Questions tagged [motivation]
For questions about the motivation behind mathematical concepts and results. These are often "why" questions.
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What is the motivation to introduce Tate-cohomology groups?
What is the motivation to introduce Tate-cohomology groups ?
Let $G$ be a Galois group and $M$ be a $G-$module.
Let $H^n(G,M)$ be usual Galois cohomology.
In group cohomology theory, we often ...
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How to motivate the teaching of differential equations?
I will have to teach a first course in differential equations. A good motivator might be to promulgate modelling with differential equations but I have seen some teachers have made polemic against ...
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Motivation for presentation of group $G$ in Hungerford’s Abstract Algebra
An immediate consequence of Corollary 9.3 and the First Isomorphism Theorem is that any group $G$ is isomorphic to a quotient group $F/N$, where $G =\langle X\rangle$, $F$ is the free group on $X$ and ...
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Origins behind different terms for the same thing in Linear Algebra
It seems that there are many terms in linear algebra that have multiple names. For example, unitary and orthogonal both refer to the same general idea, a Hermitian is essentially a self-adjoint matrix,...
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Conditions on open sets in the definition of topological space
The question on the definition of a topological space has been appeared many times on this site, but I was unable to get answer to a natural question which not only I but a new learner of this subject ...
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Motivation (intuition) about a formal group
A (one-dimensional) formal group over $\mathbb{C}$ is a formal power series $F(x,y)\in\mathbb{C}[[x,y]]$ such that
$$
F(x,y)=x+y + \text{terms of higher order}
$$
$$
F(x,F(y,z))=F(F(x,y),z))
$$
...
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Motivation for integrals over differential forms
I am trying to find some sort of motivation as to why we integrate manifold over differential form and why especially does it in some form corresponds to integrating the surface of the area. I have ...
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What is the motivation behind the definition of Cech cohomology?
From Hartshorne's Algebraic Geometry, chapter 3.4:
The Cech cohomology of a sheaf topological space wrt an open cover is defined as $\frac{ker(d_{i+1})}{im(d_i)}$, where $d$ is some operator involving ...
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Why inner automorphisms; why conjugation? Why not closure under other automorphims?
I've been casually reading up on group theory recently, and I want to get a really solid and motivated understanding of where all the definitions we use come from.
Notions like the center of a group ...
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Motivation for representation theory
Next year I have choose a few optative courses from a big list, but syllabi are not available yet, so I have to make a choice based on names only.
I am thinking about taking one titled "Groups ...
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Profinite completion motivation
I have started galois theory recently, and curiosity quickly leads one to the subject of profinite groups. Although I have yet to be comfortable using these, I get what they are and we define them as ...
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Some intuition behind the properties of intersection number.
In the study of algebraic curves,we define intersection number of $F$ and $G$ at a point $p$ to be $I(F\cap G,p)=\dim_K(\mathcal O_p(\mathbb A^2)/\langle F,G\rangle$.But it is a rather unintuitive ...
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Why is the function $r(n)$ of particular interest in Circle method?
I was reading Goldbach problem(Ternary version) and encountered the Hardy-Littlewood circle method.In this method,we work with a number $r(n)=\sum\limits_{n=n_1+n_2+n_3}\Lambda(n_1)\Lambda(n_2)\Lambda(...
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Motivation for the study of affine semigroups
I have recently been studying affine semigroup (= Semigroups that is commutative, finitely generated, embeddable in a lattice) in the context of toric varieties (Cox, Little & Schenck). I was ...
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Understanding fractional ideals. [closed]
I am having a course in algebraic number theory.Dedekind domain plays an important role in number theory.It is defined as an integral domain all of whose fractional ideals are invertible.So we need to ...
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Why is span defined as the linear combination with *finite* terms?
I understand that, for a vector space $V$ with scalar field $K$, the linear span of a family of vectors $S\subseteq V$ is usually defined as $$ \mathrm{sp}(S):=\left\{ \sum_{i=1}^k a_iv_i \mid k\in \...
