Questions tagged [big-picture]

Questions to get the "big picture" about a subject.

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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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7 votes
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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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4 votes
2 answers
82 views

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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3 votes
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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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1 answer
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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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6 votes
3 answers
1k views

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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3 votes
3 answers
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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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  • 870
12 votes
4 answers
538 views

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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2 votes
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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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6 votes
1 answer
184 views

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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3 votes
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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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1 answer
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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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43 votes
6 answers
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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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1 vote
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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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2 votes
1 answer
234 views

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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4 votes
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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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2 votes
1 answer
156 views

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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1 vote
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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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2 votes
1 answer
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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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3 votes
1 answer
90 views

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 ...
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2 votes
2 answers
167 views

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), ...
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2 votes
0 answers
76 views

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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4 votes
1 answer
157 views

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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1 vote
1 answer
61 views

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 ...
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1 vote
1 answer
87 views

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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13 votes
4 answers
616 views

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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8 votes
0 answers
399 views

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 ...
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2 votes
1 answer
124 views

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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1 vote
1 answer
119 views

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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2 votes
0 answers
78 views

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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9 votes
2 answers
1k views

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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1 vote
1 answer
96 views

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 \...
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2 votes
1 answer
152 views

Importance of Axiom of Choice (or its weak form) in Real Analysis

By Wikipedia article on Axiom of dependent choice, it is necessary to have it for development of real analysis. The ${\mathsf {DC}}$ says: Axiom (${\mathsf{DC}}$). For any nonempty set $X$ and every ...
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5 votes
1 answer
242 views

In algebraic geometry, what kind of theory can only be described by topos but not a site?

A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books ...
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17 votes
5 answers
2k views

Intuition Wanted: Why Define Integrals Component-Wise

In our analysis course, we just defined the following: Let $g := (g_1, \ldots, g_n) \colon [a, b] \to \mathbb{R}^n$, where $g_1, \ldots, g_n \colon [a,b] \to \mathbb{R}$ are integrable. Then we call ...
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42 votes
8 answers
3k views

Why are algebraic structures preserved under intersection but not union?

In general, the intersection of subgroups/subrings/subfields/sub(vector)spaces will still be subgroups/subrings/subfields/sub(vector)spaces. However, the union will (generally) not be. Is there a "...
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-1 votes
1 answer
137 views

What mathematics cannot be reduced to pigeonhole?

Pigeonhole is a fundamental principle without which state of mathematics will be much different. However what examples of good mathematics has not yet been proved and cannot be proved with pigeonhole ...
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4 votes
4 answers
331 views

A graphical representation of Mathematics as a whole?

Does a graphical representation of mathematics, its fields and subfields, exist ? Meaning, for instance a graph where vertexes are fields of mathematics (e.g convex geometry, Lie algebra, Kahler ...
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1 vote
3 answers
91 views

What can be learned for number theory from geometrical constructions (and vice versa)?

Even though this question of mine was not so well received at MO I'd like to pick two examples and make a question out of them here. Consider these two pairs of geometrical constructions which yield ...
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3 votes
3 answers
122 views

How do we get past how **every** outcome is very unlikely?

Edit: This question is about rejecting the null hypothesis. Last month my evil twin and I were at a game show. The rules are as follows: There is a sealed booth with two magic boxes. Box A has a ...
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13 votes
1 answer
244 views

On the clarification of Manin's remark about Gödel’s incompleteness theorems

In his paper Georg Cantor and his heritage Yuri I. Manin writes (see page 7, 3rd paragraph), Baffling discoveries such as Gödel’s incompleteness of arithmetics lose some of their mystery once one ...
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2 votes
0 answers
131 views

The shape and group structure of an elliptic curve over $\overline{\mathbf{F}_p}$ and intermediary extensions

Let $p$ be prime, let $q$ be a power of $p$ and let $E/\mathbf{F}_q$ be an elliptic curve defined over the finite field $\mathbf{F}_q$. Let $\overline{\mathbf{F}_q}$ be the algebraic closure of $\...
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6 votes
0 answers
231 views

Motivation behind Primary Decomposition

I am reading Atiyah's Commutative Algebra chapter on Primary Decomposition. I understand the proofs but I have no intuition as to How did one come up with definition of a primary ideal $q$ as $...
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  • 8,954
3 votes
1 answer
91 views

Why care distribution functions more than random variables?

This may be wrong, but I have often heard some saying " we mainly care about CDFs". Similarly, in textbooks, one sees $X \sim N(0,1)$, without any reference to sample space. But why - and how do we ...
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5 votes
1 answer
232 views

Mandelbrot and Julia fractals for $z_{n+2} = z_{n+1}^2 + z_n^2 + c$

The Mandelbrot and Julia type fractals are very Well known. But such fractals follow from $$z_n = f(z_{n-1},c)$$ In other words a recursion that only depends on the previous value and a constant. (...
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