thesis math definition

Writing Your Thesis

The thesis should be the heart of your graduate school career. It will certainly be the most involved and difficult thing you do while in grad school.

Of course, before writing the thesis, one needs to have research to report. To make things easier on yourself, it’s a good idea to record your results as you work. Don’t rely on your memory to save you when you need to write everything down in your thesis! While you needn’t have everything written in final draft, having a detailed account of your research progress is a great idea. When you start your research, you and your advisor should try to establish a goal for your thesis as soon as possible. Performing research without a goal can be very difficult and even more frustrating.

When one does mathematical research, one rarely knows exactly where they are going. Gaining mathematical intuition comes from lots of hard work, not simply being very smart. A tried and true method for doing research is to do lots of examples, and make simplifying assumptions when needed. Before you can prove a theorem, you need a conjecture; these aren’t going to just fall in your lap! The idea is that after seeing enough examples, one can make a general conjecture and then hopefully prove it.

It’s a good idea to find out who else in the community (both in and out of the department) thinks about your field. You may find it useful to contact these people from time to time. This serves multiple purposes: you’ll lessen the chance of duplicating someone else’s research; you’ll find multiple sources of advice. While your advisor will likely be the single biggest source of help in writing your thesis, they needn’t be your only source. Talking to many people about your work will give you several different perspectives on the same thing. Seeing the same thing in different ways can be invaluable in understanding something.

When you have enough results such that you and your advisor are satisfied, you need to organize your work into one coherent document. This can be a highly non-trivial task! Make sure that your problem is stated clearly, along with why it is important, and how you solved it. Your thesis shouldn’t simply be a list of definitions, theorems, and proofs; there should be quite a bit of prose to explain the mathematical ambiance of your work. What is the motivation for even thinking about this problem? The more people that find your research interesting, the better.

Please refer to this manual for guidelines on formatting your thesis:  http://grad.ucsd.edu/_files/academics/BlueBook%202017-18%20updated%204.13.18.pdf

Defending Your Thesis

Setting a time to defend your dissertation can be frustrating. Contact your committee members well in advance in order to check availability and schedule a date/time.

You would think that finding a time for 6 people to meet would be an easy task. However, it can be exceedingly difficult. You may need to be very flexible and accommodating in order to make things work. You may also need to be persistent about asking if you have a non-responsive committee member.

Please carefully review these guidelines regarding committee attendance:

Department  Policy on Graduate Examination  Format:

Effective Fall 2022, the default format of a graduate examination in the Mathematics Department is  in person , i.e.,  all the committee members and the student are physically present in the same room for a scheduled examination . (This is set by the Division of GEPA.) However, when an unexpected situation arises and affects a committee member’s ability to participate in the examination synchronously, and when the student agrees, a remote or hybrid examination is allowed and can be decided by the committee chair or co-chairs. The following guidelines should be followed to arrange a remote or hybrid, synchronous examination:

• In forming the committee, the student needs to provide different examination options, in person, remote, or hybrid, to potential faculty committee members, and based on the conversation, the student can decide whether or not they want the faculty member on their committee. If such conversation did not take place, and if an unexpected situation arises, the faculty committee member can request remote examination, and can be released from the committee duty should the student refuse the request.
• In general, the graduate student is not allowed to opt for a remote examination unless there are extenuating circumstances, such as illness, travel difficulties related to visa problems, or a graduation deadline. Under such circumstances, the committee chair can decide to reschedule an in-person examination, or have a remote or hybrid examination.
• According to the Division of GEPA, there must be sufficient expertise among present members to examine the student. If a committee member must be absent for the scheduled exam, it is permissible for one absent committee member to examine the candidate on a separate date. The committee chair, or one co-chair, must participate synchronously in the scheduled exam.

Make sure to inform the PhD staff advisor in advance if any of your committee members will not be physically present.

During this scheduling phase, you also want to schedule your “Preliminary Appointment” with Graduate Division:  https://gradforms.ucsd.edu/calendar/index.php  – this appointment is optional but highly recommended! The purpose of this appointment is for them to check the margins and the formatting of your dissertation. While the above information should get you through this part without any problem, sometimes there are minor issues that arise and must be confronted (for example, published work that shows up in your dissertation has some extra requirements associated to it). The meeting should last about 30 minutes and you’ll receive a couple questionnaires to complete before your final appointment. You will also be required to schedule a Final Appointment with Graduate Division – allow at least a few days between your defense and your final appointment in order to finalize department paperwork.

In addition, the following information is critical to you completing your thesis, defending it, and completing your PhD:

• The university requires that your committee members each have a good readable draft of your dissertation at least FOUR WEEKS before your final defense.
• It  is your responsibility  to make arrangements with each committee member for the date and time of your defense.  Room reservations should be made at the Front Desk (in person or email to  [email protected])
• The  Final Report  form must have the original signatures of all members of the doctoral committee; the  Final Report  must also be signed by the program chair. (The  Final Report  form is initiated by the graduate coordinator and signatures are obtained from each faculty member through DocuSign.). Proxy signatures are not accepted.
• After your examination, committee chair emails PhD staff advisor confirming the passing of the defense. PhD staff advisor prepares Final Report through DocuSign.
• The final version of the thesis must conform to procedures outlined in the " Preparation and Submission Manual for Doctoral Dissertations and Master's Theses "
• The student submits the final approved dissertation to the Graduate Division  at the final document review  (the  Final Report  form is routed electronically from the program’s graduate coordinator via DocuSign). Final approval and acceptance of the dissertation by the Dean of the Graduate Division (on behalf of the University Archivist and Graduate Council) represents the final step in the completion of all requirements for the doctoral degree.

A few other suggestions:

About a week before you defend, you should send an email to your committee to remind them that your defense is coming, and you might even want to send a day-before or day-of reminder.

You should discuss the details of your defense with your advisor, but it’s basically a 50-minute talk where you highlight the main results of your dissertation. The audience is usually your committee plus a few graduate students.

Once Graduate Division has signed off on your thesis, it is time to submit your thesis online to Proquest/UMI. When you do this, they give you an option to purchase bound copies of your thesis from them. This is not particularly appealing for three reasons:

• They are rather pricey, about $40-$60 per copy
• They will print it exactly as you submitted it, according to Graduate Division standards: double-spaced, 8.5×11, etc, which doesn’t make for an attractive book. (How many of the math books on your shelf are 8.5×11 double-spaced?)

Fortunately, another option is available: self-publishing services. Originally these were intended for authors who had written a book, but couldn’t find a publisher for it, so they’d have it printed at their own expense. Nowadays, there are online sites filling this market, where you submit your manuscript and design the book yourself through their site. They can print on demand, so there is no minimum number of copies to order, and they can be quite inexpensive. A former graduate student, Nate Eldredge, chose to go with Lulu, so this article will describe that service.

You can begin by creating an account on Lulu’s site, which is pretty self-explanatory. They have several different book types available. I decided to go with a 6×9 “casewrap hardcover”, which is a pretty standard size and style for a book. If you have a yellow Springer book on your shelf, that’s a pretty good facsimile of what we’re talking about here.

The main issue, then, is reformatting the thesis into a 6×9 format. Fortunately, LaTeX makes this pretty easy. Pretty much, you just need to swich from the UCSD thesis class to the standard LaTeX book class and make a few other changes. Here is a modified version of the UCSD thesis template, modified to fit this format. Nate put comments in various places indicating the relevant changes and choices he made. In several places he took advantage of the fact that he no longer had to conform to OGS’s awkward requirements to make the thesis more “book-like” and remove some things that wouldn’t appear in a book. It shouldn’t take you more than an hour or two to convert your thesis file, depending how fastidious you are. (If you don’t want to go to this trouble, Lulu will also print 8.5×11 books. You could use your existing PDF without change. It may not look as pretty, but it will still be cheaper than UMI.)

