Questions tagged [big-picture]
Questions to get the "big picture" about a subject.
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Big picture: Geodesics on a spherical surface embedded in $\Bbb R^3$
Consider the closed surface $M$ embedded in $\Bbb R^3$ with $g:=ds^2=dx^2+dy^2+dz^2$ and with $M:=\log^2 x+ \log^2 y +\log^2 z=1.$
Then restrict the metric to $M$. Here is a 3D plot of $M$ embedded in ...
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The number $2$ in cohomology theories
I've started feeling this rather curious mystique coming from an unaddressed - at least in my experience - excessive presence of the number $2$ in a few different areas of maths. My curiosity really ...
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Why does the foliation $\mathcal{F}$ of this Lorentzian manifold also solve the backwards heat equation?
Consider a linear parabolic partial differential equation:
$$t \partial_{tt}\varphi_t(x)=\pm x\partial_x \varphi_t(x)$$
which (essentially) takes the form of the backwards heat equation (minus sign) ...
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Why is studying centralizers the/a key to classifying finite groups?
In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says
One problem, as least with the current methods of ...
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Relations between the three different descriptions of 2nd cohomology group in Group Cohomology
I am coming at the 2nd cohomology group in Group Cohomology from the perspective of the Group Extension Problem (or rather the group central extension problem, which perhaps more closely related to ...
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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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What is the importance of the spectral theorem? (not the importance of diagonalization)
(In this question, $(*)$ means normal when working over $\mathbb C$ and means self-adjoint when working over $\mathbb R$.)
This question is related but despite the same title what that question ...
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Layer Cake Representation Intuition
Let $(X, \mathcal{F}, \mu)$ be a measure space. If $f: X \to [0, +\infty)$ is non-negative and measurable, then
$$
\int_X f(x) d\mu(x) =\int_0^\infty \mu(\{ x \in X: f(x)\geq t\})dt
$$
It is not very ...
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Theorems about finite sets the proof of which require the notion of infinite set
I believe that there should exist theorems about finite sets which are not provable without the notion of infinite sets.
I am curious if I am right.
What are the examples of such theorems if they ?
...
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Ruler and compass cannot trisect an angle, proof without field theory
It is well known that ruler and compass cannot trisect an angle. The standard proof of it uses field theory. It is just a contradiction of tower law.
However, I wonder if we can prove this only using ...
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Traditional and modern approaches to class field theory
I read in Gerald Janusz's book Algebraic Number Fields where he says there are two approaches to class field theory. The traditional approach, as in his book, uses L-series, Dirichlet density and ...
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A useful way of characterizing primes?
It's proved that if $P(n)$ is the sum of all different prime factors of $n$, then for all odd primes $p$ there is an inequality
$$p\ge P(p^2-1)$$
An inequality holding for odd primes
Therefore there ...
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Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?
If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
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The miraculous nature of the "matrix coefficients" $\langle Tv,v \rangle$ (especially in the context of positive type functions)
$\newcommand{\ak}[1]{\langle #1 \rangle}$I've noticed that for linear operators $T$ and an inner product $ \ak{\bullet, \bullet }$, the expression $\ak{ Tv,v}$ tends to show up a lot. For instance, it ...
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Why does Galois theory most naturally take place in the context of fields?
At least as far as I can tell, historically Galois theory was a more computational tool than it appears now, and https://hsm.stackexchange.com/questions/8099/how-did-the-modern-understanding-of-galois-...
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Build a "rich" first-order logic within a given category
I would like to know a mathematical framework with an internal logic where isomorphic objects can be considered equal.
For example, consider the rationals $\mathbb{Q}$. With this set we can construct ...
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Purely algebraic context in which homology arises naturally?
I know that homology is most commonly introduced from a topological context (and/or Stokes theorem related context), but in say a homological algebra course/text, you are just given the definitions (...
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Crux of linear algebra
What is the connection between solving systems of linear equations and vector spaces? And what do matrices have to do with all of that?
I know this is a crux of linear algebra, and therefore, not so ...
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What questions does Topology answer?
I was reading the great Thurston's article on Mathematical Education and one thing he points out is:
People appreciate and catch on to a mathematical theory much better after
they have first grappled ...
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The main idea behind Big O notation
Well, when we use Big O notation we never know even approximate number of steps of a given algorithm, right? For example, if we have $O(n)$ algorithm, then we don't know how fast this algorithm itself ...
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Why to calculate "Big O" if we can just calculate number of steps?
I never really understood how we use Big O to estimate algorithm efficiency. I know that $f\left(n\right) = O\left(g\left(n\right)\right)$ as $n \to +\infty$ if there exists some constant $C > 0$, ...
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Proof of Stewart's Theorem using elementary geometry
I was reading Stewart's Theorem which states that
Given a triangle with side lengths $a$, $b$, $c$ and a cevian of length $d$ which divides $a$ into two segments $m$ and $n$ as shown in the figure
...
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Definitions and concepts that are different for various types of spaces - sources for seeing "the bigger picture"?
I have quite unusual question. Almost everytime I am studying math, I lowkey want to "categorize" the stuff and look at it as a bigger picture.
Do you know about any resource (diagram, ...
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Trying to understand the idea behind $T$-conductor
I was reading Linear Algebra by Hoffman Kunze and I found a strange definition
Let $W$ be an invariant subspace for $T\in \mathcal L(V)$ and let $\alpha$ be a vector in $V$. The $T$-conductor of $\...
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Functors making functions natural transformations and vice-versa.
I apologize in advance if this is naive.
In this answer Conjugation in a groupoid it is said that given a groupoid $\mathcal G$, and an arbitrary function $\mu:\mathcal G_0\to \bigcup_{x\in \mathcal ...
