Teaching Portfolio Yixin Guo

Department of Mathematics

Drexel University

Contents

1. Teaching Responsibilities……………………………………………….1 2. Teaching Statements…………………………………………………….2 3. Future Goals……………………………………………………………..5 4. Evidence of Teaching Excellence……………………………………….5

4.1 Teaching Evaluation………………………………………………...5 4.2 Student comments from evaluations……………………………….6 5. Samples of Teaching Material………………………………………….13

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1. Teaching Responsibilities In my teaching career, I have taught various courses at both undergraduate and graduate levels. Since spring 2006, I have taught the following courses in the Department of Mathematics at Drexel University:

MATH 624 Graduate Differential Equations: fall 2008.

MATH 323 Partial Differential Equations: spring 2006, winter 2008.

MATH 311 Probability and Statistics I: spring 2008

MATH 301 Numerical Analysis II: summer 2011, summer 2012

MATH 210 Differential Equations: winter 2007, fall 2008, winter 2008.

MATH 201 Linear Algebra: spring 2007, spring 2008, winter 2011 (two sessions), spring 2012

MATH 123 Calculus III: fall 2006, spring 2011.

MATH 121 Calculus I: fall 2006, winter 2008, fall 2011 (two sessions).

MATH 100 Fundamental of Mathematics: fall 2007

MATH 110 Pre-calculus: winter 2008, fall 2008, winter 2008

My complete teaching history at Drexel University can be found in Appendix A.

I have served on Ph.D. thesis committee twice: first for Ph.D. student in mathematics, Amal Aafif, defended on June 27, 2007. The second time I was on Svitlana Zhuravytska’s dissertation committee. Svitlana Zhuravytska defended on May 26th 2011.

I have served on Ph.D. candidacy exams for Linge Bai on August 15, 2012; I served on Ph.D. candicacy exams for Amrit Misra, Patrick Ganzer, Marissa Powers, (all three from Karen Moxon Lab in the School of Biomedical Engineering); Walter Hinds, Honghui Zhang (both from Joshua Jacobs Lab in the School of Biomedical Engineering) on August 12, 2011; I served on Emek Kose’s candidacy exam in June 2007.

I supervised graduate Min Rong on research topic: Study on neuronal data for patients with Parkinson’s disease, from fall 2008 to spring 2010. Min Rong graduated with a Master degree in spring 2010. We have a manuscript with our collaborators from IUPUI submitted to Journal of Neurophysiology.

I have been supervising postdoctoral researcher Dennis Guang Yang on three research projects: One project was on localized states in 1-D homogeneous neural field models with general coupling and firing rate functions. We have submitted a manuscript to Journal of Mathematical Biology. The second project is on entrainment of a thalamocortical neuron to periodic sensorimotor signals. The third project is on a horseshoe structure of multi-bump standing pulses in a firing rate model. We have done significant work on the second and third project and are working intensely to finish the manuscripts. We expect to have at least two more manuscripts submitted within a year.

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2. Teaching Statement

2.1 The path to where I am now I grew up in the academic environment since my parents were both college professors. Teaching is more than a profession to me. It is the life I have known since I was a child. I set my goal to follow my parents' footsteps many years ago. I started my first position at the department where I received my BS degree in mathematics in China. The position was a combination of undergraduate advising and teaching. As an undergraduate advisor, I not only supervised students in academic performance, but was also in charge of many extracurricular activities and administrative duties for three classes of 90 some math majors. I relived my college years again since I spent most of my time inside or outside of the classroom with the same group of students until they graduated college. Even though it was a fun and youthful period of time, I often felt unsatisfied and I had this empty feeling at the end of the day. I came to realize that my passion is teaching and more challenging research activity, not so much on extracurricular activities or administrative duties. Upon the approval from my department head and university officials, I switched to a lecturer position, and used my spare time to prepare for GRE, TOEFL, TSE exams to apply for Ph. D. programs in the United States and some European counties. Looking back at those days when I taught in China, even though I did not form the complete blueprint of my teaching guidelines because the education system and style is very different from that in the US. I did learn from my experience that a good teacher should have a caring attitude toward students and their progress. In China I only worked with math majors who liked math or even were fascinated by math, I did not have the experience of dealing with students fear and anxiety toward learning math. After I came to the US, I taught much more diverse bodies of students, and most of my students were non-math majors. When I was a Ph.D. student in the Department of Mathematics at the University of Pittsburgh, I led recitations in Calculus, Linear Algebra and Business Calculus, and lectured Business Calculus. For the first time in my life, I saw many students struggled with math, and their anxiety and fear made me realize I should make effort in making the math learning environment more at ease. That was when I started developing my pedagogic philosophy. From August 2000 to May 2003, I was a research assistant on my Ph.D. advisor's grant. After receiving my Ph.D. in 2003, I took a visiting assistant professor position at Ohio State University. It was a joint position conducting research at the Mathematical Biosciences Institute and teaching for the department of mathematics. I was really happy to have the opportunity to teach again. I designed and instructed a course on differential equations twice, one in 2003 fall quarter and the second time in 2004 fall quarter. Before I came to Drexel as a tenure-track faculty, I was on a visiting assistant professor position at Harvey Mudd College for nine months during which I taught Mathematical Biology. Since I came to Drexel University, I have taught various undergraduate and graduate courses that are listed on the teaching responsibility section.

2.2 Teaching guidelines I developed critical guidelines during the time when I was a graduate teaching assistant, a visiting teaching professor and a tenure track faculty at Drexel University. They are 1) be caring and create an optimal learning environment; 2) be concrete and give specific examples; 3) always pursue growth in all aspects of teaching.

2.2.1 Be caring and create optimal learning environment Mathematics is not a lovable subject for many students. It may even be a big obstacle for those students who fear math. To help my students overcome the obstacle, I always make it clear from the first day of teaching, that I am here in class and during my office hours to help them and that I genuinely care about their progress. During my teaching, I often ask students whether they are on the same page and always watch their responses. I have experienced from teaching a wide range of students that they are individuals and that what inspires one student does not necessarily touch another. Especially in an upper level math class, such as Partial Differential Equations, half of my students were math majors and the other half were from computer science and engineering. Often times, when I prove a theorem, it is necessary to present details in theoretical math language. The computer science and engineering majors may be lost since they have different way of thinking due to different training background. Or sometime a student just stumbled on a specific detail. In my classroom, I am always flexible enough to change the pace and be willing to discuss questions that arise spontaneously. I always