Note that you should check carefully for overfull \hbox’es when you compile the thesis, because changing the paper size may have caused things to run outside the margins or off the page. You may have to manually break up long equations or reword paragraphs. Also, the book class will insert several apparently blank pages; these relate to the fact that the book will be printed double-sided, and guarantee that certain things always appear on the left- or right-hand side of a spread. If you want a book-like effect, you should not try to defeat this.

Once you’ve generated an appropriate 6×9 PDF file and uploaded it to Lulu, you can design a cover for it. They have a couple of different interfaces. For his thesis, Nate created a pretty simple cover with a UCSDish blue color scheme, and the abstract and a graduation photo on the back cover.

When you are all finished, Lulu creates a page where you or anyone else can buy copies of the book. (You have the option of keeping this private, so that only people you share it with can find it.) Then you can buy as many copies as you want to keep or give away, and you can also send the link to your parents if they want to buy lots of copies for all the relatives. (In this case, Lulu’s “revenue” option may be useful, where you select an amount to add to the price of the book, which Lulu passes along to you after each sale. The page remains up indefinitely if you want more copies later.

If you want to see what a finished product looks like, Nate Eldredge’s thesis Lulu page is located at http://www.lulu.com/content/7559872.

The book turned out quite nice looking, with quality and appearance comparable to commercially published math books. And they were only $15.46 per copy (plus tax and shipping). Overall that is a vast improvement over UMI.

Also, Nate uploaded the template as a Lulu project. It can be found at http://www.lulu.com/content/7686303.

9500 Gilman Drive, La Jolla, CA 92093-0112

(858) 534-3590

Senior Thesis Guidelines

A senior thesis can form a valuable part of a student's experience in the  Mathematics Major . It is intended to allow students to cover significant areas of mathematics not covered in course work, or not covered there in sufficient depth. The work should be independent and creative. It can involve the solution of a serious mathematics problem, or it can be an expository work, or variants of these. Both the process of doing independent research and mathematics exposition, as well as the finished written product and optional oral presentation, can have a lasting positive impact on a student's educational and professional future.

Supervision

Supervision by a qualified member of the field of mathematics at Cornell is the normal requirement for a senior thesis. Other arrangements are possible, however, provided they are made with the assistance of the student's major advisor, and with the approval of the Mathematics Major Committee.

Finding a supervisor/Encouraging students.  

It should be emphasized that both the writing and the supervising of a senior thesis are optional activities, both for students and faculty. Students interested in doing this will need to find a suitable supervisor — perhaps with the aid of their major advisor or another faculty member whom they know. Advisors and other faculty who encounter students whom they think would benefit from this activity are invited to mention this option to them and assist them in finding a supervisor.

Standard venues for senior theses . 

One obvious way in which a senior thesis can be produced is through an independent research course (MATH 4900); another way is through an REU experience, either at Cornell or elsewhere. (If the REU work was accomplished or initiated elsewhere, a "local expert" will still be needed to supervise or "vouch for" the work as a senior thesis.) In yet a third way, a student may present a faculty member with a solution or partial solution to an interesting problem. In such cases, this could form the core of a senior thesis. Faculty are invited to encourage such work from their students.

Public Lecture

A public lecture in which the results of the senior thesis are presented is welcome but optional. This should be arranged by the thesis supervisor in conjunction with the undergraduate coordinator and adequately advertised. Department faculty and graduate students are encouraged to attend these presentations.

Submission Deadlines

The supervisor must approve the student's thesis. The student will submit a completed first draft of the thesis to the thesis supervisor. If the supervisor asks the student to make changes, the student will have two weeks to do so and submit a PDF copy of the thesis in final form. The thesis will be posted on the department's web site.

For students graduating in December 2023 , the deadline for the first draft is Friday, November 17 and the final submission is due to the thesis supervisor and the undergraduate coordinator on Friday, December 1.

For students graduating in May 2024 , the deadline for the first draft is Friday, April 19 and the final submission is due to the thesis supervisor and the undergraduate coordinator on Friday, May 3.

Format of the Thesis

Ideally, the final document should be TeXed or prepared in some equivalent technical document preparation system. The document must have large left margins (one and one-half inches or slightly larger). The title page should contain:

The student's name and graduating class.

The title of the senior thesis.

The name of the faculty supervisor. (If there is more than one supervisor, list both. If one of the supervisors is not in the Mathematics Department, list the department and institution.)

The date of completion of the thesis.

This information will be used to produce a standard frontispiece page, which will be added to the document in its library copies.

Judgment as to the merit of a senior thesis will be based largely on the recommendation of the faculty member supervising the thesis. The Mathematics Major Committee will use this recommendation both in its determination of honors and in its decision on whether to place the thesis in our permanent library collection.

The senior thesis will automatically be considered by the Mathematics Major Committee as one of the ingredients for deciding on an  honors  designation for the student. Students may receive honors without a thesis and are not guaranteed honors with one. However, an excellent senior thesis combined with an otherwise excellent record can elevate the level of honors awarded.

Library Collection

Meritorious senior theses will be catalogued, bound, and stored in the Mathematics Library.

Guidelines for writing a thesis

These guidelines are intended for students writing a thesis or project report for a  Third Year Project Course ,  Honours year  or  Postgraduate Coursework Project . Postgraduate research students should see  Information about Research Theses  for postgraduate research students.

Before you start your Honours or Project year, you should speak to members of staff about possible thesis topics. Find out who works in the areas that you are interested in and who you find it easy to talk mathematics with. If at all possible, settle on a topic and supervisor before the start of the first semester of your Honours or Project year.

Most students see their supervisor about once a week, although this is usually open to negotiation between the student and the supervisor. Even if you haven't done much between visits it is a good idea to have a regular chat so that your supervisor can keep track of how you are going. You can expect your supervisor to:

• Help you select - and modify - your topic.
• Direct you to useful references on your topic.
• Help explain difficult points.
• Provide feedback on the direction of your research.
• Read and comment on drafts of your thesis.
• Help prepare you for your talk.
• Give general course advice.

Your thesis or project report is an overview of what you have been studying in your Honours or Project year. Write it as if you were trying to explain the area of mathematics or statistics that you have been looking at to a fellow student.

• Include an introduction that explains what the project is all about, and what its contents are. (It is sometimes better to leave writing this part to the end!) For many reports, a conclusion or summary is appropriate.
• Your thesis should be a coherent, self-contained piece of work.
• Your writing should conform to the highest standards of English. Aim at clarity, precision and correct grammar. Start sentences with capital letters and end them with full-stops. Don't start sentences with a symbol.
• Take great care with bibliographic referencing. Wherever some material has an external source, this should be clear to the reader. Don't just write in the introduction: 'This report contains material from [1],[2] and [3]' - give the references for the material wherever it is used. Don't gratuitously pad your reference list with references that are not referred to in the text. Check current journals for acceptable referencing styles.
• Be careful not to plagiarise. What constitutes plagiarism is perhaps a little different in mathematics and statistics compared to some other subjects since there is a limit to how different you may be able to make a proof (at least in its basic structure). We do, however, expect the report to be written in your own words. A basic rule is: if you put a fact or an idea in your report which is not your own, the reader should be able to tell where you got this fact or idea.
• The University has  policies on academic honesty and plagiarism  which all students should familiarise themselves with.

Generally, mathematics reports and theses are almost always typed in LaTeX. If you are going to type it yourself, you should allow a certain amount of time to become familiar with this software. Indeed, starting to learn LaTeX well before you actually want to write is a very good idea.

You should not underestimate the time it takes to produce a polished document. You will almost certainly need several drafts. It is very difficult to concentrate on getting the mathematics, spelling, grammar, layout, etc., all correct at once. Try getting another student to proofread what you have written - from their different viewpoint they may pick up on lots of things that you can't see.

P R Halmos (1970) in  How to write mathematics, Enseignement Math.  ((2) 16, 123-152) has the following advice: "The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly:

• you must have something to say (i.e., some ideas), and you must have someone to say it to (i.e., an audience)
• you must organize what you want to say, and you must arrange it in the order you want it said in
• you must write it, rewrite it, and re-rewrite it several times
• and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation.

