Questions tagged [motivation]
For questions about the motivation behind mathematical concepts and results. These are often "why" questions.
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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 ...
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0answers
79 views
Is there a von Neumann type theorem for the $\sigma$-algebra generated by the set of all the eigenfunctions?
Defintions
Let $(X, \mathcal X, \mu, T)$ be a measure preserving system.
Let $U_T:L^2_\mu\to L^2_\mu$ be the associated Koopman operator.
We will write $\mathcal X_0$ to denote the $\sigma$-algebra ...
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1answer
54 views
Jacobian Matrix at a non-singular point
I am currently studying singularities in Algebraic Geometry and wanted to understand why the rank of the Jacobian matrix would characterise a point of singularity/non-singularity (assuming we start ...
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1answer
47 views
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 ...
2
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1answer
80 views
Why use sequents?
In the sequent calculus, the building blocks of a proof are inference rules, which are rules for inferring the validity of certain sequents from other sequents, something like this:
$$\frac{\vec\...
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Motivation behind the definition of orthogonal latin squares. [duplicate]
Two Latin squares of the same size are said to be orthogonal if you form a square by superimposing the two squares in the following way,
$\left[\begin{array}{l}1&2&3\\3&1&2\\2&3&...
4
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1answer
81 views
What is the motivation behind the definition of adjoint of a linear operator?
Given $T:V\to V$ linear and $V$ being an inner product space, we define $T^*$ by a linear operator on $V$ such that $\langle Tx,y\rangle=\langle x,T^*y\rangle$ for each $x,y\in V$.
We later see that, ...
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1answer
89 views
Motivation for the fractional Sobolev spaces
I am strying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya, but there is ...
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45 views
Gautschi's Motivation for polynomial interpolation error
In the text Gautschi. Numerical analysis: an introduction, Birkhäuser, Boston, 1997; 2nd edition, 2012, Gautshi gives the following motivation for error in polynomial interpolation:
"It is not ...
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1answer
63 views
Motivation for Baby Rudin Theorems 2.38-2.40 (Compactness, k-cells)
I would appreciate some context around Baby Rudin's Theorems 2.38-2.40. It's in the section dealing with compactness. I find hard to give any motivations to these theorems in particular. Why are they ...
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Motivation for the definition of the Pearson correlation coefficient
Let $X$ and $Y$ be two random variables with joint distribution $P_{X,Y}$ and marginal distributions $P_X$ and $P_Y$. The Pearson correlation coefficient is defined to be $$\rho_{X,Y}=\dfrac{\mathbb{E}...
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A question on nowhere dense sets.
Consider the $2$ definitions:
A set $A$ in a topological space $(X,\tau)$ is said to be a nowhere dense set if it is not dense in any nonempty open set.
A Set $A$ in a topological space $(X,\tau)$ ...
6
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1answer
109 views
On the beginnings and motivations of certain branches of set theory
I am planning to give a talk in my university on descriptive set theory, large cardinals and inner model theory. And the target audience are undergraduate students. I am trying to roughly explain what ...
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2answers
108 views
What lies behind the definitions of split monics and epics?
Is there an easy way to memorize the definitions of split monics and split epics, and not to confuse the domains/codomains of the arrows from those definitions?
For example, is there a mnemonic rule?...
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1answer
53 views
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 ...
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1answer
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Visualization of a “Not so intuitive” problem of linear algebra.
I recently encountered a problem in Hoffmann-Kunze linear algebra:
If $(.,.)$ is the standard inner product on $\mathbb C^2$ then show that $(Tv,v)=0 \forall v\in \mathbb C^2 \implies T=0$, I think ...
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Why do functional analysts want their spaces to be complete?
Why is one in functional analysis only investigating complete spaces (like Banach or Hilbert spaces)? I heard someone saying that analysts in general like to work with limits, which makes sense. But ...
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46 views
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 ...
0
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2answers
56 views
why commutative integral with limit is important in real analysis? [closed]
why commutative integral with limit is important in real analysis ?