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encourage my students to ask question regarding details anytime in class. And I make sure that students' questions are answered immediately. After teaching in various capacities for several years at Drexel University, I have learned that it is important to conduct various courses using appropriate approaches. When I teach freshman Pre-calculus, I remind myself that my students are not mathematically mature. I emphasize the basic ideas and use simple mathematical language to explain abstract definitions. When I prepare my class, I review and design instructional materials that present key information through various examples, adopting mathematics language to more accessible language, or visual representations. I illustrate abstract definitions by problems and examples that can be observed in the real world. I carefully phrase my questions to test if my students grasp the concepts. When I teach graduate Ordinary Differential Equation and senior level Partial Differential Equations, my focus is not on definitions and formulas, but rather on logic, proofs and creativity. I encourage my students to conjecture as well as to prove. I incorporate research component into my teaching by giving projects. I ask student teams to work on projects that model tumor growth or neural networks. Each team is required to give presentations and a final written report. My students really liked my adventurous approach and were very enthusiastic to the challenge. For those seniors who will graduate and look for a job soon, this is a good opportunity to practice their presentation and communication skills. For other students, it is not only an interactive evaluation of their math and communication ability but also a flavor of creative research, which students do not see in a usual math class. I have a calm and soft voice, which can be either an advantage or a disadvantage. The advantage is that I can ease the stressful mind of those who are fraught with anxiety toward math and create a relaxed learning environment. On the other hand, I could easily lose students' attention in class. To engage students’ attention and interest, I try to prepare the teaching material in an animated and coherent way with stories of famous mathematicians or discoveries. For example, I used the history of the Euler’s number e, and the history and application of the Euler’s formula to spice up my calculus III class that is mostly about infinite series. I talked about in my differential equation class how Alan Lloyd Hodgkin and Andrew Huxley built a nonlinear differential equations model in 1952 to describe the initiation and propagation of action potential in neurons. They won the 1963 Nobel Prize in Physiology or Medicine for this work. I often incorporate applications into my courses. For example, I use basic financial math in pre-calculus class, such as calculating compound interest and principal. I also ask as many suitable questions and feedback as possible related to the material to encourage interaction and to provoke independent thinking. An optimal learning environment also includes reasonable requirements for students to fulfill, such as homework, preparation for quizzes and exams. The goal of my requirement is for the students to learn as much profound mathematics as possible for those who are keen to learn math, and keep up with the course for those who struggle with math. For different courses, I use different strategies to reach the goal. For lower level courses, I always give weekly short quizzes, several midterm exams and plenty of homework problems. I get mixed evaluations from this strategy. Those who are falling behind complain too much work, and those who are eager to learn are happy that they learn more than expected. My personal view on this is that I would rather have some complains of too much work than give student less work. Downsizing work requirements means those who struggle will quick fall into passive learning mode and eventually learn nothing. For upper level or graduate courses, I always design projects that can be applied to real life situations. I give projects to students well in advance to stir up their curiosity and learning desire. As we go along with the course, they can see step by step how we become able to solve those problems. I also change strategies according to the class size and the time when the course is offered. For example, I taught Numerical Analysis twice in the summer. Both times the class size was small with 6 or 7 students. I usually would not often give short quizzes for an upper level class like this. However, considering summer time and there are all kinds of excuses to skip classes or fall behind, I give weekly quizzes based on the material learned in the previous week. And I also make attendance mandatory for 10% of final credit.

2.2.2 Be concrete and use examples as much as possible The first thing I make sure my student know about math is that there is no ambiguity; everything must be

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concrete and precise. I often ask my students to present the detail of their work and solutions in certain ways that reflect the precision in math. For example, my students in pre-calculus and calculus I classes often drop the parentheses where they are suppose to be in their calculation. This kind of mistakes cost them minor credits. I do get complaints about why I am so strict about this. I always stay adamant not only because it is the incorrect way of representing math but also mistakes often arise from bad habits. For those students who happen to obtain the correct answer, I always demonstrate a few examples of possible mistakes and try to convince students why I impose such requirement. I always make the effort to correct every small mistake my students make even though sometime it can be tedious. For upper level classes, concreteness is not just correct math presentation, it also includes critical thinking on why and how to solve problems or prove results. In my Numerical Analysis II class, after one student presented how to solve a boundary-value problem (BVP) using two initial-value problems (IVPs), another student asked why you set up you IVPs as they were. The presenter said: ``I followed the professor’s lecture notes and she taught us the setup.” The presenter did not understand that the specific setup of the IVPs is to satisfy the boundary conditions in the original BVP, on which I did a detailed derivation. Math is all about problem solving. Teaching math is not just telling the students definitions, calculations, theorems and proofs. To me, teaching is more like guiding my students through the problem solving process. I often try to inspire my students to ask how and why questions, and ask whether there is a different path to obtain the correct answer. Examples are the best way to convey critical mathematical thinking process. I always accompany new concepts, calculation, theorems with as many examples as allowed in the limited amount of class time. These examples not only are demonstrations of problem solving process, they are also the guideline for students to follow when they solve problems independently, such as doing homework or taking exams. From comments on evaluations, my students really liked my teaching style and lecture notes full of examples.

2.2.3 Pursue growth in all aspects of teaching Teaching is a constant learning process for me. I believe that a thorough knowledge and experience of the subject of mathematics is necessary to teach well. A Chinese proverb says: `To see far, stand high.' Without profound subject knowledge I would not be able to show the connections between seemingly diverse and abstract concepts to my students very well. Not only do I expect to possess advanced knowledge myself but I also encourage my students to develop more sophisticated math skills. When I teach Calculus, I incorporate some basic ideas of analysis into my teaching. Some of my students like it because they feel this both broadens their scope of mathematics and helps them understand Calculus better. However, a good teacher needs to be more than knowledgeable. During my teaching career, I continuously try to improve my teaching skills. Thus far, I have learned how to effectively incorporate computational tools, such as Mathematica/Maple and MATLAB into my teaching. From my experience of directing Mathematica lab sessions for Calculus class, I believe that Mathematica/Maple is indispensable to help students visualize functions, especially those which define three dimensional objects. Since I came to Drexel, I started using a tablet to write lecture notes in class. I use computer projector to show lecture notes as I work through the math in class. The advantage is that all lecture notes can be saved as PDF files. Then I post all lecture notes, homework answers, quizzes and exam solutions on WebCT, now BlackBoard Learning (IRT managed course website). I often receive praise from my students for using the tablet and the accessible online notes. As student’s commented “I liked that Prof. Guo posted the exact notes online so we could review them. Often math classes can’t do that.” All the online resources are good study guide for students. If they forget how to proceed with certain problems, they can easily recover details from the online lecture notes.

Teaching a variety of mathematics courses is another important part of growth. I am motivated to teach both advanced courses and freshman level or service courses to the best of my ability. I believe all students, not just math majors, need to be competent in math. I am more than willing to help students who struggle with math. I am also very enthusiastic about developing research-oriented special topic courses using my interdisciplinary training in mathematical biology. For instance, I would like to develop a course on mathematical biology, a ourse on special functions and course on intergro-differential equations, giving students an opportunity to apply math to real world scientific problems.

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3. Future Goals 3.1 Student Mentoring

I recently received NSF funding to work on a project of multi-site feedback stimulation and chimera states. The funding covers at least half year support of one Ph.D. student or summer support for two students. I will recruit one or two Ph.D. students who have passed the qualified exam to work on this project. It is a project that can be developed into two dissertations.

3.2 Course development Mathematical biology is a new research branch that is in developing stage. It is becoming an important research direction in applied math. There are plenty opportunities for graduate students who have training in mathematical biology. They can get positions in medical research lab, pharmaceutical research and companies, national research labs or faculty at universities. I would like to develop a course on mathematical biology that can be offered on yearly basis. This course would incorporate knowledge from differential equations, dynamical systems, integro-differential equations and graph theory. This course can be rich in application. Students will learn and see how math is being applied to real life situations.

4. Evidence of Teaching Excellence

4.1 Students Evaluations The evaluation included are from spring 2006 to summer 2012 with the omission of the academic year 2008-2009, due to my maternity leave. I have continuously achieved good student ratings and student comments reflect my pedagogic philosophy. Using student evaluation survey, 202 total responses were received from seventeen sections. The courses included are MATH 100, MATH 110, MATH 121, MATH 123, MATH 201, MATH 210, MATH 301, MATH 311, MATH 323, MATH 623. The diagram in Figure 1 shows the results to responses of question 7 of the student evaluation on instructor assessment, “The instructor communicated ideas and information clearly and effectively”. The diagram in Figure 2 shows the results to responses of question 11 of the student evaluation on instructor assessment, “Overall, I would recommend this instructor”.

Figure 1: A diagram of student responses (spring 2006-summer 2012) to question 7 of the student evaluation on instructor assessment, “The instructor communicated ideas and information clearly and effectively”.

Strong Agree 47%

Agree 31%

Neutral 14%

Disagree 7%

Strong Disagree 1%

Total 202 responses

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Figure 2: A diagram of student responses (spring 2006-summer 2012) to question 11 of the student evaluation on instructor assessment, “Overall, I would recommend this instructor”.