That's all there is to it."

His other advice includes:

• Say something: "To have something to say is by far the most important ingredient of good exposition---so much so that if the idea is important enough, the work has a chance to be immortal even if it is confusingly misorganized and awkwardly expressed..... To get by one the first principle alone is, however, only rarely possible and never desirable."
• Audience: "The second principle of good writing is to write for someone. When you decide to write something, ask yourself who it is that you want to reach." Your broad audience will be fellow Masters and Honours students, who may not be experts in your thesis topic. "The author must anticipate and avoid the reader's difficulties. As he(/she) writes, he(/she) must keep trying to imagine what in the words being written may tend to mislead the reader, and what will set him(/her) right."
• Organise: "The main contribution that an expository writer can make is to organize and arrange the material so as to minimize the resistance and maximize the insight of the reader and keep him(/her) on the track with no unintended distractions". 
• Think about the alphabet: "Once you have some kind of plan of organization, an outline, which may not be a fine one but is the best you can do, you are almost ready to start writing. The only other thing I would recommend that you do first is to invest an hour or two of thought in the alphabet; you'll find it saves many headaches later. The letters that are used to denote the concepts you'll discuss are worthy of thought and careful design. A good, consistent notation can be a tremendous help".
• Write in spirals: "The best way to start writing, perhaps the only way, is to write on the spiral plan. According to the spiral plan the chapters get written in the order 1,2,1,2,3,1,2,3,4 etc. You think you know how to write Chapter 1, but after you've done it and gone on to Chapter 2, you'll realize that you could have done a better job on Chapter 2 if you had done Chapter 1 differently. There is no help for it but to go back, do Chapter 1 differently, do a better job on Chapter 2, and then dive into Chapter 3... Chapter 3 will show up the weaknesses of Chapters 1 and 2".
• Write good English: "Good English style implies correct grammar, correct choice of words, correct punctuation, and, perhaps above all, common sense."

More information on how to write mathematics:

• Lee, K. A guide to writing mathematics
• Lee, K. Some notes on writing mathematics 
• Jackson, M. Some notes on writing in mathematics
• Reiter, A. Writing a research paper in mathematics
• Honours thesis
• Postgraduate Coursework Project
• Third Year Project Courses

Our websites may use cookies to personalize and enhance your experience. By continuing without changing your cookie settings, you agree to this collection. For more information, please see our University Websites Privacy Notice .

COLLEGE OF LIBERAL ARTS AND SCIENCES

Department of Mathematics

Thesis formatting, introduction.

If you write a Ph.D. thesis, you have to follow the  specifications  of the graduate school.

Additional information and answers to frequently asked questions can be found the graduate school’s Doctoral Degree Programs page.

A thesis in mathematics is invariably written in some form of LaTeX. It is usually quite a large LaTeX project, so it should not be your first attempt at typesetting. This implies that you should have typed a couple articles, quizzes, homeworks or exams in LaTeX before embarking on this. Below, two options have been created that conform to the guidelines set forth by the graduate school. The first is a minimal option that builds on the familiar book class that comes standard with any LaTeX distribution, while the second is a class file to be used in place of the book class, but contains extra content not suitable for all users. You may choose the option that works best in your case.

Below are a few of the well-done thesis variants from our department.

• Thesis class by Waseet Kazmi – 2023
• Thesis class by Ben Salisbury – 2012
• Thesis class by Marc Corluy – 2000

After unzipping put files into  one  directory. Some browsers (e.g. Safari on the Macs in the department) will put the files on your desktop. In this case create a new folder and put them all in there. In any case, make this move before you start compiling, because LaTeX generates a multitude of files when compiling and your desktop will turn into a (more?) disorderly mess.

Also make sure that the extensions are preserved. Some browsers have a tendency to slap on “.txt” at the end.

Open a text editor or a dedicated LaTeX editor,  such as TeXShop  (installed on department Macs).  When you are typing LaTeX code, it is usually clearer to use a fixed width font so that you have a clear view of your indentations and matrices (should there be any). In TeXShop, a good choice is “Courier Bold 14pts”. This font also has different characters for “|” and lowercase L. This is particularly important if you are trying to typeset tables where both “|” and lowercase L are used in the declaration.

You will now have to open some of the files that you downloaded and edit them. Here are descriptions of the set of files prepared by Ben Salisbury, listed (as best as possible) in order of priority.

• macros.tex  This file is the heart of your personalization. If you have been using LaTeX for your other manuscripts and have developed your own set of user-defined commands, then you should put them in this file. If you are importing text into one of the later files from an older LaTeX file with predefined commands, you will want your personal macros to be loaded already before fumbling with LaTeX error messages.

so those who prefer to use, say,

• thesis.tex  Lines 8 through 17 of this file require the user to input their personal data; i.e., name, degrees, year of graduation, advisory committee, and title. Further down in the file, you’ll notice the command \input{ch1.tex} . This imports the text from  ch1.tex  to the current position in this file. By copying the command and changing ch1 to ch2 (and so on), you will be able to link all chapters of your work to this file. Essentially, this file is the glue that holds the whole project together. This is also the file on which the TeX engine is run to obtain the desired output.
• abstract.tex ,  acknowledgments.tex , and  ch1.tex  should be self-explanatory. Of course, you should have more than one chapter to their thesis. The challenge of creating LaTeX files for the subsequent chapters is left to the user. Good luck!
• thesis.bib  This is a standard BibTeX file to be used as the source for your bibliography output. This may take some time to learn, so the file comes preloaded with a sample  .bib  file. Your best bet is find some literature on BibTeX to tackle this beast. Of course, one could simply replace the bibligraphy declarations in the  thesis.tex  with a manual bibliography, as Marc Corluy does in his setup below. To each his own.
• frontmatter.tex  Simply put: don’t touch it. If you need to make changes here, you should probably consider using Marc Corluy’s template (unless you are familiar enough to make the appropriate changes).

Here are comments on the files prepared by Marc Corluy.

• bibliography.tex  is not your first concern. The references that are listed in there are there to give you an idea about the exact format in which a book or article should be entered into this file.
• chapter1.tex ,  chapter2.tex , and  introchapter.tex  are almost empty; they are merely there to show you how to use titles and sections.
• The file  definitions.tex  you can completely ignore if you want to. It contains (re)definitions of some symbols and a fairly long list of basic mathematical symbols and functions. It can be handy to get a certain feel for how to use “def” in LaTeX and you can also add whatever you end up using often in your thesis.
• Most of the static data (title of the thesis, your previous degrees, your advisors, acknowledgment etc.) are entered in  front_matter.tex . The comments in this file will explain you what to put where.
• If you want to explain some of the notation that you use in your thesis, you should do so in notationpage.tex . It is technically part of the front matter of the thesis, but it is a separate file because this is probably the only part of the front matter data that changes as the thesis evolves.
• You will probably not change much to  settings.tex . This file loads a lot of the extensions to LaTeX and defines the elements of the  amsthm.sty  package, as is explained in the comments in  settings.tex .
• As you can guess,  thesis.tex  is the file where it all comes together. The includeonly command will allow you to compile only parts of your thesis if you prefer to do so. Note that you should compile the thesis twice to make sure that all the labels are set correctly.
• In principle, you should never change anything to  uconnthesis.cls .

Write your thesis. Good Luck.

Google Custom Search

We use Google for our search. By clicking on „enable search“ you enable the search box and accept our terms of use.

Information on the use of Google Search

• TUM School of Computation, Information and Technology
• Technical University of Munich

Thesis and Completing your Studies in Mathematics

In order to successfully complete a Bachelor's or Master's degree program in the area of Mathematics, students must write a scientific work: the Bachelor’s thesis or Master's thesis. Here, you will find an overview of everything you need to know, from planning to submission:

General Information

Registration, when and where.

It is possible to register your thesis on the 1st and 15th of each month. 

All final theses starting from  1st February 2024 onwards in the PP Mathematics at the School of Computation, Information and Technology will be managed via the CIT portal .

