Questions tagged [motivation]

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

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737 views

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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1answer
68 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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2answers
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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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8answers
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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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6answers
4k views

What do cones have to do with quadratics? Why is 2 special?

I've always been nagged about the two extremely non-obviously related definitions of conic sections (i.e. it seems so mysterious/magical that somehow slices of a cone are related to degree 2 equations ...
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2answers
68 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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0answers
35 views

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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0answers
51 views

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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2answers
57 views

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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1answer
169 views

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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1answer
178 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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1answer
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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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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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3answers
59 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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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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2answers
116 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 ...
2
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1answer
51 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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1answer
70 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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0answers
38 views

What is the motivation behind naming “Marginal Likelihood”?

I gathered follwing remarks from internet - In statistics, the margin (as in “marginal distribution”) is the average or, in mathematical terms, the integral. Marginal means that we marginalised, ...
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2answers
164 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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3answers
244 views

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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34 views

Motivation for classification of diagrams in homotopy theory

I am reading the papers by Dwyer and Kan on the diagrams and their realization. What I understand that in principle if we want to classify diagrams in which the maximum sequence of composable arrows ...
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0answers
85 views

Braided vector space: motivation?

In one of my courses we were given the definition of a braided vector space: Let $k$ be a field. A braided vector space is a pair $(V,s)$ with $V$ a $k$-vector space $s:V \otimes V \rightarrow V \...
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51 views

What is the possible line of reasoning/motivation that led to the present definition of radians?

I know this might sound like a silly question at first. Let me elaborate. What I mean by 'line of reasoning; here is what the person who defined radians the way they are defined thought to arrive at ...
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3answers
127 views

Motivation of the definition of topology

In general topology the the definition of topology is the following: Let X be a non empty set. A set $\tau$ of subsets of $X$ is said to be a topology on $X$ if $X \in \tau$ and $\emptyset \in \tau$ ...
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0answers
47 views

What are the main results of sheaf theory?

I am learning some sheaf theory and sheaf cohomology from the book by Torsten Wedhorn. Sheaves seem to be quite a flexible tool, since one can have sheaves of rings, sheaves of abelian groups and so ...
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2answers
99 views

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 ...
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3answers
94 views

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

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, \...
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1answer
48 views

What is the motivation behind Binomial Distribution?

If there are $N$ repetitions of a "random experiment" and the "success" probability is $θ$ at each repetition, then the number of "successes" $x$ has a binomial ...
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1answer
66 views

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
84 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 of ...
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1answer
153 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
49 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 ...
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1answer
111 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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1answer
149 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
120 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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0answers
56 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
114 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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2answers
81 views

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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2answers
205 views

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 $(X,\tau)$ ...
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1answer
124 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
158 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
54 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
99 views

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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0answers
84 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 ...
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2answers
66 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 ?
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1answer
82 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
54 views

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
163 views

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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