4.2 Student comments from evaluations

MATH 201, Linear Algebra

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course” • The course was very thought-provoking to me. I even started exploring additional material on

my own, such as block matrices, even though is wasn’t required; The material was explained thoroughly, and each new thing we learned was accompanied by examples; The instructor would always stop to answer questions in order to enhance our understanding of the material.

• The way to teach is very well. Use the screen instead of black board. • Online notes. • Examples in notes; Any questions answered immediately and thoroughly. • The tests and homework were assigned well. • Invertible matrix theorem; setting up matrices; dependence of vectors. • Professor Guo made herself available to students for help and feedback on homework

assignments; Professor Guo made known the material that’d explicitly be covered on the exam; She gave simple examples, and later gave more difficult problems so we wouldn’t get lost; If we had any questions, she was more than willing to explain how and why.

• The lecture notes were clear and easy to follow; the professor communicated ideas clearly and effectively; the material was easy to understand; there was not an overwhelming amount of homework; the tests were straightforward.

• Easy to understand; easy to follow; good instructor. • Clear explanations; use of powerpoint/notes on bb vista; good examples. Yes, would

recommend the instructor. • I enjoyed my first attempt at a more theoretical approach to math; the textbook was

enjoyable (but the answer section was awful); I liked that it was abstract, yet grounded. • I liked that Prof. Guo posted the exact notes online so we could review them. Often math

classes can’t do that.

Strong Agree 49%

Agree 28%

Neutral 16%

Disagree 5%

Strong Disagree 2%

Total 202 responses

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• Wrote all notes cleanly, easy to refer to, gave practice exam for midterm, give good examples.

• Review before midterm; posting notes. • Determinants; Row reduction; Cramer’s rule; Invertible matrix theorem. • The pace of teaching, content, level of work, difficulty of tests. • Example problems were thorough and helpful; two hour sessions make it easy to

concentrate. • The examples were very useful, especially since they were different than the ones in the

book; Great pace. I took linear algebra before with a separate instructor and it was fun to fast paced; Prof. Guo teaches what matters, not useless stuff like proofs.

• Homework; in class examples; theorem style. • Examples in class, general guides to approach each problem set. • Very organize; clear about what is important to know. • Information conveyed efficiently; combination of theory learning and practice problems.

Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?” • Excellent teacher, taught the subject matter well. Homework was the only tedious bit. Hard

grading system. • I would recommend taking this course from this professor, I have scored significantly higher

in all the class I’ve had with this professor compared to every math class I’ve had in the last ten years.

• I would recommend Yixin Guo. She is one of the better math professors at Drexel. • Yes, good instructor, challenging material. • Yes, I would recommend Yixin, she is one of the few great math teachers I have come

across. • Yes, she is organized, fair, and teaches well. • I would highly recommend this professor. She teaches well, explains clearly, is very

understanding, encourages participation and gives a reasonable amount of work. • Yes. The professor was good and her lectures were easy to follow and helpful. • I would tell them to take the course because I found it worthwhile, and I have heard several

others criticized the other linear algebra teacher. So clearly, Prof. Guo isn’t just a good teacher, she’s the best one for this course.

• Yes, the instructor had a good grasp of the material and effective teaching and evaluation methods.

• Yes, she moves at a very good pace and obviously enjoys teaching. • I would recommend this instructor, she is very good. • I would highly recommend taking this course from this professor. Prof. Guo clearly explains

the topics at a reasonable pace, and the examples gave over in class are invaluable. • Absolutely yes. • I would recommend this instructor because the exams are straightforward.

MATH 121, Calculus I

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course” • Online notes; the quizzes. • Online notes; Reviewing in class; Examples in class; quizzes. • The course was well paced, the quizzes and tests were not too difficult and the subject

matter was all clearly presented and available.

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• Posted lecture notes; went through good examples. • Great instructor; course is not too hard as I imagined; quizzes every week are very helpful. • Practice tests and review worksheet help; problems from the book also help. • Section uses skills from previous sections; move as a good pace; learned a lot.

Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?” • I would recommend her because although she does give a lot of quizzes, she does explain

the material thoroughly. • I will recommend he/she should take this instructor because I have learned a lot from her. • Yes, the information is thoroughly explained in lectures. It’s just presented in a difficulty way,

which is what might make it hard. However, you will learn and do well on the exam with Prof. Yixin.

• I would recommend this professor because she takes the time to explain things in detail. • I would recommend Mrs. Guo for this class. She is great. • I would recommend this instructor. She gives good, clear notes and is helpful. • Yes because Yixin clearly explained how to do things and was very helpful. • Yes. She is very quiet but a good teacher.

MATH 123, Calculus III

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course”

• In depth notes that included examples; Homework problems gave more practice; Review days.

• The technology the instructor used; Putting up lessons on BBvista; Having review session. • The examples; Online notes. • I liked the pace of the course; It gave me enough time to study and practice to do well; Also I

enjoyed the smaller setting of the classroom, it was comforting for a college course. • Good notes; Good reviews. • I liked how we went through the stuff slowly but effectively. Also, the tablet used for lectures

was very good. • The concepts; Professor; Laid back atmosphere. • Notes being accessible online; The homework problems; The problems we went over in

class. • It’s not a huge range of information. It’s very detailed-which is interesting. I was thankful that

attendance was mandatory. • We actually learned theorems and rules, which was refreshing compared to my last math

class that was almost all about examples; Being able to download notes also helped greatly. • The style of testing; the style of notes. • Great explanation; good notes; fair grades; instructor was well prepared. • Good, timely feedback from teacher; many tests, quick grading. • Didn’t have notes prewritten out; went along with class; step by step explanation; went pretty

slow for new info. • Good teacher communication; good reviews for test/quizzes. • Take home exam was nice, but hard; projected notes were easier to read.

Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?”

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• Yes, very informed and easy to understand, patient instructor. • I would recommend my instructor because she explains concepts really well with being

friendly, and treating the students with respect. • Yes, because she really makes the effort to explain the topic to students. • Absolutely. I think she’s a great professor, maybe the best I’ve ever had for math. • I would recommend. She is an awesome instructor. • I would recommend her because she is the best prof. I’ve had at Drexel so far. • I would recommend this teacher because she is extremely knowledgeable and kind. • I would recommend that they do because the instructor teaches well, is respectful, and is

willing to help students often. • I would say yes because the speed was just right, and the instructor made concepts clear

and understandable. • Yes, she is fair and knows the material very well. • Yes, good teacher that communicates idea well.

MATH 323, Partial Differential Equations

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course” • Challenging and relevant homework. • Encouraged research/presentation; notes online; recitation helped. • Midterm/final project; lectures were very good; • The teacher did not rush things; the projects were interesting; the online notes were good. • I learned how PDE are used in real life situations; the recitation helped answer a lot of my

questions; the final presentation was very interesting to research. • Incorporation of mathematics in real world application. • The presentations and written reports were an excellent way to learn the material better. • Projects. • The interactive lectures, midterm and final projects were the best aspects of the course

because they gave me a fundamental understanding of the course material. • Lectures posted online; help available if asked for.

Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?” • I would recommend Dr. Guo because she is very knowledgeable and good at teaching. • Yes, a very patient and intelligent lady; knows her stuff. • Yes, though it was difficult. It was worth it, and I learned a lot. • Yes, I would recommend her; online lectures; available for help whenever needed. • Yes, she is intelligent and fair. • Yes, fun class, fruitful projects. • Yes, professor provided quick feedback, was always ready to help, and gave

comprehensive, clear lectures notes conveyed with necessary materials. • Yes, the course is taught in a way that is easy to understand.

