Questions tagged [motivation]

Question 1

I am studying Cycles, Transfers And Motivic Homology by Voevodsky chapter 5. In his lecture he defines the Tate motive by $M_{gm}{}^{\tilde{\, \,}}(\mathbb{P}^1)[-2]$ where $M_{gm}(X){}^{\tilde{\, \,}}...

Question 2

I've been reading Serre's Linear Representations of Finite Groups (GTM 42), but I’m a bit confused about the motivation and applications of two classical theorems: Artin’s Theorem: Each character of $...

Question 3

A pair of adjoint functors $F:\mathscr{C\leftrightarrows D:}G$ is said to be monadic if the comparision functor $K:\mathscr D\to \mathscr C^T$ is an equivalence, where $T=GF$ and $\mathscr C^T$ the ...

Question 4

As I understand, the study of large cardinals was motivated by Gödel, who wrote that large cardinal axioms could decide CH; later on, it was discovered by Lévy and Solovay that if $\kappa$ is ...

Question 5

When eigenvectors form a basis, they're a very good basis because they transform a given matrix into a diagonal matrix. The problem is that eigenvectors don't generally form a basis. However, if we ...

Question 6

While exploring the relationship between characteristic equations and recurrence relations in linear algebra, I discovered their connection to the Fibonacci sequence. Let's say there is a recurrence ...

Question 7

Question: Why use $\partial$, and not $-\partial$, $\pm \partial$, or $\mp \partial$, for defining the boundary operator in (simplicial) homology? To clarify, by $\pm \partial$ I mean, for example, $+ ...

Question 8

I encountered the concepts of split-complex and dual numbers through these videos, and am struggling to understand the reason/motivation for defining them. The argument presented in these videos is ...

Question 9

in Exercise 8.10 from Module Theory: An Approach to Linear Algebra by T. S. Blyth, the author defined a terminology as follows: If $M$ and $N$ are $R$-modules, an $R$-linear map $f:M\to N$ factor ...

Question 10

Number theory has been around for at least thousands of years, and it does not take much to see that the subject is pervaded with interesting and enchanting stuff. I have taught a basic course on ...

Question 11

I am beginning to explore symplectic geometry and would appreciate a clear conceptual comparison with Riemannian geometry. In Riemannian geometry, one works with a manifold equipped with a Riemannian ...

Question 12

The following definition is from Kalton's book on Banach space theory. I have a question regarding the definition; why is the space specifically $L^1([0,1])$. I know there is another definition ...

Question 13

Background While I was learning about bitangents on plane curves, I looked into bitangent numbers on cubic and quartic curves. As I kept searching, the name of a mathematician named Riemann was often ...

Question 14

I am working through "Differential Equations with Applications and Historical Notes" by George F. Simmons and have arrived at the Brachistochrone problem. Using optics (Snell's law), and a ...

Question 15

If $f: X \rightarrow [0,\infty]$ is measurable then an increasing sequence of simple functions approximating $f$ is given by $$\phi_n = \sum_{k=0}^{2^{2n}-1} k2^{-n}\chi_{E_k^n} + 2^n\chi_{F_n}\\ E_k^...

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Questions tagged [motivation]

shift by [-2] in the Tate motive definition

Motivations of Artin's theorem and Brauer's Theorem on Characters

Motivation to study Eilenberg-Moore category

What is the motivation for Large Cardinals

Motivation of decomposition theorem about linear algebra

A grand story from eigenvectors to diagonalization

Why use $\partial$, and not $-\partial$, $\pm \partial$, or $\mp \partial$, for defining the boundary operator in (simplicial) homology?

What is the motivation behind split-complex and dual numbers?

Motivations behind the terminology "factor through projectives"

Motivation and applications of quadratic residues

What are the key conceptual differences between symplectic and Riemannian geometry?

Definition and motivation for Radon-Nikodym property

How did Riemann connect plane curves and $\theta$-functions?

Motivation behind the substitution $\tan(\theta)=\sqrt{\frac{y}{c-y}}$ in the Brachistochrone problem

What motivates the upper limit of simple functions approximating a measurable function to be $2^{2n}-1$?

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