Author: Nihal Uppugunduri

Math

Sine Angle Product Formula

Post author By Nihal Uppugunduri
Post date December 1, 2022
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In this post, we address the following question: does there exist a polynomial identity for $\sin(ab)$ in terms of $\sin a,\cos a,\sin b,\cos b$ like for the single angle addition formula? Formulated rigorously, does there exist a polynomial $P$ such that $\sin(ab) = P\left( \sin a,\cos a,\sin b,\cos b \right)$ for all $a,b$ ? We suspect that $P$ does not exist, which we try to show here. (Actually, a more general question that is worth discussing is whether there exists a rational identity for $\sin(ab)$ , e.g. with $P$ replaced by a rational function. But we discuss polynomial for now.)

Tags Series: Sine Angle Product Formula, Studies

Math

Towards Automated Verification of Math

Post author By Nihal Uppugunduri
Post date December 1, 2022
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Given the proliferation of computers in so many aspects of our lives, it has always surprised me that verification of submitted math research is not yet totally automated. In practice, requirements like the conferral of doctoral degrees or the referral process for arXiv are still used today in order to judge whether a proposed research contribution is valid or not. This is despite the fact that the subject of math itself is supposed to be only dependent upon formal logic for validity. The idea of automating the verification of math is not a new one, and since the twentieth century there have been some interesting proposals and thoughts towards making it happen.

Tags General and Terminology

Math

Linear Independence in Number Theory

Post author By Nihal Uppugunduri
Post date December 1, 2022
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In this article, we discuss some questions I had regarding linear independence as it shows up in subjects like transcendental number theory.

Tags Studies

Education

The Problem of Curriculum Growth

Post author By Nihal Uppugunduri
Post date November 27, 2022
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Many subjects are cumulative. Mathematics by nature is; many sciences end up being practically cumulative, since even if a theory is proven wrong at some extreme limit and replaced by another theory there, typically the old theory is still applicable enough to the phenomena it adequately addresses that it is still useful. A typical example in physics is the continued importance of classical mechanics, despite the fact that it has shown limitations and has had to be replaced by theories such as relativity and quantum mechanics. History and analysis of literature by nature are cumulative, and even subjects that deal “mostly” with analysis of current snapshots can still benefit from learning from the past in order to make better judgments and decisions for the future, making them somewhat cumulative as well.

Math

On Logical Independence

There are multiple definitions of logical independence out there in the literature. In this article, we tease out these definitions and investigate their relationships with each other.

Tags Studies

Math

Trigonometric Identities as Functional Equations

Post author By Nihal Uppugunduri
Post date November 22, 2022
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We take some basic trigonometric identities and essentially ask whether trigonometric functions are the only ones that satisfy these identities. In other words, what are all the functions that satisfy a given trigonometric identity? Thus, we take trigonometric identities and use them as functional equation problems.

Tags Series: Trigonometric Identities as Functional Equations, Studies

Math

Field Automorphisms of the Real Numbers

Post author By Nihal Uppugunduri
Post date November 22, 2022
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Here, we try to solve the following problem (and some related ones):

Problem. Prove that if $f\mathbb{:R \rightarrow R}$ such that $f(x + y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$ , then either $f(x) = 0$ or $f(x) = x$ .

Tags Studies

Math

Hypersurfaces and Manifolds

Post author By Nihal Uppugunduri
Post date November 22, 2022
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In multivariable calculus, we study line integrals and surface integrals, as well as volume integrals and various theorems involving these concepts, including for example the gradient and divergence theorems. We then learn that manifold calculus provides vast generalizations of these concepts and results. We try to understand and tease this out further in this article.

Tags Studies

Math

Continuity on Subsets of Topological Spaces

Post author By Nihal Uppugunduri
Post date November 22, 2022
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In general topology, a function $f:T \rightarrow U$ is defined to be continuous if the preimages of open sets in $U$ are open in $T$ . This doesn’t define however functions that are continuous on just a subset of $T$ or at even a point. An application of continuity on subsets would be topological fields where the multiplicative inverse couldn’t possibly be continuous on all of $T$ , since $T$ includes 0. Since topological fields are standard and well-known concepts, the idea of generalized continuity on subsets must also be standard and well-known. In this article, I scour the Internet to understand how we define this today according to already-established standard literature.

Tags Studies

Math

Subring Generated by Subset

Post author By Nihal Uppugunduri
Post date November 22, 2022
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There are multiple equivalent definitions of the (sub)ring generated by a subset. In this article, we state these definitions and prove their equivalence.

Tags Studies