For questions about the motivation behind mathematical concepts and results. These are often "why" questions.

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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 ...
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2answers
66 views

Motivation for the study of algebraic structures

I am currently studying group theory and I realized that most concepts we study are just definitions on which we build theory. I do understand that some theorems are beautiful and don't need any ...
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0answers
42 views

Why is Grobner basis useful? [duplicate]

I don't get the motivation for calculating Grobner bases. What's good by computing a Grobner basis for an ideal of $k[X_1,...,X_n]$? Moreover, is there any theorem whose proof relies on the use of ...
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1answer
34 views

Motivation for introducing $A^{\prime} := \frac{1}{\left|G \right|} \sum_{g \in G} \Delta(g^{-1}) A \Delta(g)$ in representation theory proof

Assume $\left|G \right|= \left| G \right| \cdot 1_F$ is invertible in $F$. Let $\Delta:F \to GL_n(F)$ be a representation and $U \subseteq F^n$ be an $F$-subspace that is $\Delta$-invariant. Then ...
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3answers
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Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendieck then later gave a more abstract definition of the right derived ...
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3answers
87 views

What is the 'meaning' of nowhere dense set?

In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres). So what is the 'meaning' (i.e motivation, ...
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1answer
42 views

Providing motivation for the importance of the concept of a 'basis'.

In a few situations, I found myself being asked by younger students why the concepts of a basis was important. First, in the concept of linear spaces, it's easy to explain that having a basis allows ...
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1answer
41 views

Motivation behind Arithmetic Mean

I know that the arithmetic mean $(x_1+x_2+...+x_n)/n$ is the value that minimizes $f(x)=\sum_{k=1}^n (x_k-x)^2$; however, I'm looking for an intuitive relationship between the mean and $g(x)=\sum_{k=1}...
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1answer
32 views

Why are differential equations with sinusoidal source terms easier to solve than others?

I am a software engineer trying to wrap my tiny human brain around Fourier Transforms for a project I'm currently working on. Although I will ultimately use an open source Math library to do all the ...
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2answers
46 views

Value Proposition of Fourier Analysis?

I am a software engineer trying to wrap his head around Fast Fourier Transform (FFT). Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of ...
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1answer
56 views

Examples of physical motivation for integrals over scalar field?

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found (source): A rescue team follows a path in a danger area where for each position ...
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2answers
143 views

What is Representation Theory?

I'm beginning a course that uses representation theory, but I do not really understand what that is about. In the text I am following, I have the following definition: A representation of the Lie ...
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1answer
76 views

What is the significance of “Homomorphism”?

Certainly Homomorphism is a prerequisite to establish an “Isomorphism”(Bijection), but what does a Homomorphism tell independently when it is established between two sets? Homomorphism relates ...
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0answers
78 views

What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...
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1answer
111 views

The significance of the composition of an operator and its adjoint

As I read the literature, I have noticed that the composition $T^*T$ of a linear operator $T:H\to H$ and its adjoint frequently turns up in all kind of places. I am aware that it is Hermitian (at ...
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1answer
67 views

Why Frobenis concerned the groups which today called “Frobenius Group”?

From their work, it seems that the Ancient mathematicians were investigating a mathematical object not as a fun, but to solve some problem occurred in earlier work of someone. Lagrange, Galois, Abel ...
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8answers
1k views

What is the motivation behind the study of sequences?

I was discussing some ideas with my professor and he always says that before you work on something in mathematics, you need to know the motivation for studying/working on it. A better way to put this ...
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0answers
60 views

Motives, anabelian geometry? What are they?

I am about to delve into one of these subjects, but they had such huge prerequisite that I am hesitant, not sure which choice to make. Try as a might, I don't really understand the big picture behind ...
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1answer
32 views

Different discriminant ideal, what are their applications?

Lots of texts online on number theory do not even mention the different ideal. Some do, but then it gets ignored and is never mentioned again. I could not find a single application for it, as if it is ...
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1answer
64 views

Applications of the completeness of $L^1$

I'm teaching a measure theory class. I think one of the main motivations for the development of the Lebesgue integral is that the space $L^1(\mathbb{R})$ of integrable functions on $\mathbb{R}$ is ...
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0answers
76 views

Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic ...
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127 views

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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1answer
23 views

Particular and homogenous differential equation solution

When solving linear nonhomogeneous equations, we deal with two types of solutions: particular homogeneous Why do we have these two types of solutions for differential equations? What does each of ...
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2answers
39 views

The definition of a subspace in linear algebra

I'm trying to learn linear algebra on my own but I am stuck on the definition of a linear subspace. Let's assume I want to find out if $S$ is a subspace of $\mathbb{R}^2$, where $ S = [X_1 , X_2] $ ...
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2answers
1k views

How can I motivate myself to study math every morning? [closed]

I am jobless, so I have relatively plenty of time though life is short. I started studying math and some other things every day because I want to become an artificial intelligence researcher. When I'...
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5answers
157 views

How to make sense out of the $\epsilon-\delta$ definition of a limit?