MATH 210, Differential Equations

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course” • 1) Learned more about differential equations than in other classes where there was only an

intro to DE; 2) Learned how to solve difficult problems;3) Learned how calculus and linear

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algebra ties into differential equations. • Homework; examples; exams. • Lots of homework problems helps ot engrave the material into my brain better. • The homework problems; the teacher explained the material effectively. • Lecture notes and homework. • Solving ODE; Euler’s equation; solve I. V. P. with Laplace transformation. • Excellent notes; many examples given; all questions answered; tests were a reflection of

what was taught. • Reviews before exams; mandatory homework; classroom style lectures. • Lots of problems assigned; good lectures. • Step by step teaching; notes given if absent; excessive practice; good teacher. • The attention paid to theory and the derivation were probably the best part of lecture. • Idea and concepts were clearly expressed and the goal of each section was clear; the

homeworks were also very encompassing and helped to learn the material. • Dr. Guo taught material with the expectation that basics of Calculus 3+4 were known by all

students; communicated ideas effectively; provided several examples. Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?” • I would recommend taking this instructor because she presents the information in a clear,

understandable manner. • She is nice and always helpful when asked for assistance. • Yes, since she explained the material well and even though there are a lot of homework

problems, you know how to do it when you’re done. • I would tell him that you will learn a lot if you can keep up with the monstrous amount of

homework. • I would recommend the instructor. She is very helpful and really helps you to understand

ODE. • Yes- clear, helpful, fair, good assessment, adequate homework. • I would recommend because she taught well, and it was easy to understand the concepts. • Yes, she is good. • Absolutely, Dr. Guo was/is an excellent teacher.

MATH 100, Fundamental of Mathematics

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course” • Good teacher; good class structure; helpful homework; good atmosphere. • The instructor clearly went step by step on each problem and didn’t just assume everyone

knew how to do it. • We did a lot of practice problems in class. • The practice worksheets; the work book; the homework related to the textbook. • The weekly homework; study session. •

Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?”

• Yes, she makes sure everyone understands. • I would not recommend this course because it’s too easy; but I would recommend the

instructor because she gave clear and effective directions.

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• Yes, she explains clearly. • Yes, it’s not a hard course and the teacher practices problems a lot. • She is pretty good. • The teacher was good so I would recommend.

MATH 110, Pre-calculus

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course”

• Good note taking—very specific; teacher was timely; tests are efficient. • The workload was a little heavy, but well-paced; the amount of material covered in class was

manageable; the time spent per week in/for the course was appropriate. • Good pace; use of board for instruction. • Class discussions; doing math. • The three best aspects of this course was one a teacher who takes pride in her work, second

the skills I learned in this class, and lastly a teacher that took the time to respond to her students’ concerns.

Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?” • I would. Teacher is VERY smart and will be able to answer any questions. • I would recommend this course with this instructor. The professor is capable and competent,

and replies promptly when asked a question. • Yes, because she showed respect to her students and taught them well.

MATH 311 Probability and Statistics I

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course” • Worksheets helped reinforce material and with homework; overhead projector made note

taking easier; study of the worksheet was useful. • She gave worksheets of lectures we went over; gave answers to homework; gave answers to

test. • Worksheet+homework=good idea. • Exercise; hand-out; nice teacher. • Provide note sheets were a huge help for following along.

Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?”

• The course is boring, the instructor is good. • Instructor is thorough and I would recommend her. • Yes, good class and good teacher. • Yes, gives you idea about probability required for all sciences.

MATH 301 Numerical Analysis II

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course”

• Coding the methods in Matlab for the projects was a good way for me to grasp the material;

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teacher was very knowledgeable; interesting algorithms. • I enjoyed learning the algorithm because I’m a Matlab addict. Also, the quizzes forced me to

keep up with the class and re-read my notes. • I enjoyed the numerous numerical methods to solve ODEs, very helpful; I liked the Final

Presentation idea, creative idea for a grade. • Learned a lot; topics were concise and atomic; the teacher’s teaching style. • Projects; quizzes; notes being posted online. • Learned good programming techniques; enjoyable projects; helpful quizzes; good textbook. • The course gave me a better idea of different numerical analysis that are used in my

field/career. • Projects to apply numerical methods; quizzes on each topic covered.

Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?”

• It is a good math course for learning useful and practical numerical methods. • Yes. The instructor was nice and knowledgeable. Good way to practice Matlab and learn

useful techniques. • Yes, I would recommend taking this course from the instructor because covered mass

amount of worthy material quickly and effectively. • Yes. The workload was appropriate and the teacher’s teaching style really let me relax and

not worry too much about my grade. I think that this leaves an enjoyable learning environment instead of the usual stressful on.

• Yes. Her pace is good and her notes are easy. • Yes, she is a good teacher and gives good projects. • Professor was prepared for the class. She explained every topic clearly and was ready to

assist the students. Her way of teaching and syllabus was also good (realistic). • I would recommend taking this course with this instructor. The material is interesting and the

instructor does a good job of conveying information and structuring the class. MATH 623 Graduate Ordinary Differential Equations

Comments to the question “In terms of your learning, what were the 3-4 best aspects of this course”

• Interesting topics; very nice structure of the course; clear lectures. • I liked working on the homeworks and how they usually followed what we had learned in

class. • Homework assignments to review the material from class; notes form class posted on the

internet. Comments on question “If a good friend asked whether you would recommend taking this course from the instructor, what would you recommend and why?”

• I would recommend this instructor. She is very nice, knowledgeable. I liked the final project, also.

• Yes. • I would recommend taking this course from the instructor because the material was

presented at a reasonable pace and the instructor was very helpful with any problems that arose.

13

Sample of Teaching Material

Yixin Guo

Department of Mathematics

Drexel University

Numerical Analysis II 2012 Summer Quarter (201145) One project and one quiz with solution

Math 301Project # 2

Problems:1. (a). Give detailed derivation of 4-step Adams-Bashforth explicit method (including the formula for

wi+1 and the local truncation error).(b). Use 4-step Adams-Bashforth method with h = 0.1 to approximate the solution of the following

initial-value problem:

y′ = −5y + 5t2 + 2t with 0 ≤ t ≤ 1 and y(0) =1

3

Use Runge-Kutta method of order four to obtain starting values. Compare the results with the actualsolution y(t) = t2 + 1

3e−5t.

2. (a). Give detailed derivation of 3-step Adams-Moulton implicit method derivation (including theformula for wi+1 and the local truncation error).

(b). Use 3-step Adams-Moulton method to approximate the solution of the initial-value problem givenin 1(b). Use Runge-Kutta method of order four to obtain starting values. Compare the results with theactual solution given in 1(b).

3. (a). Transfer the following higher-order initial-value problem to a system of first order initial-valueproblems

y′′ − 2y′ + y = tet − t with 0 ≤ t ≤ 1 and y(0) = y′(0) = 0

(b) Use the Runge-Kutta method of order four for system algorithm with h = 0.1 to approximatethe solutions of the higher-order differential equation given in 3(a) and compare the results to the actualsolution y(t) = 1

6 t3et − tet + 2et − t− 2.

4. (a) Solve the following stiff initial-value problems using the Adams fourth-order predictor-correctormethod and compare the results with the actual solution.

y′ = −20y + 20 sin t+ cos t with h = 0.25 0 ≤ t ≤ 2 and y(0) = 1

actual solution y(t) = sin t+ e−20t.(b) Is your approximation accurate? If not, why? What do you need to do to obtain accurate approx-

imation?5. Consider the differential equation

y′ = f(t, y) a ≤ t ≤ b and y(a) = α

.(a). Show that

y′(ti) =−3y(ti) + 4y(ti+1) − y(ti+2)

2h+h2

3y′′′(ξi)

for some ξ, where ti ≤ ξi ≤ ti+2. (Hint: expend y(ti+1) and y(ti+2) in second Taylor polynomials abouty(ti). )

(b) Part (a) suggests the difference method

wi+2 = 4wi+1 − 3wi − 2hf(ti, wi) for i = 0, 1, ..., (N − 2)

.Use this method to solve

y′ = 1 − y 0 ≤ t ≤ 1 and y(0) = 0

,with h = 0.1. Use the starting values w0 = 0 and w1 = y(t1) = 1 − e−0.1. Compare with the exact

solution y(t) = 1 − et.

