### Selected Publications

• Expected 2018

#### Tensor amplification and decomposition and their applications in biomedicine

N. Tokcan, H. Derksen , in preparation

• Expected 2018

#### A novel tensor similarity score for the classification of cardiac index

N. Tokcan, H. Derksen, G. Jonathan, L. Hernandez, K. Najarian, in preparation

• Expected 2019

#### Low rank approximation of symmetric tensors over different fields

N. Tokcan, B. Reznick, in preparation

• January 2017

#### Binary forms with three different relative ranks

B. Reznick and N. Tokcan, accepted for publication in Proceedings of the American Mathematical Society

• March 2017

#### On the Waring rank of binary forms

N. Tokcan, acctepted for publication in Linear Algebra and Its Applications

• March 2017

#### Newton Polytopes in Algebraic Combinatorics

C. Monical, N. Tokcan and A. Yong, available on arXiv

• June 2017

#### WCET Derivation under Single Core Equivalence with Explicit Memory Budget Assignment

R. Mancuso, R. Pellizzoni, N. Tokcan and M. Caccamo, in In Proceedings of the 29th Euromicro Conference on Real-Time Systems (ECRTS 2017), Dubrovnik, Croatia. To Appear

### Affiliations

• Faculty 2017 - Present

Postdoctoral Assistant Professor in Mathematics, and Computational Medicine and Bioinformatics

University of Michigan, Ann Arbor

• Ph.D. 2012 - 2017

Ph.D. in Mathematics

University of Illinois at Urbana-Champaign Department of Mathematics

• M.S. 2011 - 2012

Master of Science in Mathematics

University of Illinois at Urbana-Champaign Department of Mathematics

• B.S. 2005 - 2009

Bachelor of Science in Mathematics

Cukurova University, Turkey

### Honors, Awards and Certificates

• 2018
Michigan Precision Health Scholars Award - 2 years, $80,000, UMICH The University of Michigan’s Precision Health Scholar Award program grants up to$80,000 for one year per recipient with the potential for renewal. The awards program aims to expand the field of precision health by cultivating a cohort of promising early-career researchers in the field and spark new collaborative research avenues by engaging early-career investigators with tools and data to support their work.
• 2017
Irving Reiner Memorial Award in Algebra, UIUC
The Irving Reiner Memorial Award was established in memory of Professor Irving Reiner by his family in 1988, with the support also of colleagues and friends. Professor Reiner, a long-time member of the University of Illinois Department of Mathematics, was a leader in the field of integral representation theory. The Reiner Prize is awarded each spring to one or more graduate students in recognition of outstanding scholastic achievement in the field of algebra.
• Spring 2017
Campus Research Board Award, UIUC
• Fall 2016
AMS Graduate Student Travel Grant to the Joint Mathematics Meetings
• Fall 2016
Nominated for Departmental TA Instructional Award, UIUC
• 2016
Graduate Teaching Certificate, Center for Innovation in Teaching and Learning, UIUC
• Fall 2015
Finalist for Departmental TA Instructional Award, UIUC
• From 2014
List of Teachers Ranked as Excellent by Students, UIUC
I was included in the List of Teachers Ranked as Excellent by Students in UIUC in the following terms: Fall 2014, Spring 2015, Summer 2015, Fall 2015, and Spring 2016. In Spring 2016 I was given outstanding rating
• 2011
Council of Higher Education of Turkey Scholarship
It is a prestigious scholarship given by the Faculty Development of Higher Institution Program of Ankara Gazi University in Turkey. The acceptance is conditional on accepting a faculty position at the issuing institution.
• 2009 - Pres.
Republic of Turkey Ministry of National Education (MEB) Scholarship
This is a scholarship offered by the Turkey Ministry of National Education that provides financial support for pursuing graduate education in the United States. The completion of this program is followed by an offer of a faculty position in Turkey.
• 2009
TUBITAK Scholarship
The TUBITAK (The Scientific and Technological Research Council of Turkey) is a scholarship that provides support for graduate studies and research in Turkey
• 2009
Top Student of the Faculty of Science and Letters Award - Cukurova University
• 2009
Top Student of the Mathematics Department Award - Cukurova University
• 2006 - 2009
Cukurova University High Honor Award

### Filter by type:

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#### Binary forms with three different relative ranks

Bruce Reznick and Neriman Tokcan
Journal Paper accepted for publication in Proceedings of the American Mathematical Society, January 2017

#### Abstract

Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \subseteq \mathbb C$. The $K$-rank of $f$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d \ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fields. The $K$-rank of a form $f$ (such as $x^3y^2$) may depend on whether -1 is a sum of two squares in $K$.

