Creative uses of the Moment Generating Function of a Random Variable
The theorem below finds the raw moments of the $\text{Lognormal}$ distribution by using the MGF of the $\text{Normal}$ distribution. I found this to be quite creative (since it is different from the...
View ArticleWhy are compact sets called "compact" in topology?
Given a topological space $X$ and a subset of it $S$, $S$ is compact iff for every open cover of $S$, there is a finite subcover of $S$.Just curiosity:I've done some search in Internet why compact sets...
View ArticleOn the mathematical convention used to talk about biconditional proofs
Given $P \Longleftrightarrow Q$, the following does apply:$P \Rightarrow Q$ is equivalent to:$P$ is a sufficient condition for $Q$,$Q$ is a necessary condition for $P$.$Q \Rightarrow P$ is equivalent...
View ArticleRoadmap for self-learning pure mathematics from high school level onward....
Motivation:- I’m making this post because I know a lot of people who are genuinely interested in learning pure math but have no formal background or experience or help. Many of them are unsure how to...
View ArticleLinear combinations of minor determinants
While thinking about properties of determinants, alternating multilinear maps, Grassmann algebras, I've stumbled upon some class of scalar- or vector-valued functions defined on spaces of matrices or...
View ArticleWhat is the mental model for (modular) lattices?
When we do computations* in abstract commutative rings, we use a mental model all the time: the arithmetic of integers. After the foundations are developed, we don't really care about the specific ring...
View ArticleReally advanced techniques of integration (definite or indefinite)
Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? Every...
View ArticleBuilding Foundations for Geometric Analysis: Advice and Strategies?
I am a first-year graduate student, and I aim to shift my focus to geometric analysis because I have recently realized that it is likely the field I want to pursue. In the past, I was more focused on...
View ArticleHelp with deciding on the right bachelors [closed]
I soon have to decide on a bachelors degree, but I'm a little torn between two. One is a business mathematics degree, which focuses on analysis, linear algebra, financial mathematics, topology,...
View ArticleIs there such a new notion of weakly sequentially continuous in the literature?
Let $G:X \rightarrow Y$ be a continuously differentiable nonlinear operator, with $X$ and $Y$ being real Hilbert spaces. I want to know what is the weakest hypothesis necessary to guarantee that, for...
View ArticleAbout the hyperbolic metric on the disk
In this article https://www.math.stonybrook.edu/~bishop/classes/math401.F09/Beardon-Minda.pdf the authors define the hyperbolic metric as follows:$\lambda_\mathbb{D}|dz| = \frac{2|dz|}{1-|z|^{2}}$....
View ArticleApplications of complex analysis?
I am a math major and I really cannot understand complex analysis. I've tried it twice before doing so poorly on the midterms that I had to drop. I gave it a go during this summer and I again ended up...
View ArticleWhy are infinite-dimensional vector spaces usually equipped with additional...
In a first course in linear algebra, it is common for instructors to mostly restrict their attention to finite-dimensional vector spaces. These vector spaces are usually not assumed to be equipped with...
View ArticleHow do I convince someone that $1+1=2$ may not necessarily be true?
Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical...
View ArticleWhy are skew fields defined the way they are?
As far as I can judge, an interesting question was asked recently. Unfortunately, it was not specific enough. Which is still, in my opinion, intrinsically linked to the problem raised.Personally, I...
View ArticleWhy are skew fields called skew fields? [closed]
As far as I can judge, an interesting question "Why are fields defined the way they are?" was asked recently. Unfortunately, it was not specific enough. Which is still, in my opinion, intrinsically...
View ArticleWhy are modular lattices important?
A lattice $(L,\leq)$ is said to be modular when$$(\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$where $\vee$ is the join operation, and $\wedge$ is the meet...
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Is open set and interiority a necessary condition for differentiability?
I am going by what I am seeing on Wikipedia: In 1D, the standard definition for differentiability is,A function $f:U\to\mathbb{R}$, defined on an open set$U\subset\mathbb{R}$, is said to be...
View ArticleWhat is mathematical research like?
I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly?...
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Why is $\Gamma(\sin(x)) \approx \csc(x)$?
Is there any deeper mathematical reason for why $\Gamma(\sin x) \approx \csc(x)$ for $(0<x<\pi)$, where $\Gamma(x)$ is the Gamma function (apart from the reasoning below)?Both $\Gamma(\sin x)$...
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