1

(c) Analyze this method for consistency, stability, and convergence.6. Use Linear Shooting method with h = π

8 to approximate the solution of the boundary-value problem

y′′ = y′ + 2y + cosx 0 ≤ x ≤ π

2, y(0) = −0.3, y(

π

2) = −0.1

,Compare your results to the actual solution y(x) = − 1

10(sinx+ 3 cosx).

Problem set 1: 1,3,5,6Problem set 2: 2,3,5,6Problem set 3: 2,3,4,6Problem set 4: 1,3,4,6You may choose one set of problems out of the four choices. Choose one problem out of the set on

which you decide to work for the oral presentation.

2

MATH 301: Project guideline

Drexel University Summer 2011-2012

Due dates: Send me your matlab file before the class on August 30. Hand in written report on

Tuesday 30 August in class. Give presentation on August 30 in class. All work is expected to

be independent! Show all work.

Prepared by Yixin Guo

In this project, there are four sets of problems. You may choose one set out of the four. Then

choose one problem out of the set you work on to give an oral presentation. Following are the

guidelines for your presentation:

• Your presentation should be a summary of the essence of your study. You may prepare

your presentation in either computer, overhead slides, black board. However, we do not

encourage you to do a board presentation.

• Total time for the presentation is 10 minutes + 3 minutes (questions from your peers).

Every student needs to be ready to ask in-class questions. Each one of you will be indi-

vidually responsible for thinking up questions to ask. You do not have to turn in the the

questions. However, during the student presentations, each student in the audience will be

required to ask at least one question of the presenting student. Asking questions will be

part of the evaluation of this project. So be prepared and pay attention.

• You may give a presentation on laptop using power point or any format you are comfortable

with, such as overhead projector. No matter how you present, you must stop when your

presentation time is up because we have others lined up and we only have limited amount

of time to use. For computer presentation, we will centralize all the slides on instructor’s

laptop so that we will not have to waste our time to setup for each presentation. If you

decide to do a laptop presentation, you must send your slides to me (yixinguo@gmail.com)

before you present in class.

Guideline for evaluation:

• The evaluation will mainly be based on three parts: the written assignment including

Matlab code, oral presentation. Bonus points will be given to those who ask questions at

the end of each presentation. The oral presentation is worth 35% of the total credit for

project 2. Written assignment is 65% out of the total credit. We do not evaluate what

questions you ask. As long as you ask questions, you receive the bonus credit.

• The detailed evaluation checklist for presentation is attached.

• Students will have their own comment sheets. They can express, anonymously, their con-

structive opinions about their fellow classmates’ performance.

• In the end, the course instructor will communicate with each student on their performance,

including grades and students comments.

1

Oral Presentation Evaluation Checklist

Name(s):

1. Introduction Points a. Introduce yourself (and team-mates) 0 10 10 b. Provide overview 0 10 20

2. Presentation a. Present correct assigned content 0 20 40 b. Communicate with correct mathematical reasoning 0 20 40 c. Present adequate support for conclusions 0 10 20 d. Respond accurately to questions 0 10 20

3. Conclusion a. Review results 0 10 20

4. Organization and style a. Timing 0 20 20 b. Quality of visual 0 10 20 c. Clarity of communication, eye contact 0 10 20 d. Apparent preparation 0 10 20

Bonus: Creativity, appropriate humor 0 10 20 Prepared by Yixin Guo

Linear Algebra 2012 Winter Quarter (201125) Final exam

Final Exam for Linear Algebra

Last Name: First name:

From 10:00AM-12:00PM on April 07, 2012

1. (4pt) Given the determinant

∣∣∣∣∣∣∣∣∣a b c

d e f

g h i

∣∣∣∣∣∣∣∣∣ = 7, calculate the following determinants.

(a).

∣∣∣∣∣∣∣∣∣a b c

d e f

5g 5h 5i

∣∣∣∣∣∣∣∣∣ =

(b).

∣∣∣∣∣∣∣∣∣a + 2d b + 2e c + 2f

d e f

5g 5h 5i

∣∣∣∣∣∣∣∣∣ =

(c).

∣∣∣∣∣∣∣∣∣g h i

a b c

d e f

∣∣∣∣∣∣∣∣∣ =

(d).

∣∣∣∣∣∣∣∣∣a b c

2d + a 2e + b 2f + c

g h i

∣∣∣∣∣∣∣∣∣ =

1

2

2. (8pt) Answer the following two questions:

(a). Find the matrix for the linear transformation

T (x1, x2, x3, x4) = (x1 − x2, 2x1 − x3 + x4, 2x2 − x3 − x4).

(b). Determine whether the linear transformation T is one-to-one.

3

3. (6pt) Answer the following two questions for matrix B =

0 2 1

1 0 1

3 1 1

:

(a). Find the inverse B−1 of B by using row reduction of the augmented matrix [BI].

(b). Use B−1 to solve Bx = b, where b = (1, 2, 3)T

4

4. (8pt) Use Cramer’s rule to compute the solutions of the following linear system.

2x1 + x2 + x3 = 4

−x1 + 2x3 = 2

3x1 + x2 + 3x3 = −2

5

5. (8pt) Find the inverse of matrix A =

0 −2 −1

−3 0 0

−1 1 1

using the inverse formula A−1 =

1detA

adjA.

6

6. (5pt) Is the set of matrices of the form A =

a 0

b c

a subspace of M2×2 that is the set of

all 2× 2 matrices? And explain why.

7. (2pt) If v has coordinates

3

0

1

with respect to basis B =

1

−4

3

,

5

2

−2

,

4

−7

0

,

then v =?

7

8.(3pt) Find the coordinate vector [x]B of x relative to the given basis B = {b1,b2}, where

b1 =

1

−2

, b2 =

5

−6

, x1 =

4

0

.

9. (6pt) Assume that the matrix A is row equivalent to B. Answer the following questions.

A =

2 −3 6 2 5

−2 3 −3 −3 −4

4 −6 9 5 9

−2 3 3 −4 1

, B =

2 −3 6 2 5

0 0 3 −1 1

0 0 0 1 3

0 0 0 0 0

(a). Find a basis for ColA.

(b). Find a basis for RowA.

8

(c). Find a basis for NulA.

(d). Determine the dimensions of NulA and the rank of A.

(e). What is the orthogonal complement of the null space of A?

9

10.(18pt) Answer the following questions for the given matrix F =

4 0 1

−2 1 0

−2 0 1

.

(a). Find the characteristic equation of matrix F

(b). Find all the eigenvalues and their corresponding eigenvectors.

10

11

(c). Is matrix F diagonalizable? If yes, what is the diagonal matrix D that is similar to F?

(d). What is the invertible matrix P such that P−1FP = D?

(e). Calculate F 3.

12

11. (8pt) Given v1 =

1

1

1

, v2 =

2

1

−3

, v3 =

4

−5

1

, x =

5

−3

1

, answer the

following questions:

(a). Do {v1, v2, v3} form an orthogonal set? Explain why.

(b). Are v1, v2, v3 linearly independent?

(c). Do they form an orthogonal basis for R3?

(d). Express x as a linear combination of v1, v2, and v3.

13

12. (8pt) Given u1 =

2

5

−1

, u2 =

−2

1

1

, y =

1

2

3

,

(a). What is the length of u2?

(b). What is the unit vector in u2 direction?

(c). Find the distance between u1 and u2.

(d). Find the orthogonal projection of y onto W=span{u1,u2}.

14

13. (6pt) Answer the following questions:

(a) If the null space of a 5 × 6 matrix A is 2-dimensional, what is the dimension of the column

space of A? What is the dimension of the row space of A

(b) If A is 5× 7, what is the largest possible rank of A? What is the smallest possible dimension

of NulA?