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What's the typical role of the constant $e^{-\gamma}$?
I often encounter this constant in my research, but I wonder what typical roles does it play in other areas of mathematics? Wikipedia mentions probability theory but nothing exact.
Also, I am ...
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Connecting the various interpretations of normal subgroup
What is a normal subgroup? I have heard all of the following:
A subgroup $N$ of $G$ is normal if $gNg^{-1} = N$ for all $g \in G$. Or equivalently, a subgroup is normal if it is invariant under ...
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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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Differential analog to the Lebesgue spaces $\mathcal{L}^p$
I am looking for a differential analogue to the Lebesgue spaces $\mathcal{L}^p$ in the following sense:
The integral behaves badly for functions with too much mass i.e. such whose tails decay too slow ...
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What does the loop integral of geodesic curvature intuitively mean in the absence of a surface normal?
The geodesic curvature $k_g$ can be interpreted as the rate of rotation of the tangent vector about the surface normal, as discussed in this article. The article goes on to say that the integral $\int ...
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What are the historical trends of contemporary mathematics?
Question: What would future historians see as the general directions of mathematics in the period of 1970s -- 2020s?
What I look for in an answer: a pointer to a review paper on this topic would be ...
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What is the motivation for studying this subgroup of $SL_n(\Bbb R)?$
Consider a diagonal matrix $A$ with entries $e^{s_1},e^{s_2},\cdot\cdot\cdot$ on the diagonal s.t. $\sum_{i \ge1} s_i=0.$ From what I understand this is a subgroup of $SL_n(\Bbb R).$ This is because ...
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Hyperbolic curve and hyperbola?
Def: A hyperbolic curve is an algebraic curve obtained by removing $r$ points from a smooth,
proper curve of genus $g,$ where $g$ and $r$ are nonnegative integers such that $2g−2+r > 0.$
How ...
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Does anyone know what this diagram could be about?
Does anyone know what this diagram could be about? I found it about a year ago on some blog and I tried to relocate the source of the picture but was unable to.
My best guess is that maybe $R(x)$ and ...
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What is the big picture behind the quotient of a ring by its maximal ideal being a field?
A ring $(X, +, \times, 0, 1)$ is an algebraic structure such that $(X, +, 0)$ is a commutative group, $(X, \times, 1)$ is a monoid, and $\times$ distributes over $+$.
A sub-ring is a subset $Y \...
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Is a specific choice of ultrafilter necessary in order to get "concrete results" in nonstandard analysis?
Suppose we have sequences of real numbers which are indexed by the natural numbers. We can then define an ultrafilter $\mathcal{U} \subset 2^{\mathbb{N}}$ (where $2^{\mathbb{N}}$ is the powerset of ...
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On two conceptions of measurability
This is a very broad question about two conceptions of a measurable function. Let $(X, \mathcal X)$ be a measurable space, and let $f$ be a real-valued function of $X$. Throughout the post, I assume ...
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On the "duality" between formation of quotient structures and formation of substructures as stated in Joy of Cats
In the book Joy of Cats, the authors write (page 118),
One of the nice insights that can be gleaned from category theory is that the formation of quotient structures (such as groups of cosets and ...
user170039
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Intuition behind the definition of initial morphism and embedding as stated in Joy of Cats
In Definition 8.6 of Joy of Cats the authors write (I have changed the wording to avoid explaining the conventions that the authors introduce earlier which would lengthen the post unnecessarily),
...
user170039
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A map (big picture) of the relationship of all areas in mathematics (including probability, stat, and information theory, etc)
Update 10/15:
What I could remember is the picture was under a section named "interesting stuff" (something like this), with a link toward the picture (if I remember it correctly). Also, the professor ...
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Classification of monoidal closed structures
The motivations for this question are somewhat vague probably. I am trying to have a better understanding of monoidal closed structures. In order to do so I would like to collect unusual examples and ...
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Alternative Proof to "Prove that it cannot be proven that "The United States had more fallow acreage than planted acreage"
Given:
A ten year comparison between the United States and the Soviet Union
in terms of crop yields per acre revealed that when only planted
acreage is compared, Soviet yields were equal to 68 ...
user685057
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General rule of thumb for tensor products.
When would I want to use a tensor product? For example, say I am trying to derive "something" but, to get to the right derivation, I should use the tensor product.
The question comes from quantum ...
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What does a topology do, and what makes a particular topology the 'right' one?
From Wikipedia:
The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.
[Aside: ...
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A definition of differentiable functions for arbitrary topological spaces
Background
It is well-known that there is no notion of derivative for arbitrary topological spaces. However while investigating the notion of derivative as we find in one variable real analysis I came ...
user170039
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Looking for intriguing applications of martingales
I've been studying martingales before I start graduate school in statistics, and I'd like to ask, what are the most intriguing or surprising applications of martingales that you've come across, either ...
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Is there a notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$?
It is accepted belief $\mbox{countable }\infty$ is not $\mbox{uncountable }\infty$. Is there notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$ (latter ...
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Do we have to redo category theory when learning about enriched categories? [closed]
I'd like to know whether there exists a common language that encompasses both categories and enriched categories, so that results pertaining to either may be proven in a uniform way. I'd prefer it if ...
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Why is Catalan's constant $G$ important?
I am aware that Catalan's constant appears in the evaluation of many definite integrals, as well as in the evaluation of certain infinite series, and is a special value of a function closely related ...
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Two questions regarding the convention concerning concrete categories
Definition of Concrete Categories
Let $\mathbf{X}$ be a category. A concrete category over $\mathbf{X}$ is a pair $(\mathbf{A}, U)$, where $\mathbf{A}$ is a category and $U : \mathbf{A} \to \...
user170039