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Why are natural transformations the morphisms between functors?
When we have a category $\mathcal C$, it is usual to define the category $\textrm{Ar}(\mathcal C)$ of morphisms of $\mathcal C$ as the one whose objects are the morphisms of $\mathcal C$ and whose ...
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Why are $p$-adic numbers ubiquitous in modern number theory?
I'm currently at a stage where I think I'm quite comfortable with the appearance of local non-archimedean fields in the maths I encounter, having seen a fair bit of technology built upon their ...
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Affine change of coordinates.
I am reading Fulton's book on algebraic curves.In the second chapter they have defined what they call affine coordinate change map between two affine spaces.It is defined in the following manner:
Let ...
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Understanding the polynomial maps between two affine varieties.
I am reading Fulton's book on algebraic curves.I am in chapter $2$ currently.After defining coordinate rings,they define polynomial maps.
If $V\subset \mathbb A^n$ and $W\subset \mathbb A^m$ are ...
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Proof of Zariski lemma.
I am studying algebraic curves but I have no background of commutative algebra.An important theorem in this topic is the weak Nullstellensatz which states that:
Any maximal ideal of $K[X_1,...,X_n]$ ...
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Find all $k$ such that there are infinitely many positive integers $a$ such that $a(a+k) + k$ is a perfect square
Problem:
For a given positive integer $k$, we call an integer $n$ a $k$-number if both of the following
conditions are satisfied:
(i) The integer $n$ is the product of two positive integers which ...
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What is the motivation behind the definition of topological dimension and Krull dimension.
Usually in algebraic geometry,we define the dimension of a topological space as follows:
$\dim(X)=\sup\{ m:\text{ there exists a descending chain of irreducible closed sets of length } m\}$
and in a ...
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Why is the equivalence relation for localization defined in this way?
I am studying localization of a commutative ring with respect to a multiplicative subset.The concept is motivated by the field of fractions of an integral domain.We introduce inverses of elements of a ...
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Visualizing the norm of a self-adjoint operator.
I am studying functional analysis.I have observed that if I do not get an intuitive idea of what a theorem/result is saying,then we have to remember it by heart.Now something similar is happening to ...
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How to remember the fact that $P^2=P$ and $P^*=P\iff P$ is an orthogonal projection.
In functional analysis,I have studied the fact that for an operator $P:H\to H$ where $H$ is a complex Hilbert space, $P^2=P$(Idempotent) and $P^*=P$(Self-adjoint)$\iff P$ is an orthogonal projection....
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Simple explanations of differences between topology and geometry
I have google but still can't find a simple explanations of differences between topology and geometry. Most of answers on internet are not easy to understand for non-math major. Based on what I found (...
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Understanding the notion of compatible basis.
I am a postgraduate student studying final year.We have a course on module theory.Our instructor told us that if $M$ is an $R$-module with finite basis $\mathcal B=\{x_1,x_2,...,x_k\}$,then we will ...
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motivation for regular schemes
Clearly regular schemes are like smooth varieties (in the sense of dimension of tangent spaces) and should be very important in algebraic geometry. Is there any big theorem focusing on regular schemes?...
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motivation for character of group in group theory
I am reading Milne's book, Group theory. And there is a section named 'the linear characters of a commutative group', but I don't see any result that help us understanding group theory itself.
Does it ...
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How to motivate definitions in self-study
This is a problem that frequently arises while I self-study. I come across a new definition in a book, and I don't understand why that definition is made.
My thought about definitions is that they ...
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Anisotropic (linear) algebraic group: etymologic origin
A pretty naive question:
An anisotropic algebraic (linear) group over a field $k$ is a
linear algebraic group
$G$ defined over $k$ and of $k$-rank zero,
i.e. not containing non-trivial $k$-split tori
...
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What should the natural norm on a quotient space be?
Suppose, $X$ is a normed linear space and $M$ be a closed subspace.Then we can define a norm on $X/M$ by $\|\tilde x\|=\|x-M\|=\inf\limits_{m\in M}\|x-m\|$.I want to know why this should be the ...