#### On the Waring rank of binary forms

Neriman Tokcan
Journal Paper accepted for publication in Linear Algebra and Its Applications, March 2017

#### Abstract

The $K$-rank of a binary form $f$ in $K[x,y],~K\subseteq \mathbb{C},$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We completely classify binary forms of Waring rank 3.

#### Newton Polytopes in Algebraic Combinatorics

Cara Monical, Neriman Tokcan, Alexander Yong
Journal Paper in submission to Selecta Mathematica, March 2017

#### Abstract

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others. Our principal construction is the Schubitope. For any subset of $[n]$ x $[n]$, we describe it by linear inequalities. This generalized permutahedron conjecturally has positive Ehrhart polynomial. We conjecture it describes the Newton polytope of Schubert and key polynomials. We also define dominance order on permutations and study its poset-theoretic properties.

#### WCET Derivation under Single Core Equivalence with Explicit Memory Budget Assignment

Renato Mancuso, Rodolfo Pellizzoni, Neriman Tokcan, Marco Caccamo
Conference Paper in Proceedings of the 29th Euromicro Conference on Real-Time Systems (ECRTS 2017), Dubrovnik, Croatia, June 2017

#### Abstract

In the last decade there has been a steady uptrend in the popularity of embedded multi-core platforms. This represents a turning point in the theory and implementation of real-time systems. From a real-time standpoint, however, the extensive sharing of hardware resources (e.g. caches, DRAM subsystem, I/O channels) represents a major source of unpredictability. Budget-based memory regulation (throttling) has been extensively studied to enforce a strict partitioning of the DRAM subsystem’s bandwidth. The common approach to analyze a task under memory bandwidth regulation is to consider the budget of the core where the task is executing, and assume the worst-case about the remaining cores’ budgets.
In this work, we propose a novel analysis strategy to derive the WCET of a task under memory bandwidth regulation that takes into account the exact distribution of memory budgets to cores. In this sense, the proposed analysis represents a generalization of approaches that consider (i) even budget distribution across cores; and (ii) uneven but unknown (except for the core under analysis) budget assignment. By exploiting the additional piece of information, we show that it is possible to derive a more accurate WCET estimation. Our evaluations highlight that the proposed technique can reduce overestimation by 30% in average, and up to 60%, compared to the state of the art.

### Teaching History

• Spring 2016

#### Merit Workshop for Partial Differential Equations

Workshop for undergraduate students with high academic potential who are members of groups, such as ethnic minorities and women, who tend to be underrepresented in STEM. The workshop involves challenging problems to encourage critical thinking and is designed around in-class activities to promote class discussion and active participation. An example of a class activity performed in the workshop is available below. Sample worksheets, solutions and lecture notes are also available/

• Fall 2015

#### MATH 221 - Calculus I

First course in calculus and analytic geometry for students with some calculus background; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.

• Summer 2015

#### MATH 220 - Calculus (Primary Instructor)

First course in calculus and analytic geometry; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.

Class material is available here.

• Spring 2015

#### MATH 234 - Calculus for Business I

A calculus course intended for those studying business, economics, or other related business majors. The following topics are presented with applications in the business world: functions, graphs, limits, exponential and logarithmic functions, differentiation, integration, techniques and applications of integration, partial derivatives, optimization, and the calculus of several variables.

• Fall 2014

#### MATH 221 - Calculus I

First course in calculus and analytic geometry for students with some calculus background; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.

• Spring 2014

#### MATH 234 - Calculus for Business I

A calculus course intended for those studying business, economics, or other related business majors. The following topics are presented with applications in the business world: functions, graphs, limits, exponential and logarithmic functions, differentiation, integration, techniques and applications of integration, partial derivatives, optimization, and the calculus of several variables.

• Fall 2013

#### MATH 220 - Calculus

First course in calculus and analytic geometry; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.

## At My Office

You can find me at my office located at University of Michigan, Ann Arbor. The address is: 2074 East Hall 530 Church Street Ann Arbor, MI 48109-1043 - Room 4863

You may consider sending an email to arrange an appointment.

## General Course Information

Lectures: 243 Atgeld Hall

Time: Mon, Tue, Wed, Thurs, Fri - 10 AM to 11.40 AM

Instructor: Neriman Tokcan

Email: tokcan2@illinois.edu

TO BE DEFINED

Course Information

Emergency Information

## Week 1 (click to expand)

### Assignment 1

Please read the section 1.1 and 1.2 . In 1.1 do #4, 7, 8, 25, 34, 41, 47, 51, 54, 71, 73, 77. You should complete this before tomorrow's lecture.The sections are found in the textbook (same aseBook) and the homework is found in the exercises at the end of each section. Homework is not to turn in . You can check your work since solutions are posted on the compass as '' Solutions Chapter 1''.