(d) A homogeneous system of twelve linear equations in eight unknowns has two fixed solutions

that are not multiples of each other, and all other solutions are linear combinations of these two

solutions. Can the set of all solutions be described with fewer than twelve homogeneous linear

equations? If so, how many? Explain why.

14. (10pt) Mark each statement True or False. Justify you answer.

(a). The dimension of the row space and the column space of A are the same, even if A is not

square.

(b). If A is a 3× 3 matrix, then det(5A) = 5detA

15

(c). If det(AT ) = −detA.

(d). Any system of n linear equations in n variables can be solved by Cramer’s rule.

(e). If ‖ u ‖2 + ‖ v ‖2=‖ u + v ‖2, then u and v are orthogonal.

(f). The length of every vector is a positive number .

(g). A vector v and its negative −v have equal lengths.

(h). If r is any scalar, then ||rv|| = r||v||.

(i). Eigenvalues must be nonzero scalars.

(j). Eigenvectors must be nonzero vectors.

Probability and Statistics 2008 Spring Quarter (200835) Final exam and one solution manual for homework

Homework 6 solutions Page 135, 2 a. f(x) = 10

1 for –5 ≤ x ≤ 5, and = 0 otherwise

a. P(X < 0) = 5.0

5 101 =∫− dx

b. P(–2.5 < X < 2.5) = 5.5.2

5.2 101 =∫− dx

c. P(–2 ≤ X ≤ 3) = 5.3

2 101 =∫− dx

d. P(k < X < k + 4) = ] 4.])4[(1014

10

4

101 =−+== ++

∫ kkdx kk

xk

k

Page 135, 4

a. ] 1)1(0);( 02/

0

2/2

2222=−−=−==

∞−∞ −∞

∞− ∫∫ θθ

θθ xx edxexdxxf

b. P(X ≤ 200) = ∫∫ −

∞−=

200

0

2/2

200 22);( dxexdxxf x θ

θθ ] 8647.11353.

2000

2/ 22=+−≈−= − θxe

P(X < 200) = P(X ≤ 200) ≈ .8647, since x is continuous. P(X ≥ 200) = 1 – P(X < 200) ≈ .1353

c. P(100 ≤ X ≤ 200) = =∫200

100);( dxxf θ ] 4712.

200

100000,20/2

≈− −xe

d. For x > 0, P(X ≤ x) = =∫ ∞−

xdyyf );( θ ∫ −x y dxe

ey

0

2/2

22 θ ] 2222 2/0

2/ 1 θθ xxy ee −− −=−=

Page 135, 6

a.

x

f(x)

4.54.03.53.02.52.0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

b. 1 = ∫∫ −=⇒=−=−−

1

1

24

2

2

43

34]1[])3(1[ kduukdxxk

c. P(X > 3) = 5.])3(1[4

3

243 =−−∫ dxx by symmetry of the p.d.f

d. ( ) 367.12847])(1[])3(1[

4/1

4/1

243

4/13

4/11

243

413

411 ≈=−=−−=≤≤ ∫∫ −

duudxxXP

e. P( |X–3| > .5) = 1 – P( |X–3| ≤ .5) = 1 – P( 2.5 ≤ X ≤ 3.5)

= 1 – 313.165])(1[

5.

5.

243 ≈=−∫− duu

Page 135, 10 a.

θ

b. 11),;( 1 ==

−⋅===

∞∞

+

∞

∞− ∫∫ k

k

kk

k

k

xdx

xkdxkxf

θθθθθ

θθ

c. P(X ≤ b) = kb

kkb

k

k

bxdx

xk

−=

−⋅=∫ +

θθθ

θθ

111

d. P(a ≤ X ≤ b) = kkb

ak

kb

a k

k

baxdx

xk

−

=

−⋅=∫ +

θθθθ 11

Page 142, 12

a. P(X < 0) = F(0) = .5 b. P(–1 ≤ X ≤ 1) = F(1) – F(–1) = 6875.16

11 = c. P(X > .5) = 1 – P(X ≤ .5) = 1 – F(.5) = 1 – .6836 = .3164

d. F(x) = F′(x) =

−+

34

323

21 3xx

dxd

= ( )22

409375.3

343230 xx

−=

−+

e. ( ) 5.~ =µF by definition. F(0) = .5 from a above, which is as desired.

Page 142, 18

21

)1(11)( =−−

=xf for –1 ≤ x ≤ 1

a. P(Y = .5) = P(X ≥ .5) = ∫1

5. 21 dx = .25

b. P(Y = –.5) = .25 as well, due to symmetry. For –.5 < y < .5, F(y) = .25 + ∫−y

dx5. 2

1 = .25 + .5(y + .5)

= .5 + .5y. Since Y ≤ .5, F(.5) = 1 and F(y) = 1 for y > .5 as well. That is,

≤<≤−+

−<=

yyy

yyF

5.15.5.5.5.

5.0)(

y

F(y)

1.00.50.0-0.5-1.0

1.0

0.8

0.6

0.4

0.2

0.0

Page 142, 20

a. For 0 ≤ y ≤ 5, F(y) = 5025

1 2

0

yuduy

=∫

For 5 ≤ y ≤ 10, F(y) = ∫∫∫ +=yy

duufduufduuf5

5

00)()()(

1505

2255

221 2

0−−=

−+= ∫

yyduuy

y

F(y)

1086420

1.0

0.8

0.6

0.4

0.2

0.0

b. For 0 < p ≤ .5, p = F(yp) = ( ) 2/12

5050

pyy

pp

=⇒

For .5 < p ≤ 1, p = )1(25101505

2 2

pyy

y pp

p −−=⇒−−

c. E(Y) = 5 by straightforward integration (or by symmetry of f(y)), and similarly V(Y)=

1667.41250

= . For the waiting time X for a single bus,

E(X) = 2.5 and V(X) = 1225

Page 142, 22

a. For 1 ≤ x ≤ 2, F(x) = ,412121121

1 2 −

+=

+=

−∫ x

xy

ydyy

xx

so

F(x) = ( )

−+

142

01xx

2211

>≤≤<

xxx

b. px

xp

p =−

+ 412 ⇒ 2xp

2 – (4 – p)xp + 2 = 0 ⇒ xp = ]84[ 241 ppp +++ To

find µ~ , set p = .5 ⇒ µ~ = 1.64

c. E(X) = 614.1)ln(2

2121122

1

22

1

2

1 2 =

−=

−=

−⋅ ∫∫ xxdx

xxdx

xx

E(X2) = ( ) ⇒=

−=−∫ 3

83

2122

1

32

1

2 xxdxx Var(X) = .0626

d. Amount left = max(1.5 – X, 0), so

E(amount left) = 061.11)5.1(2)()0,5.1max(5.1

1 2

2

1=

−−=− ∫∫ dx

xxdxxfx

Page 142, 24

a. E(X) = 11

1 1

1 −=

+−==⋅

∞+−∞∞

+ ∫∫ kk

kxkdx

xkdx

xkx

kk

kk

k

k θθθθ

θθθ

b. E(X) = ∞

c. E(X2) = 2

1 2

1 −=∫

∞

− kkdx

xk k

k θθθ

, so

Var(X) = ( )( )2

222

1212 −−=

−−

− kkk

kk

kk θθθ

d. Var(X) = ∞, since E(X2) = ∞.

e. E(Xn) = ∫∞ +−

θθ dxxk knk )1( , which will be finite if n – (k+1) < –1, i.e. if n < k.

            Graduate Ordinary Differential Equations 2008 Fall Quarter (200815) Syllabus, one Lecture, one Homework and take‐home Exam 

MATH 623: Ordinary Differential Equations

Drexel University Fall quarter 2008-2009

• Text: Ordinary Differential Equations with Applications by Carmen Chicone. Electronic copy

is available at Drexel library.

