We balance chemical equations by making sure the number of atoms of each element is the same on both sides. However, with this as the only criterion, it’s actually not immediate that a particular balancing we find would have to be the only one (up to equivalence.) In other words, we ask whether the set of coefficients we could balance with is one-dimensional (since scaling results in equivalency: 
Author: Nihal Uppugunduri
Consider a two-qubit system in quantum state 
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Defining Topological Groups and Algebras
We want to define topological groups as having a compatibility axiom that intuitively says that the “group operation is continuous.” But the group operation is a multivariable function, so how would we define this? And what about for general topological algebras? (Here, “algebra” is meant in the universal algebra sense, not in the “vector space with bilinear multiplication” sense.)
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Reacting in Programming
There are lots of different concepts in programming related to reacting when special occurrences happen. This post is an attempt at sorting these concepts out and illuminating how they relate to each other. Of course, a lot of this is based on my understanding, and it’s possible that different people will have different definitions of these terms; however, this is meant to accurately reflect the standard meanings.
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Main Branches of Math
Math, or rather our knowledge of it, is continually expanding in both volume and diversity of subfields. However, to better understand a general, high-level organization of it, let’s take a snapshot of current mathematical knowledge. If we roughly partitioned this corpus into its main branches, based on topical similarity and general amount of (current) content, what would they be?
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List of Confusing Math Terminology
There are many examples of confusing terminology in math, along with multiple reasons why they are confusing. Some terms are ambiguous, meaning that they are reused for different concepts. Other terms are “almost ambiguous,” meaning that they are not word-for-word reused, but have such similar names that it would be easy to be confused. These scenarios go from worst to best — the latter case is at least more manageable, if still not desirable.
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Suggestions for Learning Math
Sometimes, it can be overwhelming to even think about how to approach a subject. Where do you start?
Here, I list some pointers and resources that have helped me most effectively learn math. I also point out some resources which I may not have used myself, but that other people I know have had positive experiences with.
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Non-Technical Intro to Isomorphisms
There is an important concept in math called an isomorphism. This is typically introduced when you study a subject called abstract algebra, although it applies beyond abstract algebra too. While its definition generally requires some technical background, in this post we attempt to provide a non-technical introduction.
In this post, I discuss some philosophical questions concerning uniqueness and validity of theories in physics.
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My Journey Towards Partial Progress
I was born with a highly perfectionist personality. As I went through elementary school, I’d struggle at times with needing to just complete things versus making them absolutely perfect. Up until around seventh grade, I would fail to finish tests, but my teachers would recognize that the portions I had done were fully accurate. Over time, I’ve gotten better at completion, through my own efforts as well as those of my parents, teachers, counselors, friends, and other supporters. Yet being comfortable with partial progress is still something I struggle with in certain scenarios.