Why $\lim_{n\to\infty }\int f_n=\int \lim_{n\to\infty } f_n $ is important ?
3
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1answer
62 views
What motivates the arithmetic-geometric mean?
What motivates the arithmetic-geometric mean? What inspires it? I understand how to calculate this mean but do not understand what might prompt a mathematician to pursue such a mean in the first place....
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0answers
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What's the reason behind this substitution? How does it help solving second order linear ODE?
Consider the 2nd order linear ODE: $y^{\prime\prime}+p(x)y^{\prime}+q(x)y=0$
This is usually solve by performing the substitution: $y(x)=u(x)\exp(\int\frac{-p(x)}{2}dx)$.
We get $y^{\prime}(x)=u^{\...
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2answers
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Baby Rudin, Example 1.1, proving irrationality of $\sqrt{2}$
In Principles of Mathematical Analysis, trying to prove that $\sqrt{2}$ is irrational, we can read:
Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let B consist of all ...
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3answers
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Motivation for definition of free group?
Let $S$ be a set and $F_S$ be the equivalence classes of all words that can be built from members of $S$. Then $F_S$ is called the free group over $S$.
I don't understand the motivation for this ...
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5answers
129 views
The meaning of probability and random variables
I do understand pure mathematical concepts of probability space and random variables as a (measurable) functions.
The question is: what is the real-world meaning of probability and how can we apply ...
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1answer
172 views
Motivations for and applications of Matroid Theory?
I have taken an interest in this topic recently. If one is unfamiliar with matroids, I will give the definition here.
Let $M=(E,\mathcal I)$ where $E$ is a finite set called the ground set and $\...
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7answers
1k views
What is a good way to introduce Euler's totient function?
I was thinking of this question and when I googled I couldn't find any MSE results, but I found one from Reddit. I just wanted to ask the question here and post the answer as community wiki just so ...
4
votes
1answer
188 views
Turán’s Graph Theorem: Motivation behind the Weight Function
If a graph $G = (V, E)$ on $n$ vertices has no $p$-clique, $p \geq 2$,
then $$|E| \leq (1- \frac{1}{p-1}) \frac{n^2}{2} \;\;\; \;\;\; \;\;\;(1)$$ We get a proof from the book "Proofs from THE BOOK" as ...
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8answers
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What are the applications of the Mean Value Theorem?
I'm going through my first year of teaching AP Calculus. One of the things I like to do is to impress upon my students why the topics I introduce are interesting and relevant to the big picture of ...
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25 views
An idea to teach equation except balance model
I looking for an idea to start teaching the equation of algebra, but not use the balance model. I am looking for a new motivational idea or a pedagogical method to start teaching equations.
The link ...
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1answer
76 views
Topic for a math talk [closed]
I have to make a talk about mathematics for first and second year undergraduate students of maths. If someone could help me with a topic or an idea, it would be helpful.
Preferably it is something ...
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1answer
60 views
Intuition behind dual lattices
Suppose that we have a lattice in $\mathbb{R}^n$, that is, a discrete additive subgroup of $(\mathbb{R}^n, +).$
We can define dual lattice $L^*$ as
$$ L^* = \{x \in span(L) \; | \; \forall y \in L \; ...
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0answers
68 views
Motivation of considering homomorphisms
I've always found that isomorphisms are more natural than homomorphism. Isomorphisms are motivated by the desire of defining when two structures are "structurally equal". But what is the motivation of ...
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0answers
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Motivation for Construction of Jacobian
$\DeclareMathOperator{\Pic}{Pic}$
Hello everybody,
in a course on the Jacobians of Curves, the lecturer gave the following Motivation for the Construction of the Jacobian:
Let $D$ be a divisor on a ...
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3answers
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Why $p$th power are not allowed in radical extensions?
A field $R$ containing a field $K$ is defined to be radical extension of height $1$ if $R = K(u)$ with $u^p \in K$ for some prime $p$ and $u^p$ is not the $p$-th power of some element in $K$. Then we ...