• References:

– Ordinary Differential Equations by Wolfgang Walter

– Differential Equations and Dynamical Systems by Lawrence Perko

– Differential Dynamical Systems by James Meiss

• Prerequisites: Multi-variable Calculus, Linear Algebra and undergraduate level Differential

Equations.

• Contact: Yixin Guo, Korman center 263, 215-895-1410, yixin@math.drexel.edu.

• Office hours: Tue 9:00-10:00pm or by appointment.

• Grading: The grades for this course will be determined as follows:

20% assigned homework problems + 30% midterm exam + 40% take home project + 10%

attendance

– Homework problems: Assigned in class.

– Midterm exam: in class exam around third or fourth week

– project: One take-home project will be assigned a few weeks before the tenth week. Each

student will write a report and give a presentation for the project. The presentation will

take place during the lecture of the tenth week. The report is due the same day.

– Attendance and participation of lectures are required to pass this course. You may have

one unexcused absence. Each absence there after may reduce your course grade. This will

take into effect on the third week the course.

• Computing skills: All the students are strongly encouraged to use computer to assist learning

the material. Programs such as Matlab, Maple or Mathematica.

* Course related information, such as homework assignment, lecture notes, handouts will be posted

on WebCT.

1

HW 2, due Oct 7 at the end of the class

1. Construct the Picard iterates for the IVP y′ = 2t(y + 1), y(0) = 0 and show that they

converge to the solution y(t) = et2 − 1

2. Consider the initial-value problem

y′ = t2 + y2, y(0) = 0 (1)

Let R be the rectangle 0 ≤ t ≤ a, −b ≤ y ≤ b.

(a). Show that the solution y(t) of (1) exists for

0 ≤ t ≤ min(a,b

a2 + b2).

(b). Show that the maximum value of b/(a2 + b2), for a fixed, is 1/(2a).

(c). Show that α = min(a, 1

2a) is largest when a = 1/

√2.

(d). Conclude that the solution y(t) of (1) for 0 ≤ t ≤ 1/√

2

3. Prove that y(t) = −1 is the only solution of the IVP

y′ = t(1 + y), y(0) = −1. (2)

4. Find the solution of the IVP y′ = t√

1 − y2, y(0) = 1, other than y(t) = 1. Does this

violate Theorem 2.2 (the uniqueness of solution of IVP) on lecture 2.

5. Here is an alternate proof of Lemma 2.2 on lecture 2. Let w(t) be a nonnegative

function with

w(t) ≤ P∫

t

t0

w(s)ds (3)

on the interval t0 ≤ t ≤ t0 + α. Since w(t) is continuous, we can fine a constant A such

that 0 ≤ w(t) ≤ A for t0 ≤ t ≤ t0 + α.

(a). Show that w(t) ≤ PA(t − t0).

(b). Use this estimate of w(t) in (3) to obtain

w(t) ≤AP 2(t − t0)

2

2

.

(c). Proceeding inductively, show that w(t) ≤ AP n(t − t0)n/n!, for every integer n.

(d). Conclude that w(t) = 0 for t0 ≤ t ≤ t0 + α.

1

Midterm, due October 28 in class

1. Let v(t) represent the scaled voltage of a neuron. v(t) satisfies the differential equation

v′(t) = − 1

τv(t)(1−v(t)). Suppose three inputs to v(t) happens at time 0, x, l and each input

increases the voltage v by w. Find the voltage v(t) when 0 ≤ t < x, x ≤ t < l and t = l.

(Hint: v(0) = w.)

2. Let g ∈ C[0,∞] with∫∞

1t|g(t)|dt < ∞. Show that y′′ + g(t)y = 0 has solutions φ1(t)

and φ2(t) such that

φ1(t) → 1, φ′

1(t) → 0, φ2(t)/t → 1, φ′

2(t) → 1

as t → ∞. Hint: Use successive approximations to prove that the equivalent integral equa-

tions have bounded solutions over α ≤ t < ∞ when α is chosen sufficiently large.

3. Suppose that X is a set and n is a positive integer. Prove: If T is a function,

T : X → X, and if T n has a unique fixed point, then T has a unique fixed point.

4. Let r, k and f be real and continuous functions which satisfy r(t) ≥ 0, k(t) ≥ 0, and

r(t) ≤ f(t) +∫

t

a

k(s)r(s)ds, a ≤ t ≤ b.

Show that

r(t) ≤∫

t

a

f(s)k(s)exp[∫

t

s

k(u)du]ds + f(t), a ≤ t ≤ b.

5. Find a fundamental matrix solution of the system

x =

1 −1/t

1 + t −1

x, t > 0

Hint: x =

1

t

is a solution.

1

 

 

 

 

 

 

 

 

 

Partial Differential Equations    2006 Spring Quarter (200535) 

Syllabus, two lectures, one maple worksheet, one homework, 

and final project 

 

MATH 323: Partial Differential Equations

Drexel University Spring quarter 2005-2006

• Text: Partial differential equations for scientists and engineers by Stanley J Farlow.

• Other references:

– Boundary value problems by David L. Power.

– Applied partial differential equations by J. David Logan.

– An introduction to partial differential equations with Matlab by Matthew P. Coleman.

• Prerequisites: MATH 200 (Multi-variable Calculus) Minimum Grade: D and MATH 201

(Linear Algebra) Minimum Grade: D and MATH 210 (Differential Equations) Minimum

Grade: D

• Schedule: M T W lectures 12:00-12:50pm, F recitation.

• Instructors: M T W Yixin Guo, F recitation is given by Emek Kose. Amal Aafif helps

us with computational tasks on Matlab.

• Contact: Yixin Guo, Korman center 263, 215-895-1410, yixin@math.drexel.edu.

• Office hours: M W 2:00-3:00pm, also by appointment.

• Outline: Provide a rigorous introduction to partial differential equations including their

origins, applications and how to solve them in certain cases by useful and well-know tech-

niques.

• Grading: The grades for this course will be determined as follows:

30% assigned homework problems + 30% midterm presentation + 30% take home project

+ 10% attendance

– Homework problems: Assigned weekly except the midterm week and project week.

– Exam and project: There will be one midterm presentation and one take home project.

Midterm presentation will take place in the fifth week of the class. Take home project

will be assigned on May 30th, 2006.

– Attendance and participation of lectures and recitations are required to pass this

course. You may have one unexcused absence. Each absence thereafter may reduce

your course grade. This will take into effect on the fourth week the course (starting

from April 24th 2006).

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• Computing skills: All the students are strongly encouraged to use computer to as-

sist learning the material. Programs such as Matlab, Maple or Mathematica. Some of

the homework assignments will be related to performing basic operations of calculus and

differential equations, especially those problems that are well suited to visualization and

graphics.

* Course related information, such as homework assignment, lecture notes, handouts and

solution sheets will be posted on the course website http://www.math.drexel.edu/ yixin

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Worksheet: lecture 6

Lecture 6: First-order PDEs with variable coefficients

Yixin Guo

This worksheet contains maple code for plotting solutions of covection equation u_t+ xu_x + tu=0 ,u(x,0)=sin x.

restart:with(plots):

Warning, the name changecoords has been redefined

U:=(x,t)->exp(-0.5*t^2)*sin(x*exp(-t));

U := x, t /eK0.5 t2 sin x eKt

animate(plot, [U(x,t),x=-10..10],t=0..5,frames=100,numpoints=1000);

xK10 K5 0 5 10

K1

K0.5

0.5

1t = 0.

plot3d U x, t , x =K15 ..15, t = 0 ..5, grid = 50, 50 , orientation =K50, 40 , shading = xy, axes = boxed ;

-1.0-0.50.00.5

-15

1.05

-5 4

3x

5 2 t

1

15 0

HMC Math 323 Yixin Guo, Emek KoseSpring, 2006

Math 323 Homework Set 4

Due on Friday, May 19th, 2006

(30pt) D6: Consider the solution to the Cauchy problem for the diffusion equation,

ut = kuxx, −∞ < x < ∞, 0 < t,

u(x, 0) = φ(x), −∞ < x < ∞

where the initial condition φ(x) is non-negative and of finite mass,

φ(x) ≥ 0,∫ ∞

−∞φ(x) dx = M0 < ∞.

a) Show the heat kernel,

G(x, t) =1√

4πkte−x2/(4kt),

has unit mass, that is ∫ ∞

−∞G(x, t) dx = 1,

for t > 0.

b) Show, using the Poisson integral representation of the solution,

u(x, t) =∫ ∞

−∞φ(y)G(x− y, t) dy

that the mass of the solution is conserved, that is∫ ∞

−∞u(x, t) dx = M0

for t > 0.