Once you have found a topic and a supervising chair for your thesis, you will be registered by the supervising chair . You will receive an e-mail asking you to confirm your thesis registration. Only after you have confirmed your registration the Academic Programmes Office will be able to check the admission requirements and you will receive an email confirming your binding registration for your thesis.

For more information, see  Thesis and Completing your Studies .

Further documents?

If you have completed extracurricular activities or a stay abroad during your studies, these can be included in the Diploma Supplement. To do this, we need the  Route Card for the Diploma Supplement   by the end of your studies at the latest. Please send the completed form as a PDF or photo/scan (readable) to  bachelor(at)ma.tum.de  or  master(at)ma.tum.de . We recommend you to hand in the Route Card when registering your thesis or shortly afterwards. Please also note the Information on the Diploma Supplement .

For more information, see  Thesis and Completing your Studies

Note: The  Diploma Supplement  is an English-language supplement to the university diploma and describes the qualifications linked to the study program. Under the heading “Additional Information” in this document, you can request the incorporation of specific details of extracurricular activities completed within the framework of your degree course.

Composition

The thesis can be written in German or English language. The Bachelor's thesis muss be prefaced with an abstract in the other language, not the one in which it is written.

Formal considerations

Cover page and page 1

Here you indicate the topic of the work as well as your name and that of the supervisor. In addition, you indicate the date of submission (see example).

Example of a cover page

Here you make the following declaration: 

“I hereby declare that this thesis is my own work and that no other sources have been used except those clearly indicated and referenced.” (Place, date, signature either on a tablet or as a scan of your original signature).

If you do not wish to use a digital signature, you may alternatively submit a separate page with the declaration and your original signature at the Infopoint.

Change of title

Please write an e-mail containing the new thesis title to  bachelor(at)ma.tum.de  or  master(at)ma.tum.de  and CC your supervisor and co-supervisor.

Submisson and extension of theses starting until and including 15.01.2024

Digital submission.

The Bachelor's or Master's thesis has to be submitted digitally. Please submit the thesis as a PDF file to  bachelor(at)ma.tum.de  or  master(at)ma.tum.de   within the deadline . If you have to hand in code or other files, please e-mail all files collected in a zip folder. Printed copies and USB sticks will no longer be accepted.

Should the submission date fall on a weekend or a public holiday, submission on the subsequent workday is also possible. Theses can of course also be submitted prior to the planned submission date. 

In the event that you need more time to work on your thesis, you must submit an application to  bachelor(at)ma.tum.de  or  master(at)ma.tum.de  for an extension, at least seven days prior to the planned submission date. This application must have been signed by your thesis supervisor (or you have to send the consent of the supervisor in electronic form). In the event of a second extension, the Chair of the Examination Board must sign the application as well as the thesis supervisor.

Bachelor's Thesis: Application for an extension Master's Thesis: Application for an extension

Submisson and extension of theses starting from 01.02.2024 onwards

Theses registered from 01.02.2024 onwards are managed entirely via the CIT portal . This portal is also used for submission within the deadline (including all required files, such as codes). If necessary, you can also apply for an extension of the submission deadline.

Tools and tips

The typesetting system  LaTeX  is the standard program worldwide for the compilation of theses that contain mathematical formulas. It is recommended that you familiarize yourself with LaTeX as early as possible during the course of your studies. It is a good exercise to prepare sheets for seminar presentations using LaTeX. 

You can find further information, templates and helpful advice on our  LaTeX info page .

Corporate design and use of  TUM logos

Thesis Templates for LaTeX

LaTeX-Template Bachelor / Master

The  TUM Library  does not only offer outstanding access to a wide range of books, newspapers, and electronic media, it also offers courses on, for example, bibliographical research and how to correctly cite works. 

Please also note subject-specific practices when citing works.

Further seminar papers and tools specific to mathematics

For licensing and contractual reasons, some of these databases are only accessible via domains belonging to the TUM Departments of Mathematics and/or Informatics. Please address queries where applicable to your supervisor or the TUM Library:

• MathSciNet  – Mathematical reviews on the web
• JADE  – Journal Articles Database
• ERAM  – Electronic Research Archive for Mathematics
• Online database at  Zentralblatt MATH

English theses

The  English Writing Center  offers all TUM members free one-to-one advice on writing English texts and helps you to enhance your writing skills.

Any questions?

For questions relating to the content of your Bachelor's or Master's thesis, please contact your thesis supervisor as defined in the General Academic and Examination Regulations (APSO), co-supervisor or the Student Advisory Office responsible.

Bachelor's Thesis Details

When to start.

Do you have at least 8 credits in Advanced Courses? Then it is the right moment to commence your Bachelor’s thesis. By the eighth semester of enrollment in your current degree program, at the latest, you must commence your thesis as long as there are no reasonable grounds preventing you from doing so in accordance with section 10 (6) of the  APSO .

Time frame & length

You have three months in which to complete your Bachelor’s thesis. This period begins on the date of registration. During this period, you should spend an average of at least 30 hours per week on the Bachelor's thesis – an effort equivalent to 12 credits.

The Bachelor’s thesis should not exceed 35 pages.

Choice of topic

In order to reduce the time needed to familiarize yourself with the topic, it is recommended to choose a topic for your Bachelor's thesis which corresponds to the theme of the Advanced Seminar taken.

Examiner and assessment

The Bachelor’s thesis must be evaluated by someone who is authorized to do so (thesis supervisor as defined in the APSO) at the TUM School of CIT. You will find an overview of other eligible persons outside the School of CIT in the list below. It is the written work that is assessed. The talk you give concerning the content does not affect the grading.

Persons outside the School of CIT who are authorized to examine theses

Master's Thesis Details

You have six months to complete your Master’s thesis. This period begins on the date of registration. You normally complete the Master’s thesis in the fourth semester of your Master’s program after you have fulfilled all other academic and exam requirements.

Examiners and assessment

The Master’s thesis must be approved and evaluated by someone who is authorized to do so (thesis supervisor as defined in the APSO) at the TUM School of CIT. You will find an overview of other eligible persons outside the School of CIT in the list below:

Completing your Studies

Release of final certificate.

Once you have completed all the requirements for your degree, we will contact you by email and ask for the approval to issue your diploma certificate.

Graduation documents and preliminary certificates

After you have given us the approval for the generating of your diploma certificate, your graduation documents will be generated within 4 - 6 weeks at the Graduation Office and Academic Records Campus Garching . You will be notified by post as soon as your certificate has been issued. Please make sure that your current study address in TUMonline is up to date.

If required, you can also apply to the Graduation Office and Academic Records Campus Garching for a preliminary certificate .

Transition Bachelor – Master

If you have applied for a consecutive mathematics Master's program at our school after completing your TUM mathematics Bachelor's degree, please let us know (at bachelor(at)ma.tum.de ). It is possible to forward your Bachelor's degree to the Admissions and Enrollment Office for enrollment (not for the application!). The graduation documents are therefore not necessary for enrollment. A green checkmark will then appear in the online application portal for your degree certificate and diploma ( Application Status ). Please note that it may take a few days until the documents are updated in the portal. If you do not see these two green check marks one week before the enrollment deadline, please contact bachelor(at)ma.tum.de as soon as possible.

Diploma Supplement

If you have completed extracurricular activities or a stay abroad during your studies, these can be included in the Diploma Supplement. To do this, we need the  Route Card for the Diploma Supplement   by the end of your studies at the latest. Please send the completed form as a PDF or photo/scan (readable) to  bachelor(at)ma.tum.de  or  master(at)ma.tum.de . Please also note the Information on the Diploma Supplement .

Please find more information under  graduation .

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base
• How to Write a Thesis Statement | 4 Steps & Examples

How to Write a Thesis Statement | 4 Steps & Examples

Published on January 11, 2019 by Shona McCombes . Revised on August 15, 2023 by Eoghan Ryan.

A thesis statement is a sentence that sums up the central point of your paper or essay . It usually comes near the end of your introduction .

