Tag: Studies

Categories
Math

Field Automorphisms of the Real Numbers

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.

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Categories
Math

Hypersurfaces and Manifolds

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.

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Categories
Math

Continuity on Subsets of Topological Spaces

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.

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Categories
Math

Subring Generated by Subset

There are multiple equivalent definitions of the (sub)ring generated by a subset. In this article, we state these definitions and prove their equivalence.

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Categories
Chemistry

Uniqueness of Chemical Equation Balancing

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: O_{2} + 2H_{2} \rightarrow 2H_{2}O and 2O_{2} + 4H_{2} \rightarrow 4H_{2}O are equivalent.) Does this require actual chemistry, or is it a more general mathematical fact (under maybe generally mild assumptions?) Can we show this to still be true for all the chemical equations we’d ever actually balance?

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Categories
Physics

Expectation of Single-Qubit Property in Two-Qubit System

Consider a two-qubit system in quantum state \psi. This system may be entangled, so it may not be possible to “separate it out” into a tensor product of individual qubit states. The first qubit has a property given by operator M; what is the expectation of M?

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Categories
Math

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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