# Tagged Questions

For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

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### Calculus for Proving Properties of Discrete Objects

I posted a question earlier about a proof in graph theory I was trying to figure out. In my attempt I used Calculus to prove a part of the theorem. In the comments people kept saying how you shouldn't ...
17 views

### Style guide/typeface for handwritten mathematics

When writing math on graph paper, it's a small struggle to make my work as legible as possible and also use the page as efficiently as possible. I've read a little online about how latex typesets, ...
15 views

### Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
36 views

### What is the proper usage of $f: X \to Y$ and $f: \mathcal{P}(X) \to \mathcal{P}(Y)$ in proof writing.

I have read somewhere that suppose we are given a $$f: X \to Y$$ then $f$ is further associated with $$f: \mathcal{P}(X) \to \mathcal{P}(Y)$$ Does "associated" here means extended to a set valued ...
55 views

### Simple examples for motivation of topology [on hold]

It is easy to see motivation for groups and fields, as abstractions of operations defined on integers, rationals, reals etc. and how the results from those abstractions apply to integers, reals etc. ...
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### Why Characteristic function is given such a name?

I've come to know that, For random variable $X$ ,with a Probability mass function P, $\phi_X(t)$ defined by : $\phi_X(t) : \mathbb R \to \mathbb C$ $\phi_X(t) = E(e^{itX}) =E[\cos tX + i\sin tX]$ ...
41 views

### Can there be a metric space where no contraction has a fixed point?

We know that: If $X$ is a metric space, then every contraction has at most one fixed point. (Note: if metric space is complete, then we have existence and uniqueness) I wonder if there can be a ...
106 views

### Origins of Baby/Papa/Mama/Big Rudin

Recently, I was looking for the reviews of some Analysis books while encountered terms such as Baby/Papa/Mama/Big Rudin. Firstly, I thought that these are the names of a book! But it turned out that ...
169 views

### Why would I learn modern category theory if my interest mainly is structured sets, what would I have to gain? [closed]

A long time ago I studied mathematics at the University of Stockholm. I had a romantic view of modern algebra and manage to make the first two algebra courses by self studies in order to immediately ...
55 views

### What does it mean in general to show something is well defined? [duplicate]

There is another post that addresses this but quickly fix the problem to be something in arthmetics, and in turn what it means for that arithematics problem to be well defined. I have never ...
68 views

### Studying for grad school qualifying exams; need a little help on how to effectively study higher math. [on hold]

This is entirely embarrassing to admit, but I'm realizing, one year into my doctorate program, I don't know how to effectively study math. I feel like a failure and a fraud for even having to come ...
39 views

### Mean Value Theorem used to prove “global properties”?

This is a soft question: I am puzzled by this statement on Wikipedia: https://en.wikipedia.org/wiki/Mean_value_theorem: The theorem is used to prove global statements about a function on an ...
122 views

### On the separation axiom in a Lawvere or “generalized” metric space

According to the nLab, Lawvere metric spaces occur rather naturally (that is as certain enriched categories). A Lawvere metric space is a set $X$ equipped with a function $d : X\times X \to [0,\infty]$...
34 views

### Notation of the square (or other power) of a function $f(x)$

How do you notate the square (or other power) of a function $f(x)$? Is it $f^2(x)$ (similar to $\sin^2(x)$ for example), $f(x)^2$ or do you have to use $(f(x))^2$? Thanks in advance.
50 views

### Remove 2 matches to get a triangle? [closed]

Can you help me solve the following question? [![enter image description here][1]][1]
81 views

### Infinite sums vs infinite unions

Why is it that For every set $S$, there exists a set $\bigcup S$. is something we take for granted (even though $S$ could be infinite), while For every sequence $a_1,a_2,\dots$ of numbers, ...
139 views

### Are all theorems usable? [closed]

The (revised) question to answer: Can anyone give an example of a serious proof using this funny (revised) theorem? For any natural number $n$ and prime $p<n-1$ there exist a prime $q$ ...
2k views

### Amazing isomorphisms [closed]

Just as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether ...
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### How math help reduce terms and conditions of someone's dying wish?

Good morning everyone... This is my very first question here, so I apologise in advance for any wrongdoing which I possibly make unintentionally. So here is a little background story. I'm working at a ...
52 views

### Is Perelman comparable with Cantor? [closed]

Well, it seems Cantor is fundamental in set theory whereas Perelman extended the topological ideas of Poincaré. But set theory and topology are intimately connected, so that my question makes some ...
1k views

### Is math an exact science? [closed]

The day before yesterday I talked with a friend of mine about math. He is also a PhD student like me, and in his opinion math cannot be consider an exact science, because the same statement could be ...
208 views
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### Origins of Differential Geometry and the Notion of Manifold

The title can potentially lend itself to a very broad discussion, so I'll try to narrow this post down to a few specific questions. I've been studying differential geometry and manifold theory a ...
51 views

### A canonical example for spaces that aren't $1^{\text{st}}$ countable

To get the feeling for metric spaces (or $1^{\text{st}}$ countable spaces, in general), I always find that visualizing them as $\Bbb R^3$ gives a good intuition of what is going on (locally, of course)...
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### Motivation for the definition of continuous maps on topological spaces [duplicate]

In any category where the objects are sets equipped with certain relations and operations, the notion of "morphism" arises perfectly naturally. (Generally, a morphism between objects is one that ...
For simplicity, consider $X=\{a,b\}$. Let $Y$ be another set in bijection with $X$, and write its elements to be $a^{-1},b^{-1}$. Let $W(X)$ be the collection of all words in $a,b,a^{-1},b^{-1}$, ...