# Uncategorised

$2^2$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$\frac{x^2}{a^2}+\frac$

$\frac{x^2}{a^2}+\frac$

Mathematics Teachers' Circles consist of groups of mathematicians and K-12 teachers who meet regularly to discuss interesting mathematical problems.

The Rutgers Math Teachers’ Circle (RUMTC) was founded in 2014 and is a member of the national-level Math Teachers’ Circle Network (http://www.mathteacherscircle.org/), as well as of the National Association of Math Circles (https://www.mathcircles.org/). Our circle is mainly for middle school teachers, but we welcome high school and upper elementary teachers as well.

Mathematics Teachers' Circles consist of groups of mathematicians and K-12 teachers who meet regularly to discuss interesting mathematical problems.

The Rutgers Math Teachers’ Circle (RUMTC) was founded in 2014 and is a member of the national-level Math Teachers’ Circle Network (http://www.mathteacherscircle.org/), as well as of the National Association of Math Circles (https://www.mathcircles.org/). Our circle is mainly for middle school teachers, but we welcome high school and upper elementary teachers as well.

Each semester, the Mathematics Department hires undergraduate and graduate students in good standing to grade homework assignemtns, Maple labs, and Matlabs for several undergraduate and a few graduate courses.

Graders are typically given a few days to grade an assignment, so the working hours of graders are flexible. The pay for all courses, except Math 244 and Math 251, is \$11-13 per hour per course (depending on the level); the work load is a total of 3-4 hours per week, on average. The time commitment for each Math 244 or Math 251 is 3 hours per week, on average.

Before applying, please ask yourself if you will realistically have time to take on this commitment. It is extremely important that all deadlines the professors set for completing the grading are met. If you are interested in being considered for this position, please complete the entire form below. Please note there is often considerable interest in these grading positions, so not every applicant could be hired. To be considered for a course, an applicant needs to have successfully completed the course (with an A grade) at Rutgers, and have a GPA of at least 3.4.

This Spring, we will be hiring graders for the following courses only:
103, 104, 106, 152H(only)
244, 250, 251, 252, 285, 292
300, 311, 312, 336, 338, 348, 350, 351, 354, 356, 357, 373
403, 412, 421, 423, 428, 432, 437, 452, 477, 478, 481, 487
528

We will NOT hire graders for any other courses. Please do not request them.

The Spring 2020 application

To apply for a Grader position for any course in Spring 2020, please complete the Spring 2020 Grader application.

=============

** If you have any questions, please email Professor Mavrea at grader_coordinator@math.rutgers.edu.

*** Due to the large number of applications, only applicants who are selected will be contacted (via email).

## 01:640:350:H Linear Algebra Section

 H1 12832 Woodward, Christopher Lecture MW4 0140P-0300 HLL-009 BUS

## This course is a proof-based continuation of Math 250, covering Abstract vector spaces and linear transformations, inner product spaces, diagonalization, and canonical forms.

Prerequisites:

CALC4, Math 250 and Math 300

Text: Linear Algebra (4th ed.), by Friedberg, Insel and Spence,
Prentice Hall, 2003   ISBN 0-13-008451-4.   For this section, any recent edition of the textbook should be sufficient.

• Class  MW4 0140P-0300 HLL-009 BUS
• Office Hours:  Tues 2-3 pm, Hill 726
• Contact Information:  e-mail ctw@math.rutgers.edu

The course is strongly based on Math 250. However, we'll work axiomatically, starting from the abstract notions of vector space and of linear transformation. Much of the homework and many of the exam and quiz problems will require you to write precise proofs, building on your proof-writing experience in Math 300. From this more abstract viewpoint, we'll be developing linear algebra far beyond Math 250, with new insight and new applications.

Class attendance is very important. A lot of what we do in class will involve collective participation.  We will cover the topics indicated in the syllabus below, but the dates that we cover some of the topics might be adjusted during the semester, depending on class discussion, etc. Such adjustments, along with the almost-weekly homework assignments, will be announced in class and also posted on this webpage, so be sure to check this webpage regularly.  Absences from a single class due to minor illnesses should be self-reported using the university system; for longer absences, students should email me with the situation.   I reserve the right to lower the course grade up to one full letter grade for poor attendance.

Make-ups for exams are generally not given; if a student has an extremely good reason (e.g. documented medical emergency) I may re-arrange the grading scheme to accomodate.

Problem sets are due on most Tuesdays. There are no problems due on the two midterm-exam Tuesdays.

Note that we will cover significant material from all the chapters in the book, Chapters 1-7.

Grading policy: First midterm exam: 100 points; Second midterm exam: 100 points; Problem sets and quizzes: 100 points; Final exam: 200 points (Total: 500 points).

