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...
View ArticleComputing 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...
View ArticleWhy 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...
View ArticleSummation 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,...
View ArticleHow 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...
View ArticleProof 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...
View ArticleWhy 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...
View ArticleCan 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...
View ArticleAlgebraic 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"...
View ArticlePoset 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...
View ArticleConfusion 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 $E$...
View ArticlePrerequisites 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...
View Articleonline 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...
View ArticleWhat 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...
View ArticleWhat 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...
View ArticleUnexpected 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...
View ArticleWhat 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...
View ArticleChallenging 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...
View ArticleWhat 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...
View ArticleAre 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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