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6answers
738 views
Could *I* have come up with the definition of Compactness (and Connectedness)?
Ok, buckle up for a rather long question. I've spent a large portion of today learning about compactness, stemming mainly from this wikipedia article about point-set topology. The article mentions ...
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1answer
73 views
What are the motivations and examples of P. Hall Family?
I am reading J. P. Serre's Lie Algebras and Lie Groups and here is how a P. Hall Family is defined:
I failed to understand the motivations of this definition. Could anyone explain it to me? Also, ...
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1answer
40 views
Motivating the multivariable definition of the derivative
When I ask "What is the derivative?" the answer I find I get the most (and the answer I think is most satisfying) is: $$ \text{The derivative at a point is a local, linear approximation of the ...
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0answers
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Why and how algebraic structures emerged in mathematics? [closed]
For example, why we study Group Theory to prove general results for all instead of specifically studying $\mathbb Z$ (or any other set) closed under some operation?
What makes algebra and those ...
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1answer
63 views
What are the application of universal property of subspace topology?
I come across following theorem:
Universal property of subspace topology: $X$ is any topological space $Y\subset X$ $Z$ is any another topological space if there is continuous map $g:Z\to X$ such ...
2
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1answer
109 views
The idea of a bundle chart and bundle atlas.
The definition of a bundle charts and bundle atlas is rather obscure in my opinion. Is it fair to say that:
1) the purpose of a bundle chart is to give coordinates to each tangent space?
2) the ...
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2answers
189 views
Motivation behind point set topology [closed]
Why should I study point-set topology?
What initially interested me in topology was the pop-sci rubber sheet stuff or coffee cup-donut stuff or proving fundamental theorem of algebra using curves but ...
3
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2answers
235 views
What's the geometric interpretation of this “vector cross product”?
This answer on StackOverflow answers a question about intersection of two segments. Right at the beginning, it introduces a “vector cross product”.
Define the 2-dimensional vector cross product $\...
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2answers
151 views
What is the link between topology and graphs, if one exists?
In spirit, topology and graph theory seems fairly similar - you have points/vertices, and a notion of "how they are connected", loosely. However, it's not obvious how these fields relate, despite ...
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1answer
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Motivation for defining filter convergence
I've just learned about a filter converging in a topological space, but I just can't understand what's the motivation to define such a thing...
I get that it is a generalization of a sequence ...
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0answers
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What is the usage of the fact that $Prob(X)$ is Polish if $X$ is Polish?
Let $X$ be a Polish space and $Prob(X)$ be the set of Borel probability measures on $X$, and let $Prob(X)$ be equipped with the weak-* topology (So that a sequence $\mu_m$ converges to $\mu$ in $Prob(...
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2answers
429 views
Kernel and image of matrix: What are they? Why do they exist?
I've been trying to get an understanding of the Kernel of image of matrices. I'm studying them in college right now, but the problem is, while I can find a ton of resources on how to find them given a ...
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2answers
93 views
Motivation of the Logarithm
Suppose that someone wants to calculate approximately the product of 101,123,958,959,055 and 342,234,234,234,236 without using a computer. Since these numbers are so long, completely carrying out ...
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2answers
96 views
Interpretation of Fourier Transformation - what is it?
What exactly is Fourier Transformation? For functions on the Schwartz Space $S(\Bbb R^n)$, we may define,
$$ \hat{u}(\xi) := \int e^{-ix\xi} u(x) \, dx $$
This formula seems to come out of no where ...
2
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1answer
600 views
Why is the polarization identity important, intuitively?
The polarization identity states, roughly, that a norm satisfying the parallelogram law induces a vector space inner product (and vice versa). This has many nice applications, such as a simple ...
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Brownian motion : what is it exactly and why is it so important?
My question is simple : what is it exactly a Brownian motion and why is it so important ?
So, I read the the wiki page of the Brownian motion, and the definition is : continuous stochastic process ...