The informal intuition for the limit of a function is this: What is the value of the function $f$ as $x$ gets infinitely close to $c$? How on earth does this monster $$ \lim_{x \to c} f(x) = L ...
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7answers
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What is the purpose of showing some numbers exist?

For example in my Analysis class the professor showed $\sqrt{2}$ exists using Archimedean properties of $\mathbb{R}$ and we showed $e$ exists. I want to know why it's important to show their existence?...
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How to naturally justify sigma additivity in Kolmogorov's PT axiomatics?

The last axiom in the axiomatization of probability theory by Kolmogorov, that states: Any countable sequence of disjoint (synonymous with mutually exclusive) events E$_1, E_2$, ... satisfies $$P\...
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3answers
74 views

Purpose of a vector space [closed]

I am currently studying linear algebra and I've seen what vector spaces are, but I can't seem to understand what their purpose is. What do they allow us to do?
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2answers
115 views

What (previously and currently unsolved) problems motivate the study/development of analysis?

As I had ever know there are at least two (previously unsolved) problems motivate the study/development of abstract algebra: (1) the ancient Greeks' three problems in compass-and-straightedge ...
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Application of abstract algebra for computer science/software engineering [duplicate]

What are some cool applications of abstract algebra for computer science/software engineering? A friend of mine has been invited to my university (most of the students are soon to be software ...
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0answers
52 views

Motivation for Definition of Tensor Product of Modules

Let $A$ be a commutative ring and $M,N,P$ be A-modules.I know that tensor product of $M$ and $N$ is a universal object ($ M \otimes N$,u) (where $M \otimes N$ is a $A$-module and $u: M\times N \to M\...
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1answer
52 views

The (matrix) definition of a positive-definite function

In the definition of a positive-definite function, why are the elements of the n$\times$n matrix chosen as $f(x_i-x_j)$ for $i, j = 1,...,n$? Also, it says that "for any real numbers $x_i$". Does ...
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9answers
649 views

Motivation for the Definition of Compact Space

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} \...
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0answers
46 views

What does a functional integral evaluation look like?

I've read the Wikipedia page on functional integration, but it really isn't very easy to understand. There don't seem to be any online videos on the subject either. In addition, when I search online, ...
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1answer
65 views

The motivation for considering exponential families of distributions [closed]

I saw problems of the form: "show that the distributions ... form an the exponential family". Why is this property, being an exponential family, important?
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1answer
181 views

What can we actually do with congruence relations, specifically?

Let $T$ denote a Lawvere theory and $X$ denote a $T$-algebra. Under my preferred definitions: A subalgebra of $X$ consists of a $T$-algebra $Y$ together with an injective homomorphism $Y \...
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1answer
101 views

Motivate why $a^{-n}$ equals to $\frac{1}{a^n}$

I have to prove that $a^{-n} = \frac{1}{a^n}$ with $\frac{3^4}{3^7}$, but before I can do that I have to understand the background. The background says: we know that $\frac{3^4}{3^7} = \frac{1}{3^3}$...
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2answers
72 views

Banach spaces containing copies of $\ell^1$

Why is it important that a Banach spaces $X$ contains (or not) copies of the space $\ell^1$? I always hear talk about it but I don't know its importance. Could someone explain this?
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1answer
90 views

Equal Categories [closed]

Let $\infty$ be the "category" of all categories, where the objects are categories and the morphisms are functors. I am trying to motivate the definition of equal categories by doing it as follows. ...
3
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1answer
152 views

What motivates the definition of a ring in abstract algebra? [closed]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for theorems?...
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0answers
46 views

Motivation for the study of units in cyclotomic fields beyond Washington's book

Right now, I am reading Larry Washington's book "Introduction to Cyclotomic Fields." In Chapter 8 of this book, the unit group of the ring of integers in a cyclotomic field (or its totally real ...
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2answers
136 views

Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...
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1answer
57 views

Justification for ordinal arithmetic at limit ordinals

Wikipedia tells me that: $$\alpha + \lambda := \bigcup_{\beta < \lambda} \left ( \alpha + \beta \right ) $$ for a limit ordinal $\lambda$. Multiplication and exponentiation, as Wikipedia says, ...
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2answers
187 views

Ordinals - motivation and rigor at the same time

Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal ...
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1answer
381 views

Traveling salesman problem: why visit each city only once?

According to wikipedia, the Traveling Salesman Problem (TSP) is: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city ...
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1answer
143 views

How to understand cocategories

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is a ...
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1answer
166 views

What is the motivation to continuous functions and measurable functions?

In topology the objects of interest are the space open sets, and a function will be continuous if the inverse image of any open set is an open set. In measure theory the objects of interest are the ...
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118 views

Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
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113 views

Motivation for Grassmannian variety

I need some information about the Grassmanian variety for my final project in algebraic geometry course that I am taking. My questions are: Why do we define the Grassmannian variety? Do we use ...