### Lecture 1

Today we started our lecture with the question ''What is the Calculus?''. We answered this question asit is the study of the change. We will mainly look at how functions change. In class we answered the following questions '' Why do we define functions?, What are the real world examples of functions?''. We had a class activity '' Find The Mystery Rule of My Function''. We discussed different ways to represent a function and we gave some simple functions; constant function, identity function, linear function, quadratic function, piecewise defined functions. We ended our discussion with the examples of odd functions, even functions,increasing-decreasing functions.

### Assignment 2

Please read section 1.3 and Appendix D (Trigonometry). In section 1.2 do  #5,10,16,18. In 1.3 do #3,9,10,18.

### Assignment 4

In Appendix D do #29,30,35,36,37,38,65,67.You will need calculator for #35-38. In section #1.5 do 2,4,11,12,13,14,15,16,17,19,20,21,22,25,29,30.

### Assignment 5 (06/19)

In section #1.6, please do #5, 7, 9, 10, 15, 17, 19, 21–26, 35–41, 51–54, 57, 58.  Read Sections 2.1 and 2.2.

## Week 2 (click to expand)

### Practice Exams for Test 1

Please check the previous exams and solutions. These exams and solutions were written by Bob Murphy. Exam will cover section 1 and section 2. We are done with Chapter 1, so you can solve some part of these questions. Your exam will be similar to these exams.

### Trigonometry Worksheet

I attached the Trigonometry Worksheet, we discussed it last week. This worksheet and solutions were written by Bob Murphy, UIUC Mathematics Instructor.

### Lecture 5 (06/22)

Today we mostly solved problems about logarithm, exponential functions and inverse functions. I didn't include exercises in the lecture notes. We had Quiz 3 which covers these topics( You can find solutionsonline ). In the second part of the lecture, we discussed inverse trigonometric functions.

### Secant Line

Today we got a nice question in the class: Where does the name of the ''secant line'' come from ? Secant is coming from the Latin word ''secare'' which means ''cut''.Secant line is passing through two points of the curve, it is actually cutting the curve.

### Assignment 6 (06/23)

In Section 2.1 do #5. In Section 2.2 do #4, 7, 8, 11, 15, 17, 23, 24, 29, 31, 32, 39, 41. Please read Section 2.3 and Section 2.5. Math majors or those who want a better understanding of proof techniques for limits should read section 2.4.

### Lecture 6 (06/23)

Today, we started with discussing ''what is the tangent line to a curve at a given point?''. We said tangent is comes from the Latin word ''tangens'' whichmeans ''touching''. Tangent line of a curve is the line that touches the curve only one specific point. We tried to find the slope of the tangent line of a curve, we approximated this slope by using the slope of secant lines. We answered these following questions: ''What is the velocity of an object?'' and ''What is the difference between velocity and speed?''. We solved problems about ''average and instantaneous velocity''. We gave definition of the limit and tried to guess limit of some functions. One-sided limits were also given in the lecture. We said if the right-hand side limit and left-hand side limit are not equal to each other, then limit doesn't exist. We continued our lecture by discussing infinite limits and vertical asymptotes. We concluded our lecture by giving some simple Limit Laws.

### Lecture 7 (06/24)

Today we started our lecture by Solving Fall 2009 Test 1 for Math 220. Then we went over some homework problems. We gave basic limit rules and gave the ''direct  substitution property'' which holds for polynomials and rational functions. We explained how to deal with the limit of rational functions if we end up with 0/0 when we make substitution. We concluded our lecture with squeeze theorem (my favorite theorem for limit ). We said we mainly use it when it is hard to get limit by applying limit rules. We said functions with sine and cosine are good candidates since we know that sine and cosine are bounded.

### Assignment 7 (06/24)

Please do #11, 13, 15, 17, 18, 20, 25, 26, 37, 39 in Section 2.3. Please read section 2.5, 2.6.

### Lecture 8 and 9 (24-25/06)

We skipped some parts of Section 2.6,2.7,2.8. We will come back these sections after Exam 1.

### Assignment 8 (06/26)

Read section 2.7, 2.8. In section 2.5 do #20,45,46. In section 2.6 #8,15,21,24,25,29,30,33,41,43. In 2.8 do # 21-31.

