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0
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0answers
14 views

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 ...
1
vote
0answers
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, ...
1
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0answers
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 ...
3
votes
3answers
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 ...
-3
votes
1answer
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. ...
0
votes
0answers
22 views

What are interesting functions in 2D that vary visually as compositionality increases?

I wanted to create a function that its shape was a function of the depth of the compositionality (on a fixed interval). For example consider some compositional function $$f(x_1, x_2) = g( g( g( h_1(...
1
vote
1answer
59 views

Does anyone know a no-nonsense intro to “logic for mathematics” that I can give to a Year 11 student?

I'm looking for material on propositional and first-order logic to give to a Year 11 student that explains how they're used "in practice." For example, I want to be able to write the null-factor law ...
3
votes
1answer
65 views

Writing Math: Is using both $e^x$ and $\exp(x)$ ok for longer works?

I wanted to know what you guys think about mixing both notations for the exponential function $e^x$(for simple argument and to save space for larger equations) and $\exp(x)$ (for more complicated ...
2
votes
0answers
80 views

What is the point of making dx an infinitesimal hyperreal?

It seems fairly common to describe $\mathrm{d}x$ in nonstandard analysis as an infinitesimal. But after thinking it through (and then skimming Keisler's text), I can't see the point and actually think ...
2
votes
2answers
69 views

Simple question: Which is the Wikipedia definition of axiom of choice

I looked up Wikipedia, and on the top of the page it says: For every indexed family $(S_i)_{i \in I}$ of nonempty sets, there exists an indexed family $(x_i)_{i \in I}$ of elements such that $...
0
votes
1answer
41 views

What subbase generates metric topology?

Let metric topology be the topology generated by metric balls of a metrizable space $X$ Is there a subbase $S$ that generates the metric topology? I am asking because in most textbooks, it seems ...
1
vote
0answers
40 views

Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
8
votes
7answers
502 views

Mathematics and Perfectionism? [on hold]

Background I am an undergraduate and I have just finished my first calculus class (Calc I) this summer. While this class has gone very well for me by any objective standard, I find myself drifting ...
1
vote
2answers
46 views

How was statistics formulated [on hold]

I'm sorry for the naive question, but I've only been taught statistics at a high school level (extremely basic) and it always remained a mystery as to how certain concepts in statistics existed (...
2
votes
2answers
87 views

A 3-valued mathematical logic?

Classical propositional logic is consistent and in conformity with human language. A formal statement is true or not true and it is possible to develope rules with which it is possible decide which ...
2
votes
0answers
50 views

Biggest Unsolved Problems In Graph Theory ( a la Riemann Hypothesis to Number Theory)

I'm not sure whether this is the right place for this question, but what are the most major unsolved problems in graph theory? (Not just a list, but something like a top 10 list or something like that)...
1
vote
0answers
69 views

Brief overview of the foundations of math?

I am very interested in finding out about the foundations of math. When I feel foolish enough to try, I visit Wikipedia and end up in a truly endless rabbit hole. There are dozens of terms that I don'...
10
votes
4answers
170 views

How do we know logic works? [duplicate]

Every time I read about a theory in mathematics, it usually starts with axiomatizing the most fundamental concepts that are going to be treated. Recently, I have started finding this troubling. In ...
1
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0answers
35 views

Metric and harmonic map (or map between manifolds)

There are many questions about what metric can be placed on a given manifold . For example , place a metric with non-negative curvature. As I know , the Gauss-Bonnet theory is useful in this question....
1
vote
2answers
46 views

How do I write $B = \left\{\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \in \boxed{?}| \ldots\right\}$ with proper notation

Let $x \in X \subset \mathbb{R}^n$, then I define a set: $$A = \{x \in X| 1^Tx = 0\}$$ Now supose I have another element $y \in Y \subset \mathbb{R}^n_{+}$ I concatenate $x,y$ in to a single vector ...
2
votes
1answer
68 views

How does a complex algebraic variety know about its analytic topology?

