Why is the definition of $\pi$ as integral by Weierstrass "inverted"?
Reading https://en.wikipedia.org/wiki/Pi#Definition I stumpled upon the following definition as an integral, presumably given by Weierstrass:$$\pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}$$However I don't...
View ArticleAttempt to generalize the notion of absolute continuity to multidimensional...
Recall the notion of absolute continuity over a 1-dimensional domain:Definition. A function $f: I \to \mathbb{R}$, where $I$ is a compact subspace of $\mathbb{R}$, is said to be absolutely continuous...
View ArticleWhat can semigroup theory do better in the study of PDEs compared to...
I've recently come across semigroup theory in my mathematical physics class and while the theory itself feels nice to work with, I have not yet understood what does the theory offer for the study of...
View ArticleUseful examples of pathological functions
What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples?For instance, if $f'(a) = b$, then $f(a)$...
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Organizing the content of Euclidean geometry with pictorial mind maps
There is an idea that has been on my mind for a while, and I would like to share it so that it turns into a snowball. Perhaps it will be useful and attractive to engineering enthusiasts. I ask you to...
View ArticleHow do you say this, 97 is the only prime number in ...... [closed]
How do you say this, 97 is the only prime number in ......in what ? , nineties?, nineties seriously?i call number 90 to 99 a nineties.
View ArticleConfused if I should switch to mathematics major from physics. [closed]
Sorry for the length of this post, but I really want some advice.I am a freshman in physics. I am confused if I should or not switch to mathematics for next year onwards. We get only one chance to...
View ArticleLogical Dependence of Induction on the Well-Ordering Principle
I know from a Discrete Mathematics class in the spring that Mathematical Induction depends on the well-ordering principle for natural numbers. The explanation in my textbook (Rosen) did not give me the...
View ArticleCan I Start Studying Probability and Statistics After Algebra $1$, Geometry,...
I have a question regarding my mathematical background and its readiness for studying Probability and Statistics. I have completed Algebra $1$, Geometry, Algebra $2$, and Linear Algebra (half...
View ArticleIs $1234567891011\ldots$ an integer?
Is $1234567891011121314151617181920212223......$ an integer?This question came from that one and from that talk where it's noted that "integers have a finite count of digits", so that the "number" in...
View ArticleIs there a relation between known constants? [closed]
Recently I studied about the Napier's constant $e$ and learned that it is known as the formula$$ e=\lim_{x\to 0} (1+x)^{\frac{1}{x}} $$But why can't we express it in terms of $\pi$, for example?Also...
View ArticleHow "big" is harmonic analysis?
I don't exactly know how to ask this question, so forgive any lack of clarity.I understand that harmonic analysis is an abstract generalization of Fourier analysis. But I am having trouble seeing why...
View ArticleBest way to think of the space $C^w((0,T); L^p(\mathbb{R})$
Let $f \in C^w((0,T); L^p(\mathbb{R}))$, i.e.$$\{f \mid f:(0,T) \rightarrow L^p(\mathbb{R}) \textrm{ is weakly continuous}\}.$$Clearly this means for fixed $t$, $f(t,\cdot) \in L^p(\mathbb{R})$. I am...
View ArticleOpen source lecture notes and textbooks
This question is inspired by the popular "Best Sets of Lecture Notes and Articles".Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality...
View ArticleLooking for (overkill) usages of indicator functions
I am going to give a presentation about the indicator functions, and I am looking for some interesting examples to include. The examples can be even an overkill solution since I am mainly interested in...
View ArticleWhy are strong deformations retract more common than their non-strong...
Category theory strongly emphasizes in that stuff usually only matters up to isomorphism. That makes it seem like the right notion of deformation retract is that of a retract in $hTop$. However it...
View ArticleSome intuition to keep in mind this theorem of Cellular Homology.
When we start reading cellular homology,we begin with this basic but important theorem:Theorem: Let $X$ be a CW-complex and $X^m$ denote the $m$-th skeleton of its CW structure.Then we have the...
View ArticleIs it Possible to Define Expectation and Variance for a Measure?
This is a slightly soft question, but I am wondering if it is sensible to talk about the moments of a given measure $\mu$. In particular, suppose that $X$ is a random variable on the probability space...
View ArticleUnderstanding the theorem of Cellular Homology through intuition.
When we start reading cellular homology,we begin with this basic but important theorem:Theorem: Let $X$ be a CW-complex and $X^m$ denote the $m$-th skeleton of its CW structure.Then we have the...
View ArticleCounterintuitive examples in probability
I want to teach a short course in probability and I am looking for some counter-intuitive examples in probability. I am mainly interested in the problems whose results seem to be obviously false while...
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