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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...

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Why 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...

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On 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...

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Roadmap 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...

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Linear 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...

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What 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...

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Really 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...

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Building 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...

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Help 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,...

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Is 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...

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About 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:$\lambda_\mathbb{D}|dz| = \frac{2|dz|}{1-|z|^{2}}$....

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Applications 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...

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Why 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...

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How 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...

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Why 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...

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Why 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...

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Why 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...

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What 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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