Food web analy­sis pro­vides im­por­tant in­for­ma­tion re­gard­ing the na­ture of com­pe­ti­tion among var­i­ous or­gan­isms. Clus­ter analy­sis in graph the­ory is a pop­u­lar method to seek par­ti­tion of a given data set into sev­eral clus­ters so that the data points within the same clus­ter are more sim­i­lar than those be­longed in the sep­a­rate clus­ters. In this pro­ject we will use clus­ter analy­sis using the con­cepts of lin­ear al­ge­bra to study the com­pe­ti­tion among var­i­ous species in a given food web, in par­tic­u­lar, com­pe­ti­tion among var­i­ous Hud­son River species. Stu­dents will find a par­ti­tion of the com­pe­ti­tion graphs based on the Hud­son River food web such that the strength of com­pe­ti­tion (for shared preys) be­tween two clus­ters (two groups of preda­tors) is as low as pos­si­ble; how­ever, the strength of com­pe­ti­tion within the same clus­ters is as high as pos­si­ble. Big Idea be­hind this pro­ject Study and an­a­lyze Hud­son River Food Web and its com­pe­ti­tion graph to in­ter­pret the strength of species com­pe­ti­tion. Upon com­ple­tion of this pro­ject, stu­dents should be able to an­swer the fol­low­ing ques­tions: • Which preda­tor species are more con­nected than oth­ers? • What hap­pens if a spe­cific species (par­tic­u­larly, a prey) dies out? Par­tic­u­larly, how does the re­moval of a par­tic­u­lar species af­fect its preda­tors and also the over­all com­pe­ti­tion among all preda­tor species? I be­lieve through this pro­ject stu­dents will gain some in­sights to the mech­a­nisms of in­ter­ac­tions and com­pe­ti­tion among var­i­ous species. Stu­dents will be able to pro­pose fur­ther mea­sures for early in­ter­ven­tion if any species dies out, share their knowl­edge, and cre­ate pub­lic aware­ness of the need to pro­mote a healthy and bal­anced ecosys­tem in their own com­mu­nity. My goals as the course in­struc­tor are: 1. To as­sist stu­dents de­velop a deep un­der­stand­ing of core math­e­mat­i­cal con­cepts and help them ap­pre­ci­ate the use­ful­ness of math­e­mat­ics to an­a­lyze and ex­plain their com­mu­nity and en­vi­ron­ment. 2. To cre­ate chal­leng­ing en­vi­ron­ment for high achiever stu­dents. 3. To pro­vide train­ing in con­duct­ing re­search in an in­ter­dis­ci­pli­nary field com­bin­ing math­e­mat­ics and ecol­ogy based on bio­di­ver­sity of the Hud­son River Es­tu­ary; a topic that is care­fully cho­sen to hold stu­dents’ in­ter­ests. 4. To mo­ti­vate stu­dents in higher stud­ies in an in­ter­dis­ci­pli­nary field. 5. To help stu­dents re­tain knowl­edge for long term. Stu­dents Learn­ing Out­comes 1. To solve sys­tems of lin­ear equa­tions using ma­tri­ces. 2. To iden­tify and use vec­tor prop­er­ties (spaces, sub­spaces, bases, inner prod­uct). 3. To iden­tify prop­er­ties of ma­tri­ces (in­evitabil­ity, eigen­val­ues, eigen­vec­tors). 4. To use com­puter tech­nol­ogy to solve prac­ti­cal prob­lems. 5. To learn how to col­lect data. 6. To learn how to apply core math­e­mat­i­cal con­cepts (par­tic­u­larly eigen­val­ues and eigen­vec­tors) in solv­ing real-world prob­lems. 7. To un­der­stand in­ter­dis­ci­pli­nary ap­proach and the sig­nif­i­cance of it in real-world ap­pli­ca­tions. 8. To write tech­ni­cal re­ports and dis­sem­i­nate the key find­ings. 9. To un­der­stand how to pre­sent re­search find­ings. 10. To learn how to work as a team. 11. To be able to use com­puter tech­nol­ogy to as­sist in the above. Gen­eral Ed­u­ca­tion Learn­ing Goals 1. To un­der­stand in­ter­dis­ci­pli­nary ap­proach and the sig­nif­i­cance of it in real-world ap­pli­ca­tions. 2. To gather, an­a­lyze, and in­ter­pret the data with sci­en­tific rea­son­ing 3. To im­prove com­mu­ni­ca­tion skills via group work and oral pre­sen­ta­tions 4. To use log­i­cal think­ing to de­liver a writ­ten re­port

