For questions about gaining and achieving motivation.

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4
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2answers
700 views

Why do we need (the abstract concept of) random variables (in discrete probability models)?

What we defined: Suppose we have a (discrete) probability model $\left(\Omega,P\right)$, where $P$ is the probability function (at least, that was the way it was introduced in a course I took; that ...
4
votes
2answers
137 views

Why a norm and not some other function that defines a metric?

If one defines on a $\mathbb{R},\mathbb{C}$-vector space a norm this gives rise to a metric. Why are particularly mappings that satisfy the norm axioms so important that in every book for beginners on ...
4
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1answer
99 views

Why doesn't the definition of (model-theoretic) conservative extension need strengthening?

In the wikipedia page dedicated to conservative extensions, we find the following sentence: $T_2$ is a model-theoretic conservative extension of $T_1$ if every model of $T_1$ can be expanded to ...
4
votes
1answer
360 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
4
votes
1answer
97 views

What is a motivation for this theorem and what is an example this theorem is applied?

If one doesn't know a motivation, it's hard to memorize such a theorem. So do I. Rudin RCA p.30 Let $(X,\Sigma,\mu)$ be a measure space such that $\mu(X)<\infty$ and $f\in L^1(\mu)$. ...
4
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2answers
306 views

Motivation for the study of the Chern connection

Given a Hermitian metric $H$ over a holomorphic vector bundle $E$ with holomorphic structure $\overline{\partial}$, there exists a unique connection $\nabla$ (named afer Chern) satisying the following ...
4
votes
1answer
398 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
4
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3answers
475 views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
4
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0answers
57 views

Why do we care about non-$T_0$ spaces?

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same ...
4
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0answers
489 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
3
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2answers
114 views

Motivation for abstractness

I'm seeking examples of concepts or theorems in school mathematics that are better understood when we generalize (when we deal with a more abstract concept where the former concept is a special case ...
3
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1answer
115 views

New ways to light the fire again

Recently I've been studying a lot of analytic geometry and this subject made my motivation drop. The thing is, the courses aren't stopping and I'm beginning to lose the passion I had before. I need ...
3
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2answers
115 views

Use of $\mathbb N$ & $\omega$ as index sets

Why all the properties of a sequence or a series or a sequence of functions or a series of functions remain unchanged irrespective of which of $\mathbb N$ & $\omega$ we are using as an index set? ...
3
votes
1answer
61 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
3
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6answers
147 views

Motivation for creation of complex exponentiation

I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived?
3
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1answer
75 views

Motivation and application for stochastic geometry.

I am starting a PhD, and there is a good chance that my project will be oriented in the study of random polytopes or/and random mosaics. I was wondering what are the motivations and applications of ...
3
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3answers
199 views

Intuition behind independence & conditional probability

This is a basic question. I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$. ...
3
votes
1answer
94 views

Motivation for definition of Mobius function

Why is the Mobius function defined the way it is? \begin{align*} \mu(n) = \begin{cases} (-1)^r & \text{ if $n$ is square-free and is of the form }n=p_1p_2\ldots p_r\\ 0 & \text{ if $n$ is not ...
3
votes
1answer
87 views

Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
3
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0answers
111 views

Do expressions like $(-1)^{2/3}$ show up naturally in pure or applied math?

Let $x$ denote an arbitrary real number. Then $x^n$ makes sense for arbitrary $n \in \mathbb{N},$ via the obvious recursive definition. We can extend this definition by asserting that if $x$ is ...
3
votes
0answers
170 views

What are the advantages of proof-relevant mathematics?

I've read that Theorems in HoTT (homotopy type theory) tend to characterize the space of proofs of a proposition, rather than simply state that the corresponding type is inhabited. So, HoTT ...
3
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0answers
173 views

Algebraic Geometry question

Why do we study projective normality of a projective variety ? Does it have anything to do with non-singularity ? Any other purpose to study this ?
2
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5answers
420 views

Why - not how - do you solve Differential Equations? [closed]

I know HOW to mechanically solve basic diff. equations. To recap, you start out with the derivative $\frac{dy}{dx}=...$ and you aim to find out y=... To do this, you separate the variables, and ...
2
votes
1answer
153 views

What motivates the study of Abelian groups?

Monoids arise naturally as endomorphism monoids, and groups arise naturally as automorphism groups. These are among the primary motivators for their study, in my opinion. What are the (main) ...
2
votes
1answer
44 views

Why do we require $X$ to be Hausdorff when defining $C(X)$?

In Efton Park's "Complex Topological K-theory", he begins a section by defining the Banach algebra $C(X)$ (continuous functions $f: X \rightarrow \Bbb C$), presupposing that $X$ is Hausdorff and ...
2
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2answers
140 views

What is the importance of the spectral theorem?

I know that the spectral theorem tells us that in the case of a real inner product space, an operator is self adjoint if and only if there is an orthonormal basis with only eigenvectors of that ...
2
votes
1answer
270 views

Intuition behind the definition of Adjoint functors

I think of adjoint functors as some sort of inverses. So, the first part of the definition looks reasonable that there exists natural transformations $$\epsilon : FG \rightarrow 1_C$$ $$\eta : 1_D ...
2
votes
2answers
71 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
2
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2answers
103 views

What is the significance of permutable subgroups? (and $X$-permutable subgroups?)