This question has two parts. The first is a reference request regarding a result I assume is standard, and the second is a soft question asking for philosophy and intuition about an issue the first ...
2
votes
1answer
30 views
+50

Upper bound for the no. of maximal ideals of a finite commutative ring ( with unity ) in terms of the order of the ring and/or its number of units?

Is there any known upper bound for the no. of maximal ideals of a finite commutative ring ( with unity ) of order $n$ , in terms of $n$ and /or in terms of the no. of units of the ring ? Say , does ...
4
votes
2answers
124 views

Gauss's thesis: Theory of equations or hypergeometric functions, or both?

I had read in multiple places, and always believed, that the first good proof of the fundamental theory of algebra (that a polynomial of degree $n$ over $\Bbb{C}$ has $n$ roots, with suitable ...
2
votes
0answers
44 views

Overly advanced techniques in simple questions: trouble in exams?

I have yet to ascend to the heights of university education, and before I do, I would like to clear something up: As I was revisiting a book on complex analysis, I see questions like the following: ...
0
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0answers
43 views

Attempt to represent gaussian integers with matrices over ${\mathbb Z_+}^{4\times4}$

Let us first consider the generating element for $C_2$ : $$M_1 = \left[\begin{array}{cc}0&1\\1&0\end{array}\right], \text{ and } P_1 = ({M_1})^2 = I_2 = \left[\begin{array}{cc}1&0\\0&1\...
2
votes
1answer
42 views

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 : $\phi_X(t) : \mathbb R \to \mathbb C$ $\phi_X(t) = E(e^{itX}) =E[\cos tX + i\sin tX]$ ...
0
votes
1answer
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 ...
5
votes
1answer
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 ...
1
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0answers
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 ...
2
votes
1answer
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 ...
4
votes
1answer
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 ...
3
votes
1answer
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 ...
4
votes
3answers
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 $d : X\times X \to [0,\infty]$...
1
vote
2answers
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.
-2
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1answer
50 views

Remove 2 matches to get a triangle? [closed]

Can you help me solve the following question? [![enter image description here][1]][1]
3
votes
3answers
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, ...
2
votes
0answers
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 $q$ ...
35
votes
11answers
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 ...
5
votes
2answers
119 views

What exactly is $\mathbb{P}_\mathbb{Z}^n$?

So, I have the following definition of $\mathbb{P}_A^n$ for an arbitrary (commutative) ring $A$, from Hartshorne: Set $S=A[x_0,\ldots,x_n]$, so that $S=\bigoplus_{d\geq 0}S_d$ as a graded ring, $S_+=\...
3
votes
0answers
62 views

Algebraic number that exponentiated with algebraic number give $\pi$

I'm not sure if an algebraic number elevated with an algebraic exponent can give rise to a transcendental number. If that's the case does anybody know a closed form for an algebraic number that ...
2
votes
1answer
64 views

Have there been mathematicians that were bad at math during school, but excelled later on through self interest? [closed]

Have there been mathematicians that were bad at math during school, but excelled later on through self interest? I don't know if I'm mathematically inclined but ever since I was a kid (before I ...
0
votes
1answer
34 views

Any “interesting” theorems for element-wise matrix product?

From the point of view of linear algebra, the "natural" multiplication operation for matrices is the usual matrix product, and there are lots of theorems involving this product---e.g. the result $det(...
8
votes
1answer
64 views

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 ...
-5
votes
0answers
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 ...
10
votes
9answers
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 ...
19
votes
2answers
208 views
+100

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 ...
1
vote
3answers
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)...
1
vote
0answers
21 views

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 ...
0
votes
0answers
17 views

Lebesgue Theory in Higher dimensions

What are some things that we have to watch out for when going to higher dimensions (greater than or equal to 2) for Lebesgue Measure Theory? For instance, is there anything that is true in Lebesgue ...
1
vote
1answer
24 views

Formal construction of free groups and objections in arguments

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 $a,b,a^{-1},b^{-1}$, ...