#### Tentative Course Syllabus

WeekLecture dates Sections   topics
1 1/23 Chapter 1 Abstract vector spaces & subspaces
2 1/28,1/30  Chapter 1 Span of subsets, linear independence
3 2/4, 2/6 Chapter 1 Bases and dimension
4 2/11, 2/13 Chapter 2 Linear transformations
5 2/18, 2/20 Chapter 2 Change of basis, dual spaces
6 2/25, 2/27 Ch. 1-2  Review and Exam 1 (10/9)
7 3/4, 3/6 Chapter 3  Rank and Systems of Linear Equations
8 3/11, 3/13 Chapter 4  Determinants and their properties
9 3/25, 3/27 Chapter 5  Eigenvalues/eigenvectors
10 4/1, 4/3 Chapter 5  Diagonalization, Markov Chains
11 4/8, 4/10 Chapter 6  Inner Product spaces
12 4/15 Chapter 6  Unitary and Orthogonal operators
13 4/17, 4/22  Ch.3,4,5,6  Review and Exam 2 (4/22)
14 4/24, 4/29 Chapter 7  Orthogonal diagonalization
15 5/1, 5/6 Chapter 7  Jordan canonical form
17 5/14 (Tues) 12-3pm Final Exam HILL 009

Main 350 course page

#### Exam Dates

The exam dates are listed in the schedule above.  Any conflict (such as with a religious holiday) should be reported to me at the beginning of the semester, so that the exam may be re-scheduled.

### Special Accommodations

Students with disabilities requesting accommodations must follow the procedures outlined at https://ods.rutgers.edu/students/applying-for-services

All Rutgers students are expected to be familiar with and abide by the academic integrity policy. Violations of the policy are taken very seriously.  In particular, your work should be your own; you are responsible for properly crediting all help with the solution.

### Problem Sets

The Problem Sets are available in the assignments directory on the course Sakai site

Problem sets should be hand-written in reasonably clear writing, with an explanation of any assistance given.   Type-written assignments are allowable only by special arrangement (disability etc.)  Scans of problem sets may be submitted electronically in emergencies (illness or accident) by upload to Sakai.

Some basic writing guidelines are as follows.   All answers must be written in complete sentences; avoid starting each sentence with a symbol; ensure that each variable or notation is defined; number sentences or formulas as necessary so that you may refer back to them.  To prove a "for all x", usually begin with "Let x be a ...".  To prove an "there exists x" statement, you must construct a particular x satisfying the given property, so "Define x to be ...".  To prove a that property A implies property B, begin  with "Assume Property A...." Then deduce Property B.    Sets are equal if they have the same elements; functions are equal if they have the same values; to prove something does not satisfy a list of axioms; it suffices to show that one of the axioms fails.   On both problem sets and exams you may use properties in the text or class (referring to them by page or date) if they come before the problem you are solving in the development of the material.

#### Problem Sets from 2017

Problems in pdf.  Solutions in pdf.)

Problems in pdf. Solutions in pdf.)

Problems in pdf. Solutions in pdf.)

Problems in pdf.  Solutions in pdf.)

Practice problems in pdf for the first midterm. Last year's exam with Answers.

Problems in pdf.

Problem in pdf.   Selected Answers to PS5, PS6, PS7.

(Problems in pdf)

Problems in pdf

Problems in pdf.

Last year's second midterm with answers.

Answers and practice problems for the final.  Last year's final and solutions.

More review problems for the second midterm.

Pratice problems for the second exam:

(Problems in pdf)

#### Recommended Practice Problems (the problem sets from 2016)

Sept. 13 1.2 #17; 1.3 #19,23; 1.4 #11,13; 1.5 #9,15
Sept. 20 1.6 # 20,21,26,29; 1.7 #5,6
Sept. 27 2.1 #3,11,28; 2.2 #4; 2.3 #12; 2.4 #15,17
October 4 2.5 #3(d),7(a,b),13; 2.6 #5,10; Show that F[x]* ≅ F[[x]].
October 18 3.1 #6,12; 3.2 #5(b,d,h),17; 3.3 #8,10; 3.4 #8,15
If an nxn matrix A has each row sum 0, some Ax=b has no solution.
October 25 4.1 #10(a,c); 4.2 #23;  4.3 #12,22(c),25(c);  4.4 #6; 4.5 #11,12
Nov. 1 5.1 #3(b),20,21; 5.2 #4,9(a),12; Show that the cross product
induces an isomorphism between R³ and Λ²(R³).
Nov. 8 5.2 #18(a),21;  5.3 #2(d,f);   5.4 #6(a),13,19,25
Nov. 15 7.1 #3(b),9(a),13; 7.2 #3,14,19(a); 7.3 #13,14;
Find all 4x4 Jordan canonical forms of T satisfying T²=T³.
Dec. 13 6.1; #6,11,12,17;   6.2 #2a,6,11;   6.8 #4(a,c,d),11

Welcome to the Department of Mathematics at Rutgers University, part of the School of Arts and Sciences (SAS). This page describes programs sponsored by the Department of Mathematics for teachers of mathematics and for pre-college students with a strong interest in mathematics.