Your thesis will look a bit different depending on the type of essay you’re writing. But the thesis statement should always clearly state the main idea you want to get across. Everything else in your essay should relate back to this idea.

You can write your thesis statement by following four simple steps:

• Start with a question
• Write your initial answer
• Develop your answer
• Refine your thesis statement

Instantly correct all language mistakes in your text

Upload your document to correct all your mistakes in minutes

Table of contents

What is a thesis statement, placement of the thesis statement, step 1: start with a question, step 2: write your initial answer, step 3: develop your answer, step 4: refine your thesis statement, types of thesis statements, other interesting articles, frequently asked questions about thesis statements.

A thesis statement summarizes the central points of your essay. It is a signpost telling the reader what the essay will argue and why.

The best thesis statements are:

• Concise: A good thesis statement is short and sweet—don’t use more words than necessary. State your point clearly and directly in one or two sentences.
• Contentious: Your thesis shouldn’t be a simple statement of fact that everyone already knows. A good thesis statement is a claim that requires further evidence or analysis to back it up.
• Coherent: Everything mentioned in your thesis statement must be supported and explained in the rest of your paper.

Prevent plagiarism. Run a free check.

The thesis statement generally appears at the end of your essay introduction or research paper introduction .

The spread of the internet has had a world-changing effect, not least on the world of education. The use of the internet in academic contexts and among young people more generally is hotly debated. For many who did not grow up with this technology, its effects seem alarming and potentially harmful. This concern, while understandable, is misguided. The negatives of internet use are outweighed by its many benefits for education: the internet facilitates easier access to information, exposure to different perspectives, and a flexible learning environment for both students and teachers.

You should come up with an initial thesis, sometimes called a working thesis , early in the writing process . As soon as you’ve decided on your essay topic , you need to work out what you want to say about it—a clear thesis will give your essay direction and structure.

You might already have a question in your assignment, but if not, try to come up with your own. What would you like to find out or decide about your topic?

For example, you might ask:

After some initial research, you can formulate a tentative answer to this question. At this stage it can be simple, and it should guide the research process and writing process .

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

• Academic style
• Vague sentences
• Style consistency

See an example

Now you need to consider why this is your answer and how you will convince your reader to agree with you. As you read more about your topic and begin writing, your answer should get more detailed.

In your essay about the internet and education, the thesis states your position and sketches out the key arguments you’ll use to support it.

The negatives of internet use are outweighed by its many benefits for education because it facilitates easier access to information.

In your essay about braille, the thesis statement summarizes the key historical development that you’ll explain.

The invention of braille in the 19th century transformed the lives of blind people, allowing them to participate more actively in public life.

A strong thesis statement should tell the reader:

• Why you hold this position
• What they’ll learn from your essay
• The key points of your argument or narrative

The final thesis statement doesn’t just state your position, but summarizes your overall argument or the entire topic you’re going to explain. To strengthen a weak thesis statement, it can help to consider the broader context of your topic.

These examples are more specific and show that you’ll explore your topic in depth.

Your thesis statement should match the goals of your essay, which vary depending on the type of essay you’re writing:

• In an argumentative essay , your thesis statement should take a strong position. Your aim in the essay is to convince your reader of this thesis based on evidence and logical reasoning.
• In an expository essay , you’ll aim to explain the facts of a topic or process. Your thesis statement doesn’t have to include a strong opinion in this case, but it should clearly state the central point you want to make, and mention the key elements you’ll explain.

If you want to know more about AI tools , college essays , or fallacies make sure to check out some of our other articles with explanations and examples or go directly to our tools!

• Ad hominem fallacy
• Post hoc fallacy
• Appeal to authority fallacy
• False cause fallacy
• Sunk cost fallacy

College essays

• Choosing Essay Topic
• Write a College Essay
• Write a Diversity Essay
• College Essay Format & Structure
• Comparing and Contrasting in an Essay

 (AI) Tools

• Grammar Checker
• Paraphrasing Tool
• Text Summarizer
• AI Detector
• Plagiarism Checker
• Citation Generator

A thesis statement is a sentence that sums up the central point of your paper or essay . Everything else you write should relate to this key idea.

The thesis statement is essential in any academic essay or research paper for two main reasons:

• It gives your writing direction and focus.
• It gives the reader a concise summary of your main point.

Without a clear thesis statement, an essay can end up rambling and unfocused, leaving your reader unsure of exactly what you want to say.

Follow these four steps to come up with a thesis statement :

• Ask a question about your topic .
• Write your initial answer.
• Develop your answer by including reasons.
• Refine your answer, adding more detail and nuance.

The thesis statement should be placed at the end of your essay introduction .

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

McCombes, S. (2023, August 15). How to Write a Thesis Statement | 4 Steps & Examples. Scribbr. Retrieved April 2, 2024, from https://www.scribbr.com/academic-essay/thesis-statement/

Is this article helpful?

Shona McCombes

Other students also liked, how to write an essay introduction | 4 steps & examples, how to write topic sentences | 4 steps, examples & purpose, academic paragraph structure | step-by-step guide & examples, "i thought ai proofreading was useless but..".

I've been using Scribbr for years now and I know it's a service that won't disappoint. It does a good job spotting mistakes”

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Definition to Remember:

• Thesis = Topic + Claim

Rules to Remember:

• A thesis statement is the main idea or subject of your paper, while a topic sentence is the main idea of a single paragraph. Sometimes the thesis may develop in your mind early in the writing process, and sometimes it will become more clear or shift as you work through the writing process.
• The thesis statement should be a single, concise sentence. An effective thesis statement has two parts: (1) the topic and (2) your claim about the topic.
• Your thesis is a contract that you establish with your readers. The voice, tone, assurances, and promises of your thesis must continue throughout the essay.
• An effective thesis statement should be as specific as possible and be limited enough to make it manageable. Keep your thesis statement specific enough to be adequately discussed within the length of your paper. If a thesis statement is too general or vague, it can be difficult to decide what to write about.
• Be wary of absolute words like all, none, everyone, no one, always, never in your thesis statement. If a reader can think of an exception to your absolute statement, he or she may set your entire argument aside. Be careful, too, to avoid claims that are too over-arching and, therefore, suspect.
• Your thesis should serve as an umbrella for the essay that follows. Every topic sentence for each of the body paragraphs must fit neatly beneath the umbrella, just as every item of evidence also must fit. If anything does not fit under the umbrella of your thesis, revise accordingly to either broaden or narrow your umbrella until the simple math works.
• Restate your topic as a question. If, for example, your topic is about the health and care of domestic cats, you might restate your topic as follows: Should domestic cats be permitted to roam freely in residential neighborhoods? Do you see how this restatement begins to give both your research and your writing better purpose?
• Answer your question with a single-sentence claim. An effective thesis statement (1) announces a topic and (2) states a claim. What assertion will you make about your topic and why? To answer our domestic cat question, we might assert the following: Domestic cats should only be permitted to roam freely in residential neighborhoods if they meet specific county-designated standards.
• Focus your thesis. Remember that the more specific you are, the easier it will be to effectively discuss and prove your thesis: Domestic cats should only be permitted to roam freely in residential neighborhoods if they have a chip inserted that identifies their home, homeowner, and vaccination record.   “Write succinctly. There is a better chance people will read and appreciate your thoughts.” Dr. Aimee Stone Cooper, Pastor

Common Errors:

• Assuming that the implied focus of an introduction is enough . Every essay must have a clear, concise thesis statement; never assume that your readers understand your intentions.
• Including a thesis that does not make a claim. With the advent of the internet, gone are the days when informational papers were a necessary exercise. Because most information is available at the touch of an app, your focus must be on the claim you intend to make about the topic you have chosen.
• Presenting a thesis that is too broad or too narrow. While the line between the two can be tenuous and difficult to locate, it is worth the effort. When a thesis is too broad, it is difficult to argue adequately without leaving notable holes in your rationale; when a thesis is too narrow, it can be challenging to find much at all to say to one another.
• Neglecting to revise the thesis umbrella as the project unfolds. The more flexible you are, the more successful your end result will be.

