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

# Questions tagged [motivation]

353 questions

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### Set theory, motivating factors

Set theory seems to pop up in many different fields of mathematics. As someone with a CS degree, I've only encountered very basic set theory; dealing with non-specific sets, and their intersections, unions, and complements. Furthermore, this turned…

Alec

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### 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 $\mathbb{C}$.
Is there a reason that we consider…

Rubertos

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### 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:
$$\text{div}(a(x,u,\nabla u)) + c(c,u,\nabla u) = g$$
where examples of some of the growth…

user116403

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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 $\frac{\partial}{\partial \overline z}$ as…

Kishalay Sarkar

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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:
$(\mu\times \nu)(E)=\int_X\nu(E_x)d\mu =\int_Y \mu(E^y)d\nu$.
I am trying to find the…

Kishalay Sarkar

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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 great answers. However, I am at an undergrad level now and…

RonY

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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 subgroup of $S_n$. It
is denoted by $A_n$ and is called the…

sloth

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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 Theorem the author derived a formula, which is…

Consider Non-Trivial Cases

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### Bayesian Statistics : Motivation and Explanation of Marginal Likelihood

$P(\theta|x)$ is the posterior probability. It describes $\textbf{how certain or confident we
are that hypothesis $\theta$ is true, given that}$ we have observed data $x$.
Calculating posterior probabilities is the main goal of Bayesian…

Consider Non-Trivial Cases

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### Why we need to mention the scalar product of $\cos (nx), \sin (nx)$?

I found the following text -
The functions $\cos (nx), n = 0, 1, 2, \cdots$ and $\sin (nx), n = 1, 2, \cdots $ which are known to be orthogonal with respect to the
standard scalar product on $(-\pi, \pi)$.
The source of the problem is …

Consider Non-Trivial Cases

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### Are there any real & decent mathematics video games?

Personally, I like to play video games from time to time, especially arcade games (e.g. Tetris, pinball) or fast-paced games like Super Hexagon, which is known to be quite challenging.
However, for mathematics (that is not intended for kids) I have…

MJW

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### Series equal to e

I'm having trouble convincing myself why
$$\sum_{k = 0}^{\infty} \frac{k}{k!} = e.$$
As I was under the impression that only
$$\sum_{k = 0}^\infty \frac{1}{k!} = e$$
by definition.
By writing out terms of the first series
$$\sum_{k = 0}^{\infty}…

Featherball

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### Gautschi's Motivation for polynomial interpolation error

In the text Gautschi. Numerical analysis: an introduction, Birkhäuser, Boston, 1997; 2nd edition, 2012, Gautshi gives the following motivation for error in polynomial interpolation:
"It is not difficult to guess how the formula for the error should…

smileemote

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### How to visualize $(A\cap B)^\mathrm{o}=A^\mathrm{o} \cap B^\mathrm{o}$?

We all know that
$(A\cap B)^\mathrm{o}=A^\mathrm{o} \cap B^\mathrm{o}$, where $A,B \subset X$ which is a metric space.
The proof is not also difficult, but actually I cannot visualize or feel intuitively that this should happen.
Can someone…

Kishalay Sarkar

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### Why Do We Need Initial Conditions to Solve PDEs?

I am looking for further clarity on why solving PDEs without any specified initial values was not "good enough." For example: say we had the ODE
\begin{equation}
y' = y
\end{equation}
without specifying any initial conditions. Solving it, we…

Jetninja

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