### Exam 1 Syllabus

Today compass is not friendly with me. I tried to post it several times. Hope you can get it now.

## Week 3 (click to expand)

### Assignment 9 (07/01)

We are done with Chapter 2. Now you should be able to solve problems from Section 2.5,2.6,2.7,2.8. I assigned them partially and skipped some of them. Please make sure that you are done with  #20, 45, 49, 51, 53 in Section 2.5,  #8, 15, 21, 24, 25, 29, 30, 33, 41, 43 in Section 2.6, #5, 6, 7, 8, 9, 10, 13, 14, 27, 28, 29, 30, 31, 32 in 2.7, #4, 5, 6, 12, 16, 17, 18, 21, 23, 25, 27, 29 in 2.8.Reading : Please read 3.1,3.2,3.3

### Worksheet 3

You should be able to solve problems 1-13. Please solve them during the weekend, so that we can discuss on Monday. We will cover trigonometric derivatives, chain rule and applications of rate of change. When we are done with these topics you will be able to solve Worksheet completely.

### Lecture 12 (07/02)

This lecture notes are longer than what we did today. We stopped at the finding derivative of ''y=a^x''. You can go ahead and read other part of the lecture notes, so that you can have an idea about what we are going to do next week.

### Assignment 10 (07/02)

In 3.1 do #3–30, 33, 35, 47, 51, 53.You should use the fact that (e^x)'=e^x and slope of a horizontal line is 0.  Other than that, today's lecture notes gives you enough information to solve these problems.Please read Section 3.2,3.3,3.4.

## Week 4 (click to expand)

### Assignment 11 (07/06)

In section 3.2 do the odd problems from #3–33.In section 3.3 do the odd problems from #1–23.Please read 3.4,3.5,3.6.

### Assignment 12 (07/07)

In section 3.4 do the odd problems from #7–55. Read Section 3.6,3.7,3.8. Please check examples in 3.5(Implicit Differentiation). There are good solved examples. Tomorrow we are gonna finish Worksheet.

### Assignment 13 (07/08)

In section 3.5 do #5, 7, 9, 11, 13, 15, 17, 19, 29, 30, 31, 32, 49, 50, 51, 57. Read Section 3.6,3.7,3.8,3.9.

### Worksheet

Please start working on this Worksheet. It will help you to revise derivative rules. Ask me if you have questions in office hours or right after class. Once, we covered enough lectures ( in orderto catch up schedule of Exam 2), we will discuss this worksheet in the class.

### Practice Exams

Please go to    http://www.math.illinois.edu/~murphyrf/teaching/M220/  . You can find old tests by clicking on Tests 2. There are also solutions. We will keep same format and same content for  Exam 2.

### Assignment 14 (07/10)

In section 3.6 do #3, 5, 6, 11, 13, 19, 31, 34, 39, 43, 45. In section 3.7 do #7, 8, 9, 10. In section 3.8 do #3, 4, 8, 9, 10, 11, 12. Please read 3.9,3.10,3.11.

## Week 5 (click to expand)

### Assignment 15 (07/13)

In section 3.9 do #6, 10, 13, 15, 20, 22, 24, 27, 28, 30, 31, 38, 41. In section 4.1 do #16–25, 30, 41, 43, 49–60, 63, 75. Please read 4.1,4.3,4.7.

### Quiz 8

Tomorrow we will have a 15 minutes quiz which covers following topics: 1) Logarithmic differentiation2) Exponential growth or decay3) Derivative of inverse trigonometric functions + Chain Rule

### Practice Exams

Please solve Spring 2015 and Fall 2014 Math 220 UIUC Exams. It would be better if you can check previous exams as well. They are the best materials to work on for Test 2.

### Quiz 8

It is a take home quiz, you should turn it it tomorrow at the beginning of the lecture. For the problem 3, you should start with setting the equationdy/dx=(1/4).y  and find the correct exponential formula for it.

### Proof of L'Hospital's Rule

We skipped the section 4.2 which includes The Mean Value Theorem. If we want to prove L'Hospital's Rule, we should use this theorem. I will come back this section after 2 weeks, so that we will be able to prove L'Hospital's Rule.

### Assignment 16 (07/14)

In section 4.1 do #16–25, 30, 41, 43, 49–60, 63, 75. In section 4.3 do #10, 13, 17, 18, 33, 39, 43, 46, 48, 53, 86.  Read 4.4 and 4,7.