Exercise 10.1

Find three thesis statements from papers you have written in the past and list them here. If you walk each through the three-step process on item 6 above, how would each one fare? What changes would you make, and why?

Exercise 10.2

Select one thesis from 10.1. Draw or download an umbrella image, and write your thesis on or above the umbrella. Next, write one topic sentence from each paragraph of your essay vertically beneath the umbrella. Keep all information beneath the shape of the umbrella. As you consider your ideas, what revisions do you need to make? Do you need to broaden your thesis in order to encompass all that falls beneath? Or do you need to focus your thesis better, so your readers are able to immediately see the logic of the ideas that come under the umbrella? Are there topics under the umbrella that could be saved for another essay? Do you need additional evidence under the umbrella to strengthen your overall claim?

Exercise 10.3

Consider an essay or writing assignment you will need to complete in the next week, whether for school, work, or home. What is the topic? What is your claim about the topic? What specific evidence will you include? What will your thesis statement be? Use the progression below to aid your thinking:

Topic    →   Claim     →    Evidence     →    Thesis

The Simple Math of Writing Well Copyright © 2017 by Dr. Jennie A. Harrop is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

• More from M-W
• To save this word, you'll need to log in. Log In

Definition of thesis

Did you know.

In high school, college, or graduate school, students often have to write a thesis on a topic in their major field of study. In many fields, a final thesis is the biggest challenge involved in getting a master's degree, and the same is true for students studying for a Ph.D. (a Ph.D. thesis is often called a dissertation ). But a thesis may also be an idea; so in the course of the paper the student may put forth several theses (notice the plural form) and attempt to prove them.

Examples of thesis in a Sentence

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'thesis.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

in sense 3, Middle English, lowering of the voice, from Late Latin & Greek; Late Latin, from Greek, downbeat, more important part of a foot, literally, act of laying down; in other senses, Latin, from Greek, literally, act of laying down, from tithenai to put, lay down — more at do

14th century, in the meaning defined at sense 3a(1)

Dictionary Entries Near thesis

the sins of the fathers are visited upon the children

thesis novel

Cite this Entry

“Thesis.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/thesis. Accessed 2 Apr. 2024.

Kids Definition

Kids definition of thesis, more from merriam-webster on thesis.

Nglish: Translation of thesis for Spanish Speakers

Britannica English: Translation of thesis for Arabic Speakers

Britannica.com: Encyclopedia article about thesis

Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free!

Can you solve 4 words at once?

Word of the day.

See Definitions and Examples »

Get Word of the Day daily email!

Popular in Grammar & Usage

The tangled history of 'it's' and 'its', more commonly misspelled words, why does english have so many silent letters, your vs. you're: how to use them correctly, every letter is silent, sometimes: a-z list of examples, popular in wordplay, the words of the week - mar. 29, 10 scrabble words without any vowels, 12 more bird names that sound like insults (and sometimes are), 8 uncommon words related to love, 9 superb owl words, games & quizzes.

We use cookies on reading.ac.uk to improve your experience, monitor site performance and tailor content to you

Read our cookie policy to find out how to manage your cookie settings

This site may not work correctly on Internet Explorer. We recommend switching to a different browser for a better experience.

Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

• Daily Crossword
• Word Puzzle
• Word Finder
• Word of the Day
• Synonym of the Day
• Word of the Year
• Language stories
• All featured
• Gender and sexuality
• All pop culture
• Grammar Coach ™
• Writing hub
• Grammar essentials
• Commonly confused
• All writing tips
• Pop culture
• Writing tips

a proposition stated or put forward for consideration, especially one to be discussed and proved or to be maintained against objections: He vigorously defended his thesis on the causes of war.

a subject for a composition or essay.

a dissertation on a particular subject in which one has done original research, as one presented by a candidate for a diploma or degree.

Music . the downward stroke in conducting; downbeat. : Compare arsis (def. 1) .

a part of a metrical foot that does not bear the ictus or stress.

(less commonly) the part of a metrical foot that bears the ictus. : Compare arsis (def. 2) .

Philosophy . See under Hegelian dialectic .

Origin of thesis

Word story for thesis, other words for thesis, words that may be confused with thesis.

• 1. antithesis , synthesis , thesis
• 2. dissertation , thesis

Words Nearby thesis

• shit will hit the fan, the
• shoe is on the other foot, the
• short end of the stick, the
• The show must go on
• thesis play
• thesis statement
• Sketch Book, The
• Skin of Our Teeth, The
• sky's the limit, the

Dictionary.com Unabridged Based on the Random House Unabridged Dictionary, © Random House, Inc. 2024

How to use thesis in a sentence

“The Saudis have been proving the thesis of the film — they do in fact have an army,” said Thor Halvorssen, founder and chief executive of the nonprofit Human Rights Foundation, which funded the movie.

It’s a hypothesis that Bush pursued in her master’s thesis , and last year she began attending virtual Goth parties in a final round of field work before defending her doctoral thesis later this year.

While this partnership was planned prior to the coronavirus outbreak, co-founder Jordana Kier said the pandemic instantly proved out the expansion thesis .

They’ve had to defend that thesis for a very, very long time in front of a variety of different customers and different people.

Over the past decade, In-Q-Tel has been one of the most active investors in the commercial space sector, with a broad investment thesis that touches many aspects of the sector.

In “Back Home,” Gil also revisits the nostalgia for the South explored in his Johns Hopkins thesis , “Circle of Stone.”

At least father and son were in alignment on this central thesis : acting “gay”—bad; being thought of as gay—bad.

Her doctoral thesis , says Ramin Takloo at the University of Illinois, was simply outstanding.

Marshall McLuhan long ago argued the now accepted thesis that different mediums have different influences on thinking.

He wrote his Master's thesis  on the underrepresentation of young people in Congress.

And indeed for most young men a college thesis is but an exercise for sharpening the wits, rarely dangerous in its later effects.

It will be for the reader to determine whether the main thesis of the book has gained or lost by the new evidence.

But the word thesis , when applied to Systems, does not mean the 'position' of single notes, but of groups of notes.

This conclusion, it need hardly be said, is in entire agreement with the main thesis of the preceding pages.

Sundry outlying Indians, with ammunition to waste, took belly and knee rests and strengthened the thesis to the contrary.

British Dictionary definitions for thesis

/ ( ˈθiːsɪs ) /

a dissertation resulting from original research, esp when submitted by a candidate for a degree or diploma

a doctrine maintained or promoted in argument

a subject for a discussion or essay

an unproved statement, esp one put forward as a premise in an argument

music the downbeat of a bar, as indicated in conducting

(in classical prosody) the syllable or part of a metrical foot not receiving the ictus : Compare arsis

philosophy the first stage in the Hegelian dialectic, that is challenged by the antithesis

Collins English Dictionary - Complete & Unabridged 2012 Digital Edition © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009, 2012

Cultural definitions for thesis

The central idea in a piece of writing, sometimes contained in a topic sentence .

The New Dictionary of Cultural Literacy, Third Edition Copyright © 2005 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.

It appears you have javascript disabled. Please enable javascript to get the full experience of gustavus.edu

Mathematics advising guide.

Mathematics encompasses the study of patterns in nature, the development of tools to understand those patterns, and the generalization of those ideas in an abstract setting. A mathematics degree teaches a student to think, to reason, to experiment, and to learn and grow. Mathematics inspires not only science, technology, and their applications, but all aspects of society.

Students learn how to ask good questions, make connections, work with others, explain their thoughts, and find evidence to back up their reasoning.

Professors provide a solid foundation in the subject, spark interest in mathematical topics, use technology in learning, use innovative pedagogical approaches, and provide students with resources to pursue research experiences. 

Graduates leave Gustavus thoroughly prepared for graduate study, secondary school teaching, a life of service, or employment in government or industry. 