New York City College of Technology Mathematics Department Fall 2012 Prof. Urmi Ghosh-Dastidar Course: Introduction to Linear Algebra MAT2580 Section (6643) (3 credits) Topics include systems of equations, matrices, determinants, eigenvalues and eigenvectors, inner products, vector spaces, and subspaces. Prereq: MAT1575 (Calculus II) Meeting Time – Tues/Thurs: 2:30 – 3:45 p.m. N723 Text: Linear Algebra and its Applications,4th Edition by David C. Lay. Publisher: Addison Wesley. Instructor’s name: Urmi Ghosh-Dastidar Office Hours: Tues/Thurs: 3:50 – 4:50 p.m. (Namm 726) Office: N726 ; Ph: (718)260-5349 Office: Pearl 616 (by appointment only) If you want to meet me other than the office hours please make an appointment. e-mail: ughosh-dastidar@citytech.cuny.edu Note: All exams will take place in-class unless stated otherwise. The final exam date and time is fixed. You have to make yourself available for all in-class exams and final exam Technology prerequisites: A graphing calculator is required: We recommend a calculator which can compute eigenvalues. E-mail: All student must use City Tech e-mail address while taking this course. Reading e-mail on a regular basis is necessary. I may need to contact you via e-mail if situation arises. City Tech has provided all students with a City Tech email address. Your email address is the first letter of your first name, followed by your last name, followed by @campus.citytech.cuny.edu. You can access your email by going to the following web site: http://campus.citytech.cuny.edu/. For help with accessing email, you can also send an email to helpdesk@campus.citytech.cuny.edu. In case of emergency, you can call 718-254-8565 or email: epak@citytech.cuny.edu or rhoque@citytech.cuny.edu for technical help. Theme: Biodiversity: Eco-Math link through Linear Algebra A Brief Introduction Biodiversity and the Hudson River Flowing from the Lake Tear of the Clouds, North the Hudson River journeys 315 miles and drops 4,322 feet in elevation before emptying itself into New York Harbor. The Hudson River is home to diverse populations of fish, birds, and mammals that cohabit and compete among themselves for resources. Recently the American shad, Atlantic sturgeon, river herring (blue back herring and alewife), American eel, and largemouth bass are in decline. Intense economic harvesting pressure and overexploitation cause coastal and marine species to decline. Therefore, harvesting and fishing should be managed properly and carefully to avoid decline of current population. Food web analysis provides important information regarding the nature of competition among various organisms. Cluster analysis in graph theory is a popular method to seek partition of a given data set into several clusters so that the data points within the same cluster are more similar than those belonged in the separate clusters. In this project we will use cluster analysis using the concepts of linear algebra to study the competition among various species in a given food web, in particular, competition among various Hudson River species. Students will find a partition of the competition graphs based on the Hudson River food web such that the strength of competition (for shared preys) between two clusters (two groups of predators) is as low as possible; however, the strength of competition within the same clusters is as high as possible. Big Idea behind this project Study and analyze Hudson River Food Web and its competition graph to interpret the strength of species competition. Upon completion of this project, students should be able to answer the following questions: • Which predator species are more connected than others? • What happens if a specific species (particularly, a prey) dies out? Particularly, how does the removal of a particular species affect its predators and also the overall competition among all predator species? I believe through this project students will gain some insights to the mechanisms of interactions and competition among various species. Students will be able to propose further measures for early intervention if any species dies out, share their knowledge, and create public awareness of the need to promote a healthy and balanced ecosystem in their own community. My goals as the course instructor are: 1. To assist students develop a deep understanding of core mathematical concepts and help them appreciate the usefulness of mathematics to analyze and explain their community and environment. 2. To create challenging environment for high achiever students. 3. To provide training in conducting research in an interdisciplinary field combining mathematics and ecology based on biodiversity of the Hudson River Estuary; a topic that is carefully chosen to hold students’ interests. 4. To motivate students in higher studies in an interdisciplinary field. 5. To help students retain knowledge for long term. Students Learning Outcomes 1. To solve systems of linear equations using matrices. 2. To identify and use vector properties (spaces, subspaces, bases, inner product). 3. To identify properties of matrices (inevitability, eigenvalues, eigenvectors). 4. To use computer technology to solve practical problems. 5. To learn how to collect data. 6. To learn how to apply core mathematical concepts (particularly eigenvalues and eigenvectors) in solving real-world problems. 7. To understand interdisciplinary approach and the significance of it in real-world applications. 8. To write technical reports and disseminate the key findings. 9. To understand how to present research findings. 10. To learn how to work as a team. 11. To be able to use computer technology to assist in the above. General Education Learning Goals 1. To understand interdisciplinary approach and the significance of it in real-world applications. 2. To gather, analyze, and interpret the data with scientific reasoning 3. To improve communication skills via group work and oral presentations 4. To use logical thinking to deliver a written report

A con­tin­u­a­tion of MAT 1475. Top­ics in­clude Tay­lor poly­no­mi­als, Mean Value The­o­rem, Tay­lor and Maclau­rin se­ries, tests of con­ver­gence, tech­niques of in­te­gra­tion, im­proper in­te­grals, areas, vol­umes and ar­clength.

