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0
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0answers
22 views

How did Guillaume de l'Hôpital “devise” his rule?

I saw on Wikipedia, the proof of general case of L'Hopital's rule was given by "Taylor, 1952". But L'Hopital was born in 1661, then how he came to know about this "rule", and if he just conjectured ...
3
votes
2answers
39 views

Why is it called linearly independent?

For a system of linear equations in $\Bbb R^n$ to be linearly independent, there must be a unique solution to the system (at least I'm pretty sure that's true). There are definitely other definitions, ...
5
votes
1answer
352 views

Most efficient way to learn mathematics

So there's a lot of advice on how to learn mathematics most efficiently, and it mostly revolves around doing problems, asking questions, and considering all possible generalizations. However, I was ...
8
votes
1answer
87 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
2
votes
3answers
33 views

What does Linear mean in Linear Space (Vector Space)

The course I'm taking defines a vector space or Linear space as The vector space $\mathbb{R}^n$ has a linear structure with two features: vector addition and scalar multiplication" What does it ...
0
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0answers
25 views

Is Nathanson's Elementary methods in number theory a good book? [on hold]

I found this book at my university's library and I have looked at some of the problems in Chapters I, II and III and I must say that most of them (maybe I am wrong) are not very difficult (even ...
24
votes
4answers
606 views

Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
6
votes
0answers
81 views

Imaginary Number in Logic

The equation $x^2 = -1$ was once said to have no solution. Then the number $i$ was discovered (or invented?) and our number system got richer. In particular, in this new wonderful world of complex ...
4
votes
5answers
243 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
0
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0answers
5 views

Prequisites for chemical reaction network theory

What are the prequisites for chemical reaction network theory? Furthermore, can anyone please suggest some introductory material into the field. I thank you in advance.
0
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0answers
25 views

Is there any solution manual to Halmos' Measure Theory?

I've spent some time on Halmos' Measure Theory and must upvote such a good book. I want to solve most exercises in this book. I'm not sure whether there is a solution manual or instructor manual that ...
0
votes
2answers
55 views

Fermat's Last Theorem - Variation with arithmetically descending exponents

Are there solution(s) to the following variant of Fermat's Last Theorem in the positive integers? $$ a^n + b^{n-i} = c^{n-2i} $$ I haven't been able to identify any trivial solutions. To my ...
-2
votes
2answers
20 views

Volume question

A box holds small cube shaped blocks that are the same size. Kim tires tk build a large cube out of the small blocks but finds that she needs 6 more blocks. Takashi builds a different sized cube out ...
63
votes
23answers
9k views

An example of a problem which is difficult but is made easier when a diagram is drawn

I am writing a blog post related to problem solving and one of the main techniques used in problem solving is drawing a diagram. Essentially, I want to illustrate that some hard problems (for example, ...
1
vote
0answers
20 views

Examples of Lp spaces in Applied Math

I was wondering if there are examples of exotic Lp spaces being used in applied mathematics. I know that the "special" p's (1,2 , infinity ) are of use, for example in statistics, L1 is mean, L2 is ...
15
votes
2answers
1k views

Who is the “father of number theory”?

I noticed that some sources state Fermat as the father of modern number theory while others say Gauss. I am trying to start a paper on the history of number theory for a presentation, but I cannot ...
14
votes
5answers
3k views

Opposite of Fermat's Last Theorem?

So Wiles' proof showed that no three positive integers $a$, $b$, and $c$ can solve the equation $a^n+b^n=c^n$ for any integer value of n greater than $2$. Now what about the opposite? What does this ...
3
votes
0answers
78 views
+100

Examples of calculus on “strange” spaces

I am interested in examples of calculus on "strange" spaces. For example, you can take the derivative of a regular expression[1][2]. Also the concept extends past regular languages, to more general ...
8
votes
3answers
649 views

What are some easily-stated recently proven theorems? [on hold]

What are some easily-stated relatively recently proven theorems? I don't mean they were necessarily easy to prove, just easy to state. Here are a few examples: The proof of Fermat's Last Theorem ...
-2
votes
1answer
28 views

Notation methods for the following things? [on hold]

I go to a high school that rushes concept and does not ever talk about notation. I want to be prepared for college, and not be swamped by all this notation I don't know. From SE, I would like to know ...
7
votes
6answers
242 views

Examples of arguments from connectedness

Suppose $X$ is a connected topological space. A typical way that we prove a property $P(x)$ holds for all $x \in X$ is to show that $P$ is an open and a closed condition, and that $P(x)$ for some $x ...
-4
votes
0answers
99 views

Do “my” notations already exist or not?

Because I'm a bit slow to write my lessons, when I was in classroom, especially in maths, I created my notations as the same way as notations normaly used. For example in maths we prefer write ...
0
votes
0answers
31 views

Is embedding a function make sense?

The embedding is defined in the wikipedia link as "In mathematics, an embedding (or imbedding [1]) is one instance of some mathematical structure contained within another instance, such as a group ...
-1
votes
0answers
26 views

On classification of directed topological spaces [on hold]

Is classification of directed topological spaces (not their homotopy equivalence classes!) an important subject in modern mathematics?
1
vote
2answers
33 views

Network theory and football?

I was reading the latest post on Azimuth, Network Theory in Turin, and I watched many of the lectures Baez posted on his site here. This might be a crazy question to ask considering it's not ...
7
votes
1answer
145 views

What did Lagrange, Euler, Gauss etc. learn in order to know what they knew?