Mathematics Major

This section lists the requirements of the Mathematics major. A grade of C- or higher is necessary in all 11 courses used to satisfy the requirements of the major. Additionally, you can use the Mathematics Major Form , This form will help you plan out your mathematics courses and requirements. To declare a major, use this form :

• MCS-122 Calculus II 
• MCS-150  Discrete Mathematics
• MCS-221 Linear Algebra
• MCS-222 Multivariate Calculus
• MCS-213 Intro to Algebra 
• MCS-220  Intro to Analysis
• MCS-142 Introduction to Statistics
• MCS-177 Introduction to Computer Science I
• MCS-313 and MCS-314  Algebra
• MCS-331 and MCS-332  Analysis
• MCS-353 and MCS-357 Dynamical Systems

Electives:  Two additional mathematics courses at the 200 or 300 level. Students should consult with their advisors to discuss which courses best fit their needs. 

Mathematics Minor

A grade of C- or higher is necessary in all courses used to satisfy the requirements of the minor, which are as follows:

• MCS-122 Calculus II or MCS-132 Honors Calculus II
• MCS-150 Discrete Mathematics
• MCS-213 Intro to Algebra
• MCS-220 Intro to Analysis
• MCS-303 Geometry
• MCS-313  Modern Algebra I
• MCS-314  Modern Algebra II
• MCS-321   Elementary Theory of Complex Variables
• MCS-331 Real Analysis
• MCS-344 Topics in Advanced Math
• MCS-353  Continuous Dynamical Systems
• MCS-355 Numerical Analysis
• MCS-357 Discrete Dynamical Systems
• MCS-358 Math Model Building

Sample Student Plans

All students should ideally lay out a schedule of their own showing what courses they plan to take, and when they plan to take them. The schedule may not accurately forecast the future, but it is helpful nonetheless. A printable sample plan can be found on the  Mathematics Major Form  

Student Starter Plan

The sample plans below are useful starting points in developing an individual plan. You can select the sample plan that comes closest to fitting your own situation and then tailor it as necessary. Note that certain courses are offered on an every-other year basis; for example MCS-314 (Modern Algebra II) is offered in the spring of odd years Courses offered every other year include MCS-313, MCS-314, MCS-331, MCS-344, MCS-355, MCS-357, MCS-358, MCS-385, and MCS-394. These courses are listed with an astrix in the sample plans below. Please keep these course alterations in mind when planning out your major. Check the college catalog for when the courses you are interested in will be scheduled.

Students interested in algebra should take *MCS-313 and *MCS-314 for their  Immersive Experience and MCS-213 as their Proofs  course along with two appropriate electives. 

Students interested in analysis should take *MCS-331 and *MCS-332 for their  Immersive Experience and MCS-220 as their Proofs course and two appropriate electives.  

Applied Mathematics

Students interested in applied mathematics should take *MCS-353 and *MCS-357 for their  Immersive Experience , *MCS-358 as an elective, and an additional  Elective .  

Thinking About Graduate School in Traditional Mathematics 

Students considering graduate school in mathematics should take *MCS-313,* MCS-314, *MCS-321, and *MCS-331 for their  Immersive Experience and Electives as well as an appropriate Collaborative Experience . 

Thinking About Graduate School in Applied Mathematics

Students considering graduate school in applied mathematics should take *MCS-353 and *MCS-357 for their  Immersive   Experience , *MCS-358 as an Elective , and either *MCS-313 or  *MCS 331 as their second Elective .

Studying Mathematics Abroad

 Students traveling abroad should speak with their advisors to discuss courses and study abroad programs. Study abroad programs are listed on the MCS Resources page. 

Honors Program

In order to graduate with Honors in Mathematics, a student must complete an application for admission to the Honors program, available through the department chair, showing that the student satisfies the admission requirements, and then the requirements of the program.

The requirements for admission to the Honors program are as follows:

• Completion of steps 1 - 3 of the Mathematics Major with a grade point average greater than 3.14. 
• Approval by the Mathematics Honors committee of an Honors thesis proposal. (Guidelines are available in the Mathematics Advising Guide.)

The requirements of the honors program after admission are as follows:

• Attainment of a GPA greater than 3.14 in courses used to satisfy the requirements of the major. If a student has taken more courses than the major requires, that student may designate for consideration any collection of courses satisfying the requirements of the major.
• Approval by the Mathematics Honors Committee of an Honors thesis. The thesis should conform in general outline to the previously approved proposal (or an approved substitute proposal), should include approximately 160 hours of work, and should result in an approved written document. Students completing this requirement will receive credit for the course MCS-350, whether or not they graduate with Honors. (See the Mathematics Advising Guide for the thesis guidelines.)
• Oral presentation of the thesis in a public forum, such as the departmental seminar. This presentation will not be evaluated as a criterion for thesis approval, but is required.

Honors Thesis Guidelines

Mathematics honors thesis proposals should be written in consultation with the faculty member who will be supervising the work. The proposal and thesis must each be approved by the Mathematics Honors Committee. These guidelines are intended to help students, faculty supervisors, and the committee judge what merits approval.

The thesis should include creative work, and should not reproduce well-known results; however, it need not be entirely novel. It is unreasonable for an undergraduate with limited time and library resources to do a thorough search of the literature, such as would be necessary to ensure complete novelty. Moreover, it would be rare for any topic to be simultaneously novel, easy enough to think of, and easy enough to do.

The thesis should include use of primary-source reference material. As stated above, an exhaustive search of the research literature is impractical. None the less, the resources of inter-library loan, the faculty supervisor's private holdings, etc. must be tapped if the thesis work is to go beyond standard classroom/textbook work.

The written thesis should sufficiently explain the project undertaken and results achieved that someone generally knowledgeable about mathematics, but not about the specific topic, can understand it. The quality of writing and care in citing sources should be adequate for external distribution without embarrassment.

The thesis must contain a substantial mathematical component, though it can include other disciplines as well. If a single thesis simultaneously satisfies the requirements of this program and some other discipline's honors program, it can be used for both (subject to the other program's restrictions). However, course credit will not be awarded for work which is otherwise receiving course credit.

The Mathematics Honors Committee will maintain a file of past proposals and theses, which may be valuable in further clarifying what constitutes a suitable thesis. In order to provide some guidance of the sort before the program gets under way, here are some possible topics that appear on the surface to be suitable:

• A student could study the history surrounding Fermat's last theorem, and discuss and explain past failed attempts and the recent successful attempt to prove this theorem.
• A student could research the topic of knot theory and discuss the implications of this theory to the study of DNA and other biological materials.
• A student could study the use of wavelets in signal analysis, and the general usefulness of orthonormal families of functions in signal analysis. 

Senior Oral Exam

 As described above, every math major must either take an additional upper level math course from a specified list or alternatively submit to oral examination during the Spring semester of their final year.

A student who chooses to take the oral examination selects, in consultation with a faculty member, a topic to research. They then present a 20-minute talk on that topic to an examining committee of three faculty members. At the conclusion of the talk, the faculty question the student about the talk, and also about fundamental topics from the student's full four years' of courses. The goal is not to require recollection of details, but rather to make sure that the student is leaving with the essentials intact.

The examination committee confers privately immediately after the examination and delivers the results to the student at the conclusion of their deliberations. The outcome is either that the student is deemed to have satisfied the requirement or alternatively that the student is requested to retry the examination at a later date. In the latter case, specific suggestions for areas of improvement are provided by the faculty committee.

More information about the oral examination procedures and schedule are provided routinely to those fourth-year majors who will likely choose to take the examination.

Home > College of Natural Sciences > Mathematics > Mathematics Theses, Projects, and Dissertations

Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2023 2023.