## 01:640:350:04 Linear Algebra Section

 04 11407 Woodward, Christopher Lecture TF2 1020 A - 1140 BE-250 LIV

## This course is a proof-based continuation of Math 250, covering Abstract vector spaces and linear transformations, inner product spaces, diagonalization, and canonical forms.

Prerequisites:

CALC4, Math 250 and Math 300

Text: Linear Algebra (4th ed.), by Friedberg, Insel and Spence,
Prentice Hall, 2003   ISBN 0-13-008451-4.   For this section 04, any recent edition of the textbook should be sufficient.

• Class  TF2 1020 A - 1140 BE-250 LIV
• Office Hours:  Monday 2:15-3:15pm, Hill 726
• Contact Information:  e-mail ctw@math.rutgers.edu

The course is strongly based on Math 250. However, we'll work axiomatically, starting from the abstract notions of vector space and of linear transformation. Much of the homework and many of the exam and quiz problems will require you to write precise proofs, building on your proof-writing experience in Math 300. From this more abstract viewpoint, we'll be developing linear algebra far beyond Math 250, with new insight and new applications.

Class attendance is very important. A lot of what we do in class will involve collective participation.  We will cover the topics indicated in the syllabus below, but the dates that we cover some of the topics might be adjusted during the semester, depending on class discussion, etc. Such adjustments, along with the almost-weekly homework assignments, will be announced in class and also posted on this webpage, so be sure to check this webpage regularly.  Absences from a single class due to minor illnesses should be self-reported using the university system; for longer absences, students should email me with the situation.   I reserve the right to lower the course grade up to one full letter grade for poor attendance.

Make-ups for exams are generally not given; if a student has an extremely good reason (e.g. documented medical emergency) I may re-arrange the grading scheme to accomodate.

Problem sets are due on most Tuesdays. There are no problems due on the two midterm-exam Tuesdays.

Note that we will cover significant material from all the chapters in the book, Chapters 1-7.

Grading policy: First midterm exam: 100 points; Second midterm exam: 100 points; Problem sets and quizzes: 100 points; Final exam: 200 points (Total: 500 points).

#### Tentative Course Syllabus

WeekLecture dates Sections   topics
1 9/4 (T)  Chapter 1 Abstract vector spaces & subspaces
2 9/7, 9/11  Chapter 1 Span of subsets, linear independence
3 9/14, 9/18 Chapter 1 Bases and dimension
4 9/21, 9/25 Chapter 2 Linear transformations
5 9/28, 10/2 Chapter 2 Change of basis, dual spaces
6 10/5, 10/9 Ch. 1-2  Review and Exam 1 (10/9)
7 10/12, 10/16 Chapter 3  Rank and Systems of Linear Equations
8 10/19, 10/23 Chapter 4  Determinants and their properties
9 10/26, 10/30 Chapter 5  Eigenvalues/eigenvectors
10 11/2, 11/6 Chapter 5  Diagonalization, Markov Chains
11 11/9, 11/13 Chapter 6  Inner Product spaces
12 11/16 Chapter 6  Unitary and Orthogonal operators
13 11/21, 11/27  Ch.3,4,5,7  Review and Exam 2 (11/27)
14 11/30, 12/4 Chapter 7  Orthogonal diagonalization
15 12/7, 12/11 Chapter 7  Jordan canonical form
17 12/21 (Fri) 8-11am Final Exam Location TBA

Main 350 course page

#### Exam Dates

The exam dates are listed in the schedule above.  Any conflict (such as with a religious holiday) should be reported to me at the beginning of the semester, so that the exam may be re-scheduled.

### Special Accommodations

Students with disabilities requesting accommodations must follow the procedures outlined at https://ods.rutgers.edu/students/applying-for-services

All Rutgers students are expected to be familiar with and abide by the academic integrity policy. Violations of the policy are taken very seriously.  In particular, your work should be your own; you are responsible for properly crediting all help with the solution.

### Problem Sets

The Problem Sets are available in the assignments directory on the course Sakai site

Problem sets should be hand-written in reasonably clear writing, with an explanation of any assistance given.   Type-written assignments are allowable only by special arrangement (disability etc.)  Scans of problem sets may be submitted electronically in emergencies (illness or accident) by upload to Sakai.