What did the great mathematician, like Cauchy, Lagrange, Euler and Gauss, learn in order to know what they knew? It seems that they were extremely good in the most basic rules/structures/issues of ...
1
vote
1answer
18 views

Does a tangent exist at $x=0$ to $y=sgn(x)$?

Yesterday my professor told me that a tangent can be constructed at $x=0$ to the signum function reasoning that the two points considered while drawing a tangent must be close horizontally and not ...
44
votes
6answers
5k views

Intuition of the meaning of homology groups

I'm studying homology groups and I'm looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible ...
2
votes
0answers
20 views

List of textbooks that take a historical approach

As the title suggests my aim in this topic is to make a big list of textbooks on any mathematical topic that take a historical approach. I will start with the ones I know: Thomas Muir - The theory of ...
1
vote
0answers
26 views

Abbreviating the definition of a tangent vector field?

Let $A \subset \mathbb{R}^{n}$ be open in $\mathbb{R}^{n}$ and let $F: A \to \mathbb{R}^{n} \times \mathbb{R}^{n}$ be continuous. Then $F$ is called a tangent vector field on $A$ if and only if $F(x) ...
1
vote
1answer
24 views

Notation and name for this function?

Let $k \geq 1$; let $V,W$ be vector spaces; and let $T: V \to W$ be linear. Then how do we call and denote the function $(v_{1},\cdots, v_{k}) \mapsto (T(v_{1}), \cdots, T(v_{k})): V^{k} \to W^{k}$?
73
votes
11answers
10k views

Using “we have” in maths papers

I commonly want to use the phrase "we have" when writing mathematics, to mean something like "most readers will know this thing and I am about to use it". My primary question is whether this is too ...
2
votes
1answer
14 views

Definition of linear independence in R-module

I am revising module theory over commutative rings with 1 and I have a "soft" question regarding to why don't we define linear independence as follows: "$v_1,...,v_n$ are linearly independent if ...
0
votes
0answers
11 views

Number of possible non crossing paths on a grid of $m$ by $n$ size?

Given two points on 2 dimensional m by n grid, moving in units of 1 in either direction, how many non intersecting paths exist between the two points? in other words, with taxi cab metric, on a m by ...
-5
votes
2answers
82 views

What is the most complex mathematical topic? [on hold]

I'm a simple man living my simple life and often I like to learn more about math and science. Today my daughter asked me about how many numbers are there and I explained that there are infinite ...
3
votes
1answer
53 views

The sections of the projection $\bigsqcup_{i:I} X_i \rightarrow I.$

I just noticed something funky. Let $X$ denote an $I$-indexed family of sets. There is a projection $$\pi_X: \bigsqcup_{i:I} X_i \rightarrow I.$$ It isn't necessarily surjective, of course, because ...
2
votes
0answers
14 views

What is the purpose of continuous and differentiable dependence

In learning Gronwall's inequality you also get to learn about continuous an differentiable dependence. I know the theorems but I have no idea about their application. What is the big idea of ...
4
votes
1answer
116 views

Soft question: what are some elementary motivations of using functional analysis to study probability theory?

Recently I've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure-theoretic ...
0
votes
0answers
33 views

Is math, in the end, only geometry [on hold]

When thinking about the Universe, or "reality", Isn't every part of mathematics a tool for expressing something geometrically further down the line? Yes, every part of math is related, but isn't ...
1
vote
0answers
26 views

How important is it that I study Probability if I like Analysis/Algebra much more?

Is it crucial to a student's undergraduate studies in Math that he/she takes a course in Probability and/or Statistics? I am much more interested in Analysis/ Algebra and I was wondering if it would ...
7
votes
8answers
16k views

Back to basics: What is the fastest way to multiply two digit numbers?

I been playing different math games on my android lately (for example: Math Cruncher). I've noticed that i'm unable to quickly (under 7-8 seconds) multiply two digit numbers (i.e $ 18 * 17$). So my ...
3
votes
1answer
61 views

What solution would you come up with for this problem?

So the question is: put numbers $1, 3, 5, 7, 9, 11, 13$ and $15$ into gaps in the following expression: $$\_\_ + \_\_ + \_\_ = 30$$ The most naive approach to use summation in the group of integers ...
-2
votes
0answers
17 views

Can we build a DFA less than 5 state for word length 4( 1100)? [on hold]

========================================================== 1 if possible kindly, help me with this question.
17
votes
1answer
1k views

What is magical about Cartan's magic formula?

Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$ called "magic"? Should it be considered a highly surprising result? Does it "magically" prove several other ...
53
votes
17answers
33k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
1
vote
4answers
63 views

Why in general there is no systematic way to find counterexamples? What kind of property do they all break that lead to this? and other things

We came across counterexamples in many areas of mathematics: For example Sum of irrational numbers not necessary being irrational The "Windmill blade" function (for lack of a better name of one of ...
5
votes
2answers
73 views

How to get the most out of attending a conference talk or a research presentation? [on hold]

I'll say up front that I imagine there's a resource for this somewhere, but I was unable to find it. I've been attending conference talks and research presentations for about a year at my university, ...
12
votes
3answers
1k views

How to take the most of math lectures in college?

I am trying to become more effective in my study habits and in my case, I feel that attending classes is a huge waste of my time. The lecturer just keeps writing formulas on the board and to be ...
1
vote
2answers
29 views

Encyclopedia of Mathematics?(non-Alphabetical)

Do you know any Encyclopedia of Mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level. And what's the difference between say, ...
3
votes
2answers
229 views

Can I follow a graduate course in PDE without having studied ODE

Hi I am considering taking the first course on Partial differential equations at my university next semester. I have already taken a first course on functional analysis . I haven't taken a proof based ...