## Principal Ideal Domains non-example

A good practice in math, especially when studying a type of mathematical objects, is by considering what doesn’t fit the definition.

## Getting started with LaTeX

There are many benefits to using LaTeX as an academic. This post aims to giving you resources on how to setup your coding environment. The aim is not a comprehensive review of LaTeX editors, but to give a starting point for novices, so apologies if your favorite editor is not included here.

## Choosing a Starting Point: Online vs. Offline

There are many ways to setup your environment, and here I’ll layout the two major ways – desktop and cloud. and their pros and cons:

Based on what you’re building, you’d likely switch between cloud and desktop environments. Thus, I’ll recommend some specific ones below.

Choice as of November 2019: Offline editor I recommend TeXmaker. For online editor Overleaf, especially if I’m working on tiny projects like homework assignments and I’m not on my own computer.

### Notable Online LaTeX Editors: Overleaf, Authorea and CoCalc

• Overleaf v2: Great document editor site, one of the most popular. Overleaf hosts many templates as well as gives fairly comprehensive customer support. Result of the merger of Overleaf and ShareLaTeX, the site unfortunately is freemium. This popular website is used at the University of Illinois of Chicago, in the department of mathematics to handle creation of exams and exam keys. Also used extensively by my cohort at San Francisco State University for homework. Highly recommended to use their free solo accounts for learning LaTeX.
• Authorea: “a platform to read, write, and publish research.” Mainly a good platform for collaboration between researchers who already know latex and those who are using other text editors. Main plus: they use Git for version control, which makes this very attractive for bigger teams working on long projects. They also have a stated goal of helping bring the power of professional typesetting using LaTeX to social sciences.
• CoCalc: This is a great site for mathematicians and statisticians who enjoy open source programming philosophy. Aside from a full LaTeX compiler, they also handle SageMath worksheets, Jupyter notebooks, R and scientific Python. From my personal experience, this is a great place for learning all these formats. before beginning my master’s thesis, I wrote prototypical code here until dealing with lattice polytopes in dimensions higher than 3 made the server eventually hate me.
• Codecogs: Basic editor for creating images of mathematical equations, useful for super quick creation, no more, no less.

### Notable Offline LaTeX Editors

First, depending on your operating system, you should install a LaTeX compiler:

Then, choose a LaTeX editor. I’ve only included dependable editors that are usable across Windows, Mac as well as Linux, as that is important to me personally because I switch between Ubuntu and Windows depending on the task often.

• TeXmaker: Install this and start coding! Texmaker is considered to be one of the best LaTeX editors for GNOME desktop environment. It presents a great user interface which results in a good user experience. It is also crowned to be one among the most useful LaTeX editor there is.
• TeXstudio: A fork of TeXmaker, with more open philosophy to modification. This program allows for more customizability than TeXmaker. The learning curve is slightly higher, due to having more features.

#### Other notable editors (harder to use, but with great potential for power-users)

• Sublime Text: Great tool if you’re using already familiar with sublime text as an editor for Python and other languages. Simply add on LaTeX plugin and set up autocomplete and snippets and you’re good to go. See: LaTeX Tools
• Visual Studio Code: Similar remark to above. See: LaTeX Workshop.
• VIM: Learning this editor is HARD, but the payoffs are great if you plan on being a high-functioning code producer one day. This is likely the most powerful editor on this list, and it’s my personal goal to integrate this editor into my workflow one day. (There’s an ongoing editor war, which I hope that my joke here won’t bring too much vitriol…) This editor exist in almost all Unix/Linux distributions by default, and is highly customizable, with a significantly different flow of editing from others. Power of snippets is super intense, and while the learning curve is sharp, the potential to generate massive amount of work with minimal keystrokes cannot be underestimated. Set up snippets, autocomplete and use a latex plugin and you’re golden. (This last sentence is a non-trivial amount of work.) See the following resources: vimtex, ultisnips, and vim-snippets. For a demonstration of the power of combining vim and latex see the following blog entries by Gilles Castel: 1, 2 and 3.
• LyX: a WYSIWYM document editor, this program attempts to bridge the plaintext source code style of LaTeX and the WYSIWYG editors. In my humble opinion, this is not really a good place to start if you wish to do more than just math homework documents. In practice, using this program eventually hinders you from using the full power of LaTeX beyond the basic equation editing.

Next in the series we will cover the basic structure of a latex document, and common resources to help you as you build your code.

## Masters Thesis

I defended my thesis, “Computational Verification of the Cone Conjecture”, in December 2018, and submitted all final edits in May 2019.

• Dr. Matthias Beck
• Dr. Serkan Hosten

Abstract:

The set of polyhedral pointed rational cones form a partially ordered set with respect to elementary additive extensions of certain type. This poset captures a global picture of the interaction of all possible rational cones with the integer lattice and, also, provides an alternative approach to another important poset, the poset of normal polytopes. One of the central conjectures in the field is the so called Cone Conjecture: the order on cones is the inclusion order. The conjecture has been proved only in dimensions up to 3. In this work we develop an algorithmic approach to the conjecture in higher dimensions. Namely, we study how often two specific types of cone extensions generate the chains of cones in dimensions 4 and 5, whose existence follows from the Cone Conjecture.