A continuation of MAT 1475. Topics include Taylor polynomials, Mean Value Theorem, Taylor and Maclaurin series, tests of convergence, techniques of integration, improper integrals, areas, volumes and arclength.

A dif­fer­en­tial equa­tion is an equa­tion that re­lates a func­tion to one or more of its de­riv­a­tives. – The above rather bor­ing de­scrip­tion does lit­tle to con­vey just how fun­da­men­tal, wide­spread, and amaz­ingly ef­fec­tive dif­fer­en­tial equa­tions are in de­scrib­ing the world around us. – Ex­am­ples: Any­thing in mo­tion. Also, many things that are not in mo­tion. Also, many ad­di­tional things to which the word “mo­tion” does not re­ally apply. – Fur­ther ex­am­ples: space­ships in orbit, pop­u­la­tions grow­ing and shrink­ing, a cup of cof­fee slowly cool­ing, springs bounc­ing, fi­nan­cial mar­kets ris­ing and falling, elec­tri­cal cur­rent flow­ing through a cir­cuit, ocean waves, sound waves, light waves, vi­bra­tions in mu­si­cal in­stru­ments and air­plane wings and sus­pen­sion bridges, – More ex­am­ples: Pretty much every­thing. Top­ics in­clude meth­ods of solv­ing or­di­nary dif­fer­en­tial equa­tions and ap­pli­ca­tions to var­i­ous prob­lems. Avatar and site header cre­ated using night­cafe stu­dio

A differential equation is an equation that relates a function to one or more of its derivatives. – The above rather boring description does little to convey just how fundamental, widespread, and amazingly effective differential equations are in describing the world around us. – Examples: Anything in motion. Also, many things that are not in motion. Also, many additional things to which the word “motion” does not really apply. – Further examples: spaceships in orbit, populations growing and shrinking, a cup of coffee slowly cooling, springs bouncing, financial markets rising and falling, electrical current flowing through a circuit, ocean waves, sound waves, light waves, vibrations in musical instruments and airplane wings and suspension bridges, – More examples: Pretty much everything. Topics include methods of solving ordinary differential equations and applications to various problems. Avatar and site header created using nightcafe studio

A dif­fer­en­tial equa­tion is an equa­tion that re­lates a func­tion to one or more of its de­riv­a­tives. – The above rather bor­ing de­scrip­tion does lit­tle to con­vey just how fun­da­men­tal, wide­spread, and amaz­ingly ef­fec­tive dif­fer­en­tial equa­tions are in de­scrib­ing the world around us. – Ex­am­ples: Any­thing in mo­tion. Also, many things that are not in mo­tion. Also, many ad­di­tional things to which the word “mo­tion” does not re­ally apply. – Fur­ther ex­am­ples: space­ships in orbit, pop­u­la­tions grow­ing and shrink­ing, a cup of cof­fee slowly cool­ing, springs bounc­ing, fi­nan­cial mar­kets ris­ing and falling, elec­tri­cal cur­rent flow­ing through a cir­cuit, ocean waves, sound waves, light waves, vi­bra­tions in mu­si­cal in­stru­ments and air­plane wings and sus­pen­sion bridges, – More ex­am­ples: Pretty much every­thing. Top­ics in­clude meth­ods of solv­ing or­di­nary dif­fer­en­tial equa­tions and ap­pli­ca­tions to var­i­ous prob­lems. Site header cre­ated using night­cafe stu­dio, avatar cre­ated using Adobe Ex­press

A differential equation is an equation that relates a function to one or more of its derivatives. – The above rather boring description does little to convey just how fundamental, widespread, and amazingly effective differential equations are in describing the world around us. – Examples: Anything in motion. Also, many things that are not in motion. Also, many additional things to which the word “motion” does not really apply. – Further examples: spaceships in orbit, populations growing and shrinking, a cup of coffee slowly cooling, springs bouncing, financial markets rising and falling, electrical current flowing through a circuit, ocean waves, sound waves, light waves, vibrations in musical instruments and airplane wings and suspension bridges, – More examples: Pretty much everything. Topics include methods of solving ordinary differential equations and applications to various problems. Site header created using nightcafe studio, avatar created using Adobe Express

The course ex­am­ines the con­tent of the sec­ondary school math­e­mat­ics cur­ricu­lum from an ad­vanced per­spec­tive. Ped­a­gog­i­cal con­tent knowl­edge is ex­am­ined in dis­cus­sions of math­e­mat­i­cal con­cept rep­re­sen­ta­tions, stu­dent er­rors, and the de­sign of ac­tiv­i­ties.

The course examines the content of the secondary school mathematics curriculum from an advanced perspective. Pedagogical content knowledge is examined in discussions of mathematical concept representations, student errors, and the design of activities.