### Lecture 18 (07/15)

Today we finished Indeterminate forms. We solved Quiz 8, We solved Optimization questions ( you can find them at Exam 2 Preparation Questions ) and We solved exponential growth questions ( you can find in the Exam 2 Preparation Questions).

### Assignment 17 (07/15)

In section 4.4 do #7, 11, 17, 18, 19, 21, 25, 33, 41, 45, 49, 50, 55, 57, 61, 62, 67.  In section 4.7 do #5, 6, 13, 14, 19, 21, 32, 34, 35, 38, 49, 54. Read section 4.9.

### Assignment 18 (07/16)

Please solve practice exams Fall 2014 and Spring 2015. Tomorrow we will discuss them in class.

## Week 6 (click to expand)

### Lecture Notes

Today we mentioned aboutantiderivatives  (4.9) and Area (5.1). Tomorrow I want to introduce definite integrals (5.2). Please check worksheet 5. Tomorrow in the second part of the lecture, we will go over this worksheet. It would be better, if you check questions before coming to class. These lecture  notes also cover tomorrow's lecture. You can print it out to follow the lecture tomorrow.

### Assignment 19 (07/ 210

Your homework is to read section 4.9 and do #1–17, 20–22, 25–33, 41–43, 65, 69, 73–75. Read section 5.1 (Areas and Distances) very carefully. Do #3, 4, 13, 14, 15, 18, 20, 30 from section 5.1. Note that problems 65, 69, 73, 74 and 75 from section 4.9 are more difficult than our previous problems concerning position, velocity and acceleration.Please read Section 5.2.

### Assignment 20 (07/22)

Please read Section 5.2 and 5.3. Make sure that you are done with your homeworks from 4.9 and 5.1. Solve 1,2,3,4,6,7,8 from the Worksheet. Make sure that you have a good understanding about them.

### Lecture 20 (07/24)

These are lecture notes for tomorrow. These notes need revising. There is a little mistake, I will post revised version on Monday.

### Assignment 21 (07/25)

n section 5.2 do #2, 11, 18, 21, 22, 29, 33, 36, 37, 41, 48, 49, 52, 53, 55, 57, 59.In section 5.3 do #23, 24, 28, 31, 32, 33, 35, 39

## Week 7 (click to expand)

### Assignment 22 (07/27)

In section 5.4 do #3, 5, 6, 15, 16, 17, 18, 27, 31, 37, 43, 53, 54, 64. Read section 5.5 and 4.2.Please revise 5.2 and 5.3. Make sure that you are fine with FTC1 and FTC2. In your textbook in section 5.4 there are applications of definite integral. I would recommend you to take a look at them.

### Assignment 23 (07/28)

In section 5.5 do #8, 16, 17, 18, 20, 21, 22, 23, 25, 28, 32, 39, 40, 41, 44, 46, 48, 54, 57, 59, 60, 61, 65, 66, 67, 69, 81, 82. Read Section 5.2.

### Assignment 24 (07/29)

Read section 6.1 (Areas Between Curves). In section 6.1 do #1, 8, 11, 12, 13, 17, 23, 25, 27, 29, 50, 51. Read Section 6.2 and 6.3. We will have a problem about the volume in the midterm 3. Lecture notes areonline, I would recommend you to take a look at them before coming to class.

### Assignment 25 (07/30)

In section 6.2 do #2, 6, 7, 9, 12, 14, 16, 17, 33, 55, 56, 58. Read Section 6.2 and section 6.3.

### Volume Methods

I found these nice notes. You can check them. Today we discussed disk/washer method and cylindrical shell method1)If you are taking cross section perpendicular to the axis of rotation= Disk/Washer Method ( Cross Section is just a circle or a washer with area= \pi (outer radius)^2- \pi(inner radius)^2 so you should integrate area over boundaries.2)If you are taking cross section horizontal to the axis of rotation= Shell Method. ( Cross section is a cylinder with the surface area= 2\pi *r* height )so you should integrate surface area over boundaries.

### Assignment 26 (07/31)

Read Section 6.5, 7.2 and 7.4. In section 6.5 do #1, 2, 4, 5, 7, 9, 10, 13, 14, 17.In section 7.2 (Trigonometric Integrals) do #1–8, 12, 14, 15, 17–31 and 34. Many of the integrals in section 7.2 require the use of some basic trigonometric identities along with substitution. In section  7.4 solve 7-15.Read 6.2 and 6.3.In section 6.2 do #2, 6, 7, 9, 12, 14, 16, 17, 33, 55, 56, 58 and in section 6.3 do #3, 5, 9, 12, 14, 15, 17, 19, 20.