A naive expectation, explicitly expressed in a recently published paper by Dr. Gubeldaze, is that these special extensions suffice to generate the desired chains. This would prove the conjecture in general and was the basis of the proof of the 3-dimesional case. Our extensive computational experiments show that in many cases the desired chains are in fact generated, but there are cases when the chain generation process does not terminate in reasonable time. Moreover, the fast generation of the desired chains fails in an interesting way—the complexity of the involved cones, measured by the size of their Hilbert bases, grows roughly linearly in time, making it less and less likely that we have a terminating process. This phenomenon is not observed in dimension 3. Our computations can be done in arbitrary high dimensions. We make a heavy use of SAGE, an open-source mathematics software system, and Normaliz, a C++ package designed to compute the Hilbert bases of cones.

Full Text is below:

The actual latex code is hosted on github.

Finally set up a github to house all the custom LaTeX files that I’ve created!

Will be updating this over time – this particular github is mostly meant for personal use, though I’m happy to share what I have and continue to update this github.

As I continue to create new tools, I’ll be hosting them here.

You’ll see currently the following repositories:

## Institute of Mathematics and its Applications at University of Minnesota: SAGE Coding Sprint Journal

April 5th, 2018

• Met with the team of SAGE & Normaliz Developers at Institute of Mathematics and its Applications at University of Minnesota.
• Discovered the issues I had connecting PyNormaliz to SAGE was due to installing SAGE using the Binary, which limits the use of custom packages. While making from source allows for the use of most recent version,
• While waiting for the fresh install of SAGE, read an article about unit testing by Jeff Knupp here.
• Learned about using parallel compiling option for SAGE install, and also learned about the System Monitor (was cool to see all the cores get used at once)
• Begin writing Unit Tests

April 6th, 2018

• Shifted to the developer branch of SAGE
• Installed the newest developer version of PyNormaliz, which installs Normaliz 3.5.3
• Rewrote Top Down algorithm using Polyhedron(rays=[[vector1],…,[vectorN]],backend=’normaliz’)
• Allows for the use of common operations like verifying if a vector is contained in the Cone.
• Resolved multiple issues (that all arose basically due to me being a newb.)

April 8th, 2018

• Added multiple githubs to record my work besides my thesis
• Begin redesign of Bottom Up using the ideas above.

## Calculus I Review: Curve Sketching

Curve sketching is an important application of derivatives. Here’s an example with a polynomial, with pictures to help assist understanding:

## Notes: Real Analysis I & II

I took Real Analysis I & II with Professor Alex Schuster in 2016-2017. These comprehensive notes were compiled using lecture notes and the textbook, An Introduction to Analysis (Fourth Edition), by William Wade.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

## Tool Sheet for Real Analysis II

I took Graduate Algebra with Professor Matthias Beck in Spring 2017. These comprehensive notes were compiled using lecture notes and the textbook, David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

• Polynomial rings,
• irreducibility criteria,
• Gröbner bases & Buchberger’s algorithm,
• field extensions,
• splitting fields,
• Galois groups,
• fundamental theorem of Galois theory,
• applications of Galois extensions,
• introduction to the polynomial method with applications in graph theory and incidence geometry.

## Precalculus Resources: Spring 2017 Midterm II Review

I have gathered quizzes that relate directly to the upcoming midterm, as well as included a sample midterm. Please feel free to use these for practice! If you have trouble with the material, visit this page for additional resources. Don’t forget to grab a free version of wolfram alpha pro to help you if you’re enrolled at San Francisco State University!

## Topics Covered:

• 2.4: Polynomials
• Vertex of a parabola
• Determine the behavior of a polynomial near $- \infty$ or $\infty$
• 2.5: Rational Functions
• Determine the behavior of rational functions near $- \infty$ to $\infty$
• Determine the vertical and horizontal asymptotes of a rational function
• Determine the holes of a rational function, if they exist.
• 3.1: Exponential and Logarithmic Functions
• Their definitions as inverses
• Practice using #23 – #32 in Precalculus – Prelude to Calculus, 3rd Ed.
• 3.2: Power Rule
• Change of Base
• 3.3: Product and Quotient Rule
• Read p.249-p.252 for applications in scientific settings
• 3.4: Exponential Growth
• Compound Interest (n times per year)
• 3.5: e and the Natural Logarithm
• Understanding the definition of e will enhance your understanding of proofs later on. For now, we know it’s a real number that is approximately equal to 2.71, and it’s associated with Continuous growth!
• 3.7: Exponential Growth Revisited
• Read p.304-p.306 for an application of the approximation methods discussed in 3.6 used in financial estimations.
• Focus on Continuous Compound Interest
• 4.1: Unit Circle
• Make sure you can convert between degrees and radians.
• 4.3: Sine and Cosine
• Definition using unit Circle
• Domain and Range of Sine & Cosine
• If you want a challenge, try answering #45 from this section.