Some basic writing guidelines are as follows.  Please write in complete sentences; avoid starting each sentence with a symbol; ensure that each variable or notation is defined; number sentences or formulas as necessary so that you may refer back to them.  To prove a "for all x", usually begin with "Let x be a ...".  To prove an "there exists x" statement, you must construct a particular x satisfying the given property, so "Define x to be ...".  To prove a that property A implies property B, begin  with "Assume Property A...." Then deduce Property B.    Sets are equal if they have the same elements; functions are equal if they have the same values; to prove something does not satisfy a list of axioms; it suffices to show that one of the axioms fails.   On both problem sets and exams you may use properties in the text or class (referring to them by page or date) if they come before the problem you are solving in the development of the material.

#### Problem Sets from 2017

Problems in pdf.  Solutions in pdf.)

Problems in pdf. Solutions in pdf.)

Problems in pdf. Solutions in pdf.)

Problems in pdf.  Solutions in pdf.)

Practice problems in pdf for the first midterm. Last year's exam with Answers.

Problems in pdf.

Problem in pdf.   Selected Answers to PS5, PS6, PS7.

(Problems in pdf)

Problems in pdf

Problems in pdf.

Last year's second midterm with answers.

Answers and practice problems for the final.  Last year's final and solutions.

More review problems for the second midterm.

Pratice problems for the second exam:

(Problems in pdf)

#### Recommended Practice Problems (the problem sets from 2016)

Sept. 13 1.2 #17; 1.3 #19,23; 1.4 #11,13; 1.5 #9,15
Sept. 20 1.6 # 20,21,26,29; 1.7 #5,6
Sept. 27 2.1 #3,11,28; 2.2 #4; 2.3 #12; 2.4 #15,17
October 4 2.5 #3(d),7(a,b),13; 2.6 #5,10; Show that F[x]* ≅ F[[x]].
October 18 3.1 #6,12; 3.2 #5(b,d,h),17; 3.3 #8,10; 3.4 #8,15
If an nxn matrix A has each row sum 0, some Ax=b has no solution.
October 25 4.1 #10(a,c); 4.2 #23;  4.3 #12,22(c),25(c);  4.4 #6; 4.5 #11,12
Nov. 1 5.1 #3(b),20,21; 5.2 #4,9(a),12; Show that the cross product
induces an isomorphism between R³ and Λ²(R³).
Nov. 8 5.2 #18(a),21;  5.3 #2(d,f);   5.4 #6(a),13,19,25
Nov. 15 7.1 #3(b),9(a),13; 7.2 #3,14,19(a); 7.3 #13,14;
Find all 4x4 Jordan canonical forms of T satisfying T²=T³.
Dec. 13 6.1; #6,11,12,17;   6.2 #2a,6,11;   6.8 #4(a,c,d),11

Michael Weingart

Associate Teaching Professor of Mathematics
weingart [at] math [dot] rutgers [dot] edu

Fall 2018 Teaching:

Math 104:01 Introduction to Probability

Math 104:03 Introduction to Probability

# Courses

## 01:640:135 - Calculus I

Textbook:  For current textbook please refer to our Master Textbook List page

Math 135 provides an introduction to calculus. It is taken primarily by students interested in the biological sciences, business, economics, and pharmacy. Math 135 may be followed by Math 136.

There is another calculus sequence, Math 151-152-251, which is taken by students in the mathematical and physical sciences, engineering, and computer science. Although it is possible to take Math 152 after Math 135, this is not a recommended sequence. More importantly, the prerequisite for Math 251 is Math 152; Math 136 does not satifsy this prerequisite.

Students who may need to take Math 152 or 251 should start their study of calculus with Math 151, and students who decide after taking Math 135 that they may wish to take Math 251 should follow Math 135 with Math 152.

In addition to the standard 4-credit format of the course, a a 5-credit format has been used for some of the sections, but only the 4-credit format is now being offered.

# 01:640:135 Schedule of Sections

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Abstract:  The Bianchi-Egnell Stability Estimate is a stability estimate or quantitative version of the Sobolev Inequality – it states that the difference of terms in the Sobolev Inequality controls the distance of a given function from the manifold of extremals of the Sobolev Inequality with distance measured in the gradient square or $\dot{H}^1$norm. In this talk, we present an extension of the Bianchi-Egnell Stability Estimate to Bakry, Gentil, and Ledoux’s Generalization of the Sobolev Inequality to Continuous Dimensions. We also demonstrate a deep link between the Sobolev Inequality and a one-parameter family of sharp Gagliardo-Nirenberg (GN) inequalities and how this link can be used to derive a new stability estimate on the one-parameter family of sharp GN inequalities from our stability estimate on Bakry, Gentil, and Ledoux’s Generalization of the Sobolev Inequality to Continuous Dimensions.

Here's a $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$test equation.

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