DNA SELF-ASSEMBLY OF TRAPEZOHEDRAL GRAPHS , Hytham Abdelkarim

An Exposition of the Curvature of Warped Product Manifolds , Angelina Bisson

Jackknife Empirical Likelihood Tests for Equality of Generalized Lorenz Curves , Anton Butenko

MATHEMATICS BEHIND MACHINE LEARNING , Rim Hammoud

Statistical Analysis of Health Habits for Incoming College Students , Wendy Isamara Lizarraga Noriega

Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

Distance Correlation Based Feature Selection in Random Forest , Jose Munoz-Lopez

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil

KNOT EQUIVALENCE , Jacob Trubey

Theses/Projects/Dissertations from 2022 2022

SYMMETRIC GENERATIONS AND AN ALGORITHM TO PROVE RELATIONS , Diddier Andrade

The Examination of the Arithmetic Surface (3, 5) Over Q , Rachel J. Arguelles

Error Terms for the Trapezoid, Midpoint, and Simpson's Rules , Jessica E. Coen

de Rham Cohomology, Homotopy Invariance and the Mayer-Vietoris Sequence , Stacey Elizabeth Cox

Symmetric Generation , Ana Gonzalez

SYMMETRIC PRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Samar Mikhail Kasouha

Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez

LATTICE REDUCTION ALGORITHMS , Juan Ortega

THE DECOMPOSITION OF THE SPACE OF ALGEBRAIC CURVATURE TENSORS , Katelyn Sage Risinger

Verifying Sudoku Puzzles , Chelsea Schweer

AN EXPOSITION OF ELLIPTIC CURVE CRYPTOGRAPHY , Travis Severns

Theses/Projects/Dissertations from 2021 2021

Non-Abelian Finite Simple Groups as Homomorphic Images , Sandra Bahena

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez

SYMMETRIC REPRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Connie Corona

Symmetric Presentation of Finite Groups, and Related Topics , Marina Michelle Duchesne

MEASURE AND INTEGRATION , JeongHwan Lee

A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

Sum of Cubes of the First n Integers , Obiamaka L. Agu

Permutation and Monomial Progenitors , Crystal Diaz

Tile Based Self-Assembly of the Rook's Graph , Ernesto Gonzalez

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda

DNA COMPLEXES OF ONE BOND-EDGE TYPE , Andrew Tyler Lavengood-Ryan

Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder

ASSESSING STUDENT UNDERSTANDING WHILE SOLVING LINEAR EQUATIONS USING FLOWCHARTS AND ALGEBRAIC METHODS , Edima Umanah

Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila

VANISHING LOCAL SCALAR INVARIANTS ON GENERALIZED PLANE WAVE MANIFOLDS , Brian Matthew Friday

Analogues Between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle , Lacey Taylor James

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

Pascal's Triangle, Pascal's Pyramid, and the Trinomial Triangle , Antonio Saucedo Jr.

THE EFFECTIVENESS OF DYNAMIC MATHEMATICAL SOFTWARE IN THE INSTRUCTION OF THE UNIT CIRCLE , Edward Simons

CALCULUS REMEDIATION AS AN INDICATOR FOR SUCCESS ON THE CALCULUS AP EXAM , Ty Stockham

Theses/Projects/Dissertations from 2018 2018

PROGENITORS, SYMMETRIC PRESENTATIONS AND CONSTRUCTIONS , Diana Aguirre

Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan

MODERN CRYPTOGRAPHY , Samuel Lopez

Progenitors, Symmetric Presentations, and Related Topics , Joana Viridiana Luna

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN

THE STRUGGLE WITH INVERSE FUNCTIONS DOING AND UNDOING PROCESS , Jesus Nolasco

Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager

MANUAL SYMMETRIC GENERATION , Joel Webster

Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari

CONSTRUCTION OF HOMOMORPHIC IMAGES , Erica Fernandez

Making Models with Bayes , Pilar Olid

An Introduction to Lie Algebra , Amanda Renee Talley

SIMPLE AND SEMI-SIMPLE ARTINIAN RINGS , Ulyses Velasco

CONSTRUCTION OF FINITE GROUP , Michelle SoYeong Yeo

Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova

GEODESICS IN LORENTZIAN MANIFOLDS , Amir A. Botros

REALIZING TOURNAMENTS AS MODELS FOR K-MAJORITY VOTING , Gina Marie Cheney

Solving Absolute Value Equations and Inequalities on a Number Line , Melinda A. Curtis

BIO-MATHEMATICS: INTRODUCTION TO THE MATHEMATICAL MODEL OF THE HEPATITIS C VIRUS , Lucille J. Durfee

ANALYSIS AND SYNTHESIS OF THE LITERATURE REGARDING ACTIVE AND DIRECT INSTRUCTION AND THEIR PROMOTION OF FLEXIBLE THINKING IN MATHEMATICS , Genelle Elizabeth Gonzalez

LIFE EXPECTANCY , Ali R. Hassanzadah

PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS , Sean M. Hearon

A Dual Fano, and Dual Non-Fano Matroidal Network , Stephen Lee Johnson

Mathematical Reasoning and the Inductive Process: An Examination of The Law of Quadratic Reciprocity , Nitish Mittal

The Kauffman Bracket and Genus of Alternating Links , Bryan M. Nguyen

Probabilistic Methods In Information Theory , Erik W. Pachas

THINKING POKER THROUGH GAME THEORY , Damian Palafox

Indicators of Future Mathematics Proficiency: Literature Review & Synthesis , Claudia Preciado

Ádám's Conjecture and Arc Reversal Problems , Claudio D. Salas

AN INTRODUCTION TO BOOLEAN ALGEBRAS , Amy Schardijn

The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015

SYMMETRIC PRESENTATIONS AND RELATED TOPICS , Mashael U. Alharbi

Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

Discovering and Applying Geometric Transformations: Transformations to Show Congruence and Similarity , Tamara V. Bonn

Symmetric Presentations and Generation , Dustin J. Grindstaff

HILBERT SPACES AND FOURIER SERIES , Terri Joan Harris Mrs.

SYMMETRIC PRESENTATIONS OF NON-ABELIAN SIMPLE GROUPS , Leonard B. Lamp

Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz

CONSTRUCTIONS AND ISOMORPHISM TYPES OF IMAGES , Jessica Luna Ramirez

Unique Prime Factorization of Ideals in the Ring of Algebraic Integers of an Imaginary Quadratic Number Field , Nolberto Rezola

ALGEBRA 1 STUDENTS’ ABILITY TO RELATE THE DEFINITION OF A FUNCTION TO ITS REPRESENTATIONS , Sarah A. Thomson

Advanced Search

• Notify me via email or RSS
• Department, Program, or Office
• Disciplines

Author Corner

• Mathematics Department web site

A service of the John M. Pfau Library

Home | About | FAQ | My Account | Accessibility Statement

Privacy Copyright Acrobat Reader

Help | Advanced Search

Mathematical Physics

Title: study on a quantization condition and the solvability of schrödinger-type equations.

Abstract: In this thesis, we study a quantization condition in relation to the solvability of Schrödinger equations. This quantization condition is called the SWKB (supersymmetric Wentzel-Kramers-Brillouin) quantization condition and has been known in the context of supersymmetric quantum mechanics for decades. The main contents of this thesis are recapitulated as follows: the foundation and the application of the SWKB quantization condition. The first half of this thesis aims to understand the fundamental implications of this condition based on extensive case studies. It turns out that the exactness of the SWKB quantization condition indicates the exact solvability of a system via the classical orthogonal polynomials. The SWKB quantization condition provides quantizations of energy, which we call the direct problem of the SWKB. We formulate the inverse problem of the SWKB: the problem of determining the superpotential from a given energy spectrum. The formulation successfully reconstructs all conventional shape-invariant potentials from the given energy spectra. We further construct novel solvable potentials, which are classical-orthogonal-polynomially quasi-exactly solvable, by this formulation. We further demonstrate several explicit solutions of the Schrödinger equations with the classical-orthogonal-polynomially quasi-exactly solvable potentials, whose family is referred to as a harmonic oscillator with singularity functions in this thesis. In one case, the energy spectra become isospectral, with several additional eigenstates, to the ordinary harmonic oscillator for special choices of a parameter. By virtue of this, we formulate a systematic way of constructing infinitely many potentials that are strictly isospectral to the ordinary harmonic oscillator.

Submission history

Access paper:.

• Other Formats

References & Citations

• INSPIRE HEP
• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .