Questions tagged [motivation]

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

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intuition behind the construction of the map for showing associativity of $\pi_1(X,x_0)$.

I am studying algebraic topology.I have started with the chapter on fundamental group.Fundamental group at $x_0$ is defined to be the set of all equivalence classes of loops based at $x_0$ together ...
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Intuition behind the notation of differential operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline z}$.

I am a graduate student of Mathematics.In this semester I am studying complex analysis.Stein Shakarchi's complex analysis book defines differential operators $\frac{\partial}{\partial z}$ and $\frac{\...
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Motivation for Derivatives of Measures

Let $\mu$ be a Borel measure on $\mathbb R^n$ with compact support, and $0 < \mu(\mathbb R^n) < \infty$. We define the lower derivative and derivative of $\mu$ at $x\in \Bbb R^n$ by $$\underline{...
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1 vote
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Motivation behind the definition of paracompact.

I am self-studying topology.I encountered the definition of a paracompact space.A collection $\mathcal A$ of subsets of $X$ is said to be locally finite if each point in $X$ has an open neighborhood ...
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Motivation behind definition of product measure.

Consider two $\sigma$-finite measure spaces $(X,\mathcal F,\mu)$ and $(Y,\mathcal G,\nu)$.In standard measure theory books they define product measure as follows: $(\mu\times \nu)(E)=\int_X\nu(E_x)d\...
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2 votes
3 answers
95 views

If $\pi$ is a representation of a lie algebra $\mathfrak{g}$, why is $\pi(x)$ not required to be invertible?

The definition of a representation of a group $G$ is a homomorphism $\pi: G\to GL(V)$. So here $\pi(x)$ is an invertible linear map $V \to V$. The definition of a representation of a lie algebra $\...
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6 votes
4 answers
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Significance of Cauchy's theorem.

I am studying complex analysis. In the beginning of the chapter on integration theory there is a theorem known as Cauchy's theorem. It states that: If $\Omega\subset \mathbb C$ be open and $f:\Omega\...
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1 vote
1 answer
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Motivation for the definition of orbifold euler characteristic

Let $S$ be a normal projective surface with quotient singularities only. For each singular point $p\in S$, $(S,p)$ is locally analytically isomorphic to $(\Bbb C^2/G_p,0)$ for some finite subgroup $...
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2 answers
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What is the motivation behind the definition of Lebesgue measurable set?

We study the definition of Lebesgue measurable set to be the following: Let $A\subset \mathbb R$ be called Lebesgue measurable if $\exists$ a Borel set $B\subset A$ such that $|A-B|=0$,where $|.|$ ...
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Intuition modular equivalence of elements in general groups with element in subgroup (Herstein)

Let $G$ be a group, $H$ is a subgroup of $G$; for $a,b \in G$ we say $a$ is congurent to $b \mod H$, written as $a \equiv b \mod H$ if $ab^{-1} \in H$ Herstein, Topics in abstract algebra, page 34. ...
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Is there a connection between free–forgetful adjunctions and tensor-hom adjunctions?

In the Wiki article on adjunction https://en.wikipedia.org/wiki/Adjoint_functors, there is a motivation section that talks about how adjunctions can be viewed as "Solutions to optimization ...
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Condition for measurable function.

We define measurable functions as follows: Definition Let $(X,\mathcal S)$ be a measurable space.Then $f:X\to \mathbb R$ is called measurable function if $f^{-1}(B)\in \mathcal S$ for any Borel set $B\...
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18 votes
3 answers
598 views

How can construction of the gamma function be motivated?

The gamma function extends the factorial function. This can be proved inductively using integration-by-parts. But if you didn't already know that the gamma function had this property, and you wanted ...
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1 answer
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Help With the Similarity Solution Method

I am having trouble understanding the similarity solution method for solving partial differential equations. I have been able to replicate the highly spoon-fed example, but all of the other, more ...
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2 votes
1 answer
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Morphisms set of sub-$\infty$-category closed under equivalences

In Markus Land's Introduction to Infinity-Categories, he defines sub-$\infty$-categories the following way (p. 56): Definition. A sub-$\infty$-category $\mathscr{C}'$ of an $\infty$-category $\mathscr{...
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What does linear equivalence geometrically mean for varieties?

Suppose we have a sufficiently nice scheme or say we are working with an abstract nonsingular variety $X$ over an algebraically close field. In this setting, one can study (Weil) divisors. It is then ...
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1 answer
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Motivation for the norm of the quotient space

I'm studying Functional Analysis, more specifically, the quotient space. Since more often we see the $\sup$ as some norm definitions, I was wondering, what is the motivation for the quotient norm to ...
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3 votes
1 answer
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Motivation behind this proof of the Basel problem?

On a past paper from our examination committee there seems to be a proof to the Basel problem that is plucked out of thin air. It doesn't match any of the proofs on Wikipedia (at least by first glance)...
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2 answers
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Why do you calculate the dot product that way?

I am learning dot product these days. I understand the geometric meaning of one vector's interpretation in the same direction of the other to calculate the work in terms of force and distance in the ...
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3 votes
3 answers
140 views

Why is it important that the $p$-adic absolute value satisfy multiplicativity?

I suppose my question is really "why do we require norms in general to satisfy multiplicativity?". I ask this because for the usual absolute value on $\mathbb R$, I never feel like ...
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6 votes
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Doubts on roster notation of sets

Simple examples of sets are often described via roster notation (aka enumeration notation) like $\{0,1,4,9\}$ - simply write down the elements of the set between the two set delimiters (curly ...
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1 vote
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What is a $\sigma$-algebra, really?

I know that a $\sigma$-algebra is a suitable generalization of the notion of sample space, in the following sense: Consider a sample space $\Omega$ and a collection $\mathscr{F}$ of subsets of $\...
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1 vote
1 answer
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What are the important things to study in infinity categories?

I am reading about infinity categories. My source is http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf My aim is to think categorically so that all the constructions I deal with are natural (...
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Definition of a pseudo-free circle action

Let us consider $S^1$ as a Lie group, and suppose we are given a smooth $S^1$-action on a closed manifold. According to this paper (https://www.jstor.org/stable/60608), the action is said to be pseudo-...
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14 votes
3 answers
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Why choose sets to be the primitive objects in mathematics rather than, say, tuples?

Sets are defined in such a way that $\{a,a\}$ is the same as $\{a\}$, and $\{a,b\}$ is the same as $\{b,a\}$. By contrast, the ordered pair $(a,a)$ is distinct from $(a)$, and $(a,b)$ is distinct from ...
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Reference request for Separation axioms in Topology.

I am an undergraduate student of Mathematics and I want to study the topic "Separation Axioms" of general topology.I have already studied Basis,Subbasis,Product topology,Countability axioms,...
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3 votes
1 answer
113 views

What is the motivation behind defining continuity at isolated points?

The naive conception of a continuous function is that it is a function whose graph can be drawn without lifting your pencil off the page. In introductory analysis, we often define continuity at a ...
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3 votes
2 answers
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What is the intuition behind product topology?

Let $X$ denote the cartesian product of the sets $X_i,i\in \mathbb N$ where $(X_i,\tau_i)$ are topological spaces. Define product topology by defining $\mathcal S=\bigcup\limits_{i\in \mathbb N}\...
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5 votes
2 answers
185 views

Motivation for the Nijenhuis tensor

I'm learning about complex and almost complex structures on smooth manifolds, in particular the Newlander-Nirenberg theorem. Recall that for an almost complex structure $J$ on a smooth manifold, the ...
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26 votes
3 answers
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is it possible to motivate higher differential forms without integration?

You have been teaching Dennis and Inez math using the Moore method for their entire lives, and you're currently deep into topology class. You've convinced them that topological manifolds aren't quite ...
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2 votes
0 answers
109 views

Simplicial schemes vs simplicial presheaves

My question is not very precise, I apologize in advance. Essentially I am wondering in the context of "homotopy theory for schemes, in the broad sense, be it in motivic homotopy theory or $\...
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16 votes
4 answers
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Why are abelian groups of interest? What is their usefulness?

I am reading about Abelian groups So apparently it is a set, with an associative binary operation, and identity element, an inverse operation and the binary operation must also be symmetric. But it is ...
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6 votes
1 answer
178 views

Intuition of Algebraic Partition of Unity

In Vakil's Rising Sea (Nov. 18, 2017 Draft) on p. 130 - 131, he gives a proof that his definition of the structure sheaf on affine schemes indeed defines a sheaf. He coins this proof an argument by &...
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2 answers
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Why are Fuchsian groups interesting?

I am recently reading the book "Fuchsian groups" by Katok and now on Chapter $2$. I am curious about why Fuchsian groups are interesting. I look it up online and find answers here. Those are ...
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100 votes
8 answers
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Is Bayes' Theorem really that interesting?

I have trouble understanding the massive importance that is afforded to Bayes' theorem in undergraduate courses in probability and popular science. From the purely mathematical point of view, I think ...
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4 votes
2 answers
104 views

Why do fixed point theorems appear all over mathematics?

For example, the Banach fixed-point theorem is applied in the proof of the Picard–Lindelöf theorem about the uniqueness of solutions of ordinary differential equations and the Lefschetz fixed-point ...
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The motivation for approaching problems using discretizing tools

Introduction of the question I'm a student in mechanical engineering and we do have a lot of mathematic. We have algebra, calculus and numerical analysis classes. We are tending to learn how to ...
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1 vote
0 answers
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Motivating the multiplication of real numbers

What are some of the ways in which one can motivate the multiplication of real numbers? The sum of two real numbers can be thought in terms of jumps along a horizontal line. For example, (3)+(-2) may ...
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2 answers
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Motivation behind studying Alternating Groups

Going through "A Book of Abstract Algebra" by Charles Pinter now. At the end of Chapter 8 Permutations of a finite set, he says that: "The set of all even permutations in $S_n$ is a ...
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1 answer
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Combinatorial Technique in Selberg's Symmetry Formula

Before I ask my question, I would like to inform, I am new to this topic from a non-math background, I am trying to understand the topic. In Selberg, A. (1949). An Elementary Proof of the Prime-Number ...
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5 votes
1 answer
387 views

What is the natural motivation for smooth/étale/unramified morphisms restricting from formally smooth/étale/unramified morphisms?

Also at MO. Smooth (resp. étale) morphisms are just locally finitely presented + formally smooth (resp. étale) morphisms. For unramified morphisms, it is originally defined in EGA as locally finitely ...
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-1 votes
1 answer
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Tricky questions on one-point compactifications

We understand the one-point compactification of a topological space $X$ is the special way to build a compact space from $X$ by adjoining just one additional point such that $X$ is densely embedded. I ...
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3 votes
1 answer
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probability and generating functions...motivations and analogies

I have a question similar in spirit to this one. In essence, what does a generating function (moment, probability, characteristic, other?) "do" to a random variable $X$, and how are the ...
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1 vote
3 answers
103 views

Motivation to define $\limsup$ and $\liminf$ of sets

Let $\{A_n\}$ a collection of set. We define $$\limsup_{n\to \infty }A_n:=\bigcap_{n\in\mathbb N}\bigcup_{m\geq n}A_m\quad \text{and}\quad \liminf_{n\to \infty }A_n:=\bigcup_{n\in\mathbb N}\bigcap_{m\...
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4 votes
0 answers
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Geometric Motivation for Inner Product

I think some background will make the kind of answer I'm looking for clearer. I'm trying to think of an elementary proof of the Pythagorean Theorem. I don't like the geometric proofs because they all ...
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2 votes
2 answers
171 views

What is the motivation for Liu's definition of an algebraic variety?

I'm currently trying to understand the motivation for Liu's definition of an algebraic variety and in particular, how it arises from and generalises Milne's definition. The question is actually ...
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2 votes
1 answer
260 views

Application of the Bertrand integral

I'll have to explain to my students in an exercise class for what values of $\alpha,\beta\in \mathbb{R}$ the so-called "Bertrand integral" converges $$ \int_e^\infty \frac{dt}{t^\alpha (\ln(...
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1 vote
1 answer
150 views

Motivation for the Notion of Regular Function in Algebraic Geometry

I am studying algebraic geometry, and naturally came across the notion of regular functions as a way to make any algebraic variety a space with functions. My problem with the notion of regular ...
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2 votes
2 answers
436 views

Intuition about Euler's Theorem on homogeneous equations

I wonder, what would be the intuition or motivation to studying Euler Formula for homogeneous function $f:\mathbb{R}^k \to \mathbb{R}$ such that $f(tx) = t^n f$, for all $t>0$ . $\sum x_i \frac{\...
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6 votes
3 answers
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How do I go about self-studying Maths? Do I create a routine? Answer a bunch of Q's? [closed]

Im 19F & I'm starting my Computer Science course this year. Before starting it, I wanted to make sure I had good GCSE & A level Maths knowledge. The problem is I don't know how to go about ...
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