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In mathematics, is there a conjecture that disproved by the existence of a...

In mathematics, is there a non-trivial conjecture that can be disproved by the existence of a counterexample, without explicitly constructing the counterexample itself, because of this construction is...

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Computing eigenvectors without floating point operations

Given a matrix $A$ with elements $a_{i,j}\in \mathbb C$ I am looking for the eigenvectors.The question has a background in computing with a computer, and I really despise floating point values with all...

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Why does 3blue1brown use the "around a point" to describe a derivative?

In this article (which includes a link to the video version of the article as well), Grant Sanderson aka 3blue1brown describes a derivative. He says at the end of the passage headed "The Paradox",Since...

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Summation of uncountable sets

This is a very soft question, but I am wondering if the summation operation, often symbolized with $\sum$, is independent of countability. Typically, summations often imply countability, in any series,...

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How to relate precisely the definition of “power series” to the general...

In order to understand the concept of power series, I’d like to relate it precisely to the general notion of infinite series. My understanding of infinite series is as follows:If $\{a_n\}$ is a...

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Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we...

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Why do we want probabilities to be *countably* additive?

In probability theory, it is (as far as I am aware) universal to equate "probability" with a probabilistic measure in the sense of measure theory (possibly a particularly well behaved measure, but...

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Can the golden ratio accurately be expressed in terms of $e$ and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the infamous...

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Algebraic structure of the extended real line $\overline{\Bbb R}$.

The extended real line $\overline{\Bbb R}$ is defined to be the set $\overline{\Bbb R}=\Bbb R\cup\{\infty,-\infty\}$, where the adjoined symbols $\{\infty,-\infty\}$ represents the "points at infinity"...

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Poset structure for mathematical constants/variables/functions

In mathematical writing, constants usually come before variables, such as $2x$ or $\pi x$. In integration, most mathematicians write $f(x)dx$ instead of $dx f(x)$ to prevent confusion of where the...

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Confusion regarding Murphy's definition of a spectral measure

In Murphy's $C^*$-Algebras and Operator Theory, he defines a spectral measure in Section 2.5 as follows:Let $\Omega$ be a compact Hausdorff space and $H$ a Hilbert space. A spectral measure $E$...

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Prerequisites for Studying a Book on PDE's

I'm taking a course on Electromagnetism that will cover the boundary-value problems -- the solutions to Laplace's and Poisson's equations for various symmetries, ranging from cartesian to spherically...

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online classes for math masters pre reqs

I'm currently getting my masters in video game development and next year want to start on my math masters. Some of the pre-reqs I need to still take are calc 3 and linear algebra, does anyone know of...

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What book is good in studying beginning optimization?

Recently, I heard some talks about Optimization. And I am beginning to love that field.I want to study beginning optimization, what book can you recommend for me? Alsowhat tips can you give to a...

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What are some must-read math research papers for undergraduate students?...

I'm an undergraduate student looking to go beyond standard coursework. I want to explore mathematical research papers which are both accessible and impactful. I'm interested in papers offering deep...

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Unexpected examples of natural logarithm

Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common...

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What conditions must a sequence of functions​ satisfy to represent any “nice”...

What conditions must a sequence of functions​$ \{f_n\}_{n=1}^{\infty} $ must have in order to generate any "nice" function $F(x)$ as $F(x)=\sum\limits_{-\infty}^\infty a_n f_n(x)$ .For example:The...

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Challenging Integrals for High School Students

I am now in my last year of high school. We have covered all the techniques useful for indefinite integration that are included in our Maths and Further Maths courses. This includes:Integration by...

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What other tricks and techniques can I use in integration?

So far, I know and can use a reasonable number of 'tricks' or techniques when I solve integrals. Below are the tricks/techniques that I know for indefinite and definite integrals separately.Indefinite...

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Are all blackboard bold capital letters used in standard mathematical...

As I’ve progressed in mathematics, I’ve noticed more and more blackboard bold symbols showing up: $\mathbb{R}$ for the reals, $\mathbb{Q}$ for rationals, $\mathbb{C}$ for complex numbers, $\mathbb{Z}$...

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