Let $G$ be a group and $H$, $K$, $X$ be subgroups of $G$. We say $H$, $K$ are permutable if $HK=KH$. or we say $H, K$ are X-permutable if $‎\exists x, x\in X$ such that $H^{x}K=KH^{x}.$ Why are ...
2
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1answer
206 views

Why do we need continuous random variables since they can be approximated by discrete ones?

I do not understand the motivation of developing the theory of continuous random variables. Given simple discrete random variables, the continuous ones can be well approximated.
2
votes
1answer
74 views

What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?

The definition of the Lebesgue measure on $\mathbb R^n$ is fundamentally tied up with the following assumption: The measure of the cartesian product of $n$ intervals should be the product of the ...
2
votes
2answers
67 views

Practical motivation for analysis at school

I want to show pupils at an age of 14 to 17 and who do not like math what they can use analysis/analytic geometry in a job. I could not remember an impressive example for most topics. Let's assume ...
2
votes
1answer
317 views

Is RCA-Rudin one of the worst textbooks? [closed]

Someone told me that "Real and complex analysis - rudin" is actually rated a bad textbook among researchers, since it gives no motivation. Is it true? I agree that this text provides less ...
2
votes
1answer
69 views

For what reason, the surface measure represents the surface area?

Let $||\cdot||$ be a norm on $\mathbb{R}^n$ (It's an arbitrary norm, not 2-norm) Define $S^{n-1}=\{x\in\mathbb{R}^n : ||x||=1\}$ Let $\mu$ be the n-dimensional Lebesgue measure. Define ...
2
votes
4answers
91 views

Motivating complex structure on $\mathbb{R}^2$

I'm giving a talk to a group of bright but not all that mathematically sophisticated students on the subject of complex numbers. I'd like to introduce complex numbers via geometric considerations ...
2
votes
1answer
161 views

Why should we accept the existence of subsets $A$ such that neither $A$ nor $A^c$ are recursively ennumerable? And how can we persuade others?

Encode every pair $(t,x)$ (where $t$ is a Turing machine and $x$ is an input string) as a distinct natural number. Then the halting subset $H$ fails to be recursive. $$H := \{(t,x) \in \mathbb{N} ...
2
votes
1answer
147 views

Motivations for Prime Factorizaton

I'm at the beginning of some middle school math sessions on divisors, gcd, lcm, and prime numbers. It's the first place in the curriculum that the students encounter the three latter concepts ...
2
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0answers
42 views

What is the motivation behind the study of pattern-avoiding permutations?

There is a ton of research on pattern-avoiding permutations (permutations that do not contain some designated permutation pattern). We're figuring out how to enumerate them, what random ones are ...
2
votes
1answer
171 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
1
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2answers
81 views

Relation between long exact sequences and Derived functors

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...
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vote
5answers
100 views

What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...
1
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3answers
83 views

Mean Value Theorem Motivation

I am currently practicing presenting mathematics to various audiences and am considering the example of the mean value theorem. I was wondering how would I be able to motivate this theorem to a ...
1
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1answer
53 views

It is not possible to introduce multiplication in $v_n$(For $n>2$) so as to satisfy all field properties

In the book Calculus Vol 1- Tom M. Apostol .Before beginning to define the dot product of two vectors he tells It can be shown that except $n=1, 2$, it is not possible to introduce multiplication ...
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2answers
88 views

Motivation of vector bundle of a manifold

I am studying about vector bundle from M.Lee but not getting the feel of it. Can someone explain me about the importance of vector bundle? Why do we need to study about it? Thanks! Also, i have to ...
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3answers
136 views

The relation between been the quotient ring of a prime ideal and its localization

Let $A$ be a ring and $\mathfrak{p} \subset R$ be a prime ideal. Set $A_\mathfrak{p}=R[U^{-1}]$, where $U= A-\mathfrak{p}$. What is the relation between $A/\mathfrak{p}$ and $A_\mathfrak{p}$? My ...
1
vote
2answers
633 views

Grassmann Variables and Complex Conjugate

While dealing with Grassmann Variables, the complex conjugate is defined as $$ (\phi \psi)^{\dagger} = \psi^{\dagger} \phi^\dagger $$ and why not $ \phi^{\dagger} \psi^\dagger $. I want to know the ...
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1answer
42 views

Why do we consider only real or complex Banach spaces?

In wikipedia, normed vector space is defined as a vector space over a subfield of $\mathbb{C}$ equipped with a norm. However, Banach space is defined as a complete normed space over $\mathbb{R}$ or ...
1
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1answer
28 views

Growth conditions for partial differential equations

Hi I am interested in what the exact purpose is of growth conditions associated with solving partial differential equations. For example the following pde: $$\text{div}(a(x,u,\nabla u)) + c(c,u,\nabla ...
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1answer
95 views

Linear systems of differential equations

I would like to see an example of a real physical situation where one can find a set of variables evolving according to a system of linear differential equations. I wasn't able to find any such ...
1
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1answer
31 views

Why are special numbers important? (Such as fermat prime, mersenne prime)

Whenever I studied topics in mathematics, I found those topics are important in purely mathematical sense and I could see some motivations. However, I cannot see neither motivation nor importance of ...