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40pt D6: We have verified (homework # 1 problem A4) that the Green’s functionG(x, t) = 1√

4πkte−x2/(4kt) is a solution to the Cauchy problem for the diffusion equa-

tion:ut = kuxx, −∞ < x < ∞, 0 < t,

u(x, 0) = φ(x), −∞ < x < ∞

(a). Show that

u(x, t) = CG(x− x, t + t) =C√

4kπ(t + t)e−(x−x)2/4k(t+t)

where t, x and C are constants, also solves the Cauchy problem for the diffusion equa-tion.

(b). If x = 0, t = 1/4k and C =√

4kπ t in w(x, t) = C√4kπ(t+t)

e−(x−x)2/4k(t+t),

show that w(x, t) is the solution of the Cauchy problem for the diffusion equation withu(x, 0) = e−x2

.(c). Show that wx(x, t) is a solution to the diffusion equation if w(x, t) solves the

diffusion equation.(d). Solve the Cauchy problem for the diffusion equation with u(x, 0) = xe−x2

.

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30pt Using Duhamel’s principle, what is the solution of the IBVP

ut = α2uxx −∞ < x < ∞, 0 < tu(0, t) = 0 0 < t < ∞u(1, t) = sin tu(x, 0) = 0 0 ≤ x ≤ 1

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MATH 323: Project guideline

Drexel University Spring quarter 2005-2006

Due dates: Hand in written reports and give presentations. June 5 Monday; June 6 Tuesday;

June 7 Wednesday.

Prepared by Yixin Guo

In this assignment, you will work on a project related to tumor growth. You are given papers

from both biology and modeling literature. You write a report and create a presentation that

summarizes your studies. Following are the guidelines:

• It is recommended that you will be working in teams of two. You will choose your own

team-mate. If there is a special situation that you prefer to work individually, please talk

to the instructor first.

• You are responsible to read the provided papers on modeling brain tumor growth and

the immunotherapy. You should also keep in mind that you are not confined to these

papers provided to you. You are encouraged to conduct your own literature search. You can

use any other papers as long as they stay in the same topic. You need to develop creative

ideas on modeling brain tumor growth with immunotherapy. Do not worry about

if your ideas or your modeling approaches are right or wrong. You are encouraged to come

up with something novel and interesting. If you could not come up with your own original

ideas, at least do a thorough study on both the biology and modeling literature.

• This assignment includes two parts, written report and in-class 20 minute presentation.

Written report should not be a copy of the presentation slides. It must contains more

details on how you develop your project. In your written report, you need to write about

your study in details about the following aspects:

– What are the questions you want to ask?

– What modeling approach is being used in the provided papers?

– What can you do to revise the PDE model given in the paper to accommodate the

immunotherapy treatment? Or if you want to come up with a completely new model,

feel free to do so.

– How do you formulate your model or the model given in the paper?

– How is the problem solved in your model or the model given int the paper (analysis,

computation, verification)?

– What is the answer to the question you want to ask?

– What can be done further in the future if you would continue your study (future

direction)?

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• Your presentation should be a summary of the essence of your creative study. You may

prepare your presentation in either computer or overhead slides. We do not encourage you

to do a board presentation this time.

• Total time for the presentation is 20 minutes. You should give 5 minutes at least for

questions from your peers. Every student needs to be ready to ask in-class questions. Each

one of you will be individually responsible for thinking up questions to ask. You do not

have to turn in the the questions. However, during the student presentations, each student

in the audience will be required to ask at least one question of the presenting team. Asking

questions will be part of the evaluation of this project. So be prepared and pay attention.

Guideline for presentation:

• You may give a presentation on laptop using power point or any format you are comfortable

with, such as overhead projector. No matter how you present, you must stop when your

presentation time is up because we have others lined up and we only have limited amount

of time to use. For computer presentation, we will centralize all the slides on instructor’s

laptop so that we will not have to waste our time to setup for each presentation. If you

decide to do a laptop presentation, you must send your slides to the course instructor Yixin

Guo (yixin@math.drexel.edu) before you present in class.

• You must show your collaboration as a team and take equal responsibility during the

presentation.

Guideline for evaluation:

• The evaluation will be based on three parts: the written assignment, oral presentation and

asking questions during each presentation. The written assignment will worth 55% of the

total credit. It will be graded based on creativity. The presentation will worth 35% of

the total credit. Another 10% goes to the questions you are required to ask during each

presentation. We do not evaluate what questions you ask. As long as you ask questions,

you receive the credit.

• The detailed evaluation checklist for presentation is attached.

• Students will have their own comment sheets. They can express, anonymously, their con-

structive opinions about their fellow classmates’ performance.

• In the end, the course instructor and the recitation instructor will communicate with each

team on their performance, including grades and students comments.

• Team-mates will be evaluated together and receive the same grade.

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List of papers

Modeling papers:

• A quantitative model for differential motility of gliomas in grey and white matter by K. R.

Swanson, E. C. Alvord Jr, and J. D. Murray

• Virtual and real brain tumors: using mathematical modeling to quantify glioma growth

and invasion by Kristin R. Swanson, Carly Bridge, J. D. Murray and Ellsworth C. Alvord

Jr.

Biology papers:

• Results of a phase I clinical trial of vaccination of glioma patients with fusions of dendritic

and glioma cells by Tetsuro Kikuchi, Yasuharu Akasaki, Masaki Irie Sadamu Homma,

Toshiaki Abe, Tsuneya Ohno

• Dendritic cell immunotherapy for brain Tumors by Stephane Vandenabeele, Linda M. Liau,

and David Ashley

Other papers that are not required to read but might be useful:

• Determining control parameters for dendritic cell-cytotoxic T lymphocyte interaction by

Burkhard Ludewig, Phillippe Krebs, Tobias Junt, Helen Metters, Neville J. Ford, Roy M .

Anderson and Gennady Bocharov

• Dynamic response of cancer under the influence of immunological activity and therapy by

Harold P de Cladar, Jorge A. Gonzalez

. . .

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Service

NSF panel review June 2-3, 2008.

Referee for Journals

SIAM Journal of Applied Mathematics (4 times), SIAM Journal on Applied Dynamical Systems (3 times), Journal of Computational Neuroscience (twice), Journal of Mathematical Biology (once), Physica D (twice), Journal of Dynamics and Differential Equations (once), Dynamics of Partial Differential Equations (once).

Departmental Committee

Graduate committee, 2011-2012 full academic year.

Teaching Faculty promotion committee, 2010-2011 full academic year.

Graduate committee, 2008-2009 full academic year.

Computing committee: 2007-2008 full academic year.

Hiring committee: 2006-2007 full academic year.

Graduate committee: 2006-2007 fall quarter.

University Service

Serve as a judge to evaluate student posters at the CoAS (College of Arts and Science) research day on April 3 2012.

Serve as a judge to evaluate student posters at the Biomed Talent and Technology Showcase, November 2, 2010.

Open house spring 2006 and 2008, Meet and Greet (College of Art and Sciences) 2008, Convocation.