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Is it possible to visualize the theorems and proofs in functional analysis?
I am an MSc.second year student and just started studying functional analysis.I understand the proofs of the theorems in this topic but since I cannot visualize those things,so I forget the proofs ...
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Motivation behind Fatou's lemma.
I am a graduate student of Mathematics and currently studying measure theory.There is a lemma which is used to prove the Lebesgue dominated convergence theorem called Fatou's lemma.It states that:
If ...
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Motivations for the eigenvalue problem of an elliptic operator
I was looking at the chapter 6.5 of Evans’ book about the eigenvalue problem for (anti-)symmetric elliptic operators, and I was wondering what were the motivations for such a problem.
I guess there ...
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Motivation behind the definition of semi-algebra.
I am studying measure theory and I have come across the term semi-algebra.The definition is as follows:
Let $X$ be a non-empty set.A collection $\mathcal S\subset \mathcal P(X)$ is called a semi-...
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Significance of Hahn-Banach theorem.
I am doing a course on functional analysis. One of the pillars in functional analysis is Hahn-Banach theorem. But in different books I find different versions of the theorem and it is not easy to ...
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Motivations behind finding the maxima and minima of functions?
In my textbook, a historical motivation for the development of differentiation is given, starting with Fermat trying to find the maxima and minima of functions. What I wanted to ask is why Fermat was ...
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Motivation behind approximation of Lebesgue measurable sets from outside by open sets and inside by closed sets.
I am planning to teach a course on measure theory in MSc. first year.I always like to give motivation behind what I am teaching.I am taking the following as the definition of Lebesgue measurable set.
$...
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Why is the essential numerical range defined as $W_e(T) = \bigcap_{K\in \mathcal K(H)} \overline{W(T+K)}$?
I have been introduced to the following definition of the essential numerical range of a bounded, linear operator on a separable, infinite-dimensional Hilbert space:
$$W_e(T) := \bigcap_{K\in \mathcal ...
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Motivation behind the definition of measurable functions. [closed]
In measure theory,we work with non-pathological sets and functions called measurable sets and functions.The definition of measurable function $f:(X,\mathcal S)\to \mathbb R$ is defined as a function ...
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What is the motivation of weighted integral?
The weighted integral of a function $f:\Omega\rightarrow\mathbb{R}$ is something like $\int_\Omega f(x)w(x)dx$.
What is the motivation of it?
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What does a 'good' approximation mean?
I am a graduate student and currently studying functions of several variables.I am mainly following Paliogiannis and Moskowitz.When they are introducing differentiability for function of several ...
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Group Action: Definition & Axioms [duplicate]
Why is a "group action" defined as it is? Namely, a group $G$ acting on a set $S$, that is associative and has a unit element. This defines a monoid. But, authors go on to claim that the ...
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Motivation in considering abelian groups in modules
I am leaning about modules and I did not understand what is the motivation behind an $R$ module $M$ as an abelian group in the definition of the module.
What difference does it make if we consider $M$ ...
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Why are "smooth" varieties so defined?
The Arithmetic of Elliptic Curves by Joseph H. Silverman defines a smooth (or non-singular) variety as follows:
Let $V$ be a variety, $P\in V$, and $f_1, \ldots, f_m \in \overline{K}[X]$ a set of ...
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Why countable union and not countable disjoint union in definition of $\sigma$-algebra.
I was trying to find the motivation behind the definition of $\sigma$-algebra.The idea actually came from the fact that we can measure the whole set and if we could measure a set $A$,then we could ...
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Physical significance of argument principle.
I have studied complex analysis in which there is a milestone theorem called argument principle.It states that if $\Omega\subset \mathbb C$ is an open connected set and $f$ be a meromorphic function ...
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Initial component lax terminal object
Given a bicategory $K$, Niles Johnson and Donald Yau define an initial component lax terminal object in $K$ to be an object $T$ together with a lax transformation $k: 1_K\to \Delta_T$ such that $k_T$ ...
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