Questions tagged [big-picture]

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

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43 views

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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32 views

What objects the concept of determinant has been generalized to?

Determinat is defined on square matrices. There is a generalization to block matrices. In complex, split-complex and dual numbers determinant corresponds to the square of modulus, because these ...
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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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42 views

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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17 views

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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51 views

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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59 views

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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1answer
76 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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130 views

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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1answer
134 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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3answers
53 views

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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31 views

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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1answer
74 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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2answers
102 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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65 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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1answer
93 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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1answer
59 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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1answer
68 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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4answers
507 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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285 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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1answer
73 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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1answer
109 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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68 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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2answers
731 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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1answer
94 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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1answer
102 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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1answer
204 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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5answers
2k views

Intuition Wanted: Why Define Integrals Component-Wise

In our analysis course, we just defined the following: Let $g := (g_1, \ldots, g_n): [a, b] \to \mathbb{R}^n$, where $g_1, \ldots, g_n: [a,b] \to \mathbb{R}$ are integrable. Then we call the ...
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8answers
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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1answer
121 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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270 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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3answers
86 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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3answers
111 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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1answer
229 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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99 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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159 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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1answer
82 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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1answer
210 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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58 views

Does the defintion of chain equivalence have anything to do with liebniz product rule of differentiation?

I have this vague/ big picture question which might be too vague to answer (I don't mind it being deleted in this case). Do chain equivalences have anything to do with Leibniz product rule of ...
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1answer
200 views

The context & motivation for the Tits alternative in combinatorial group theory

The Details: Definition 1: A class $\mathcal{G}$ of groups satisfies the Tits alternative if for any $G$ in $\mathcal{G}$ either $G$ has a free, non-abelian subgroup or $G$ has a solvable subgroup of ...
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1answer
125 views

There is a natrual connection on the tangent bundle?

I come up with a (maybe stupid) question: let $M$ be a smooth manifold, then the exterior differential $d$ is a natural connection on $\Omega^k(M)$, hence by dualizing we get a natural connection on $...
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1answer
77 views

On multiplicative and additive properties of cyclotomic polynomials

Is there explicit relation between $\Phi_{a+b}(x)$, $\Phi_{ab}(x)$, $\Phi_{a}(x)$ and $\Phi_{b}(x)$ at general coprime or non-coprime $a,b\in\Bbb Z$? If $a,b$ are distinct primes then we have $x^{ab}-...
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115 views

What makes a math problem important?

Given that pure mathematics is, by definition, not concerned with applications, how does one decide that one problem is more valuable than another? Is it just a matter of certain topics becoming ...
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54 views

Analogies and differences between polynomial evaluation and modulo operations

Given a polynomial $f(x)\in\Bbb Z[x]$, prime $p$ and integer $a$ are there analogies and differences between evaluating $f(x)$ at $x=a$ and computing $a\bmod p$? Both look like some kind of function ...
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35 views

Formulas without efficient algorithms?

Suppose the fastest known method to multiply two decimal numbers is an exponential time algorithm. Would it still be beneficial to mathematics having formulas like the following for positive integers? ...
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84 views

Why, intuitively, are propositional resolution proofs so long?

I'm trying to gain an intuitive understanding of why propositional resolution proofs tend to be so long. As every essential prime implicant can be produced via resolution, intuitively I would have ...
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1answer
465 views

Mathematical disciplines with high thresholds

Are there mathematical disciplines that are extremely inaccessible, with very high thresholds even for those who have the necessary prerequisites? Number theory and graph theory, for example, are ...
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2answers
366 views

Books on the philosophy of geometry

I am looking for recent books ( say published after 2000) on the philosophy of geometry, most books on the philosophy of mathematics seem to ignore or bypass geometry at all or am I just looking with ...
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2answers
251 views

$\pi$ when not in base 10

Very novice amateur mathematician here. My daughter (8 yo) is a math junkie and is trying to wrap her head around irrational numbers. We were talking about $\pi$, and I rambled on about how folks have ...
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3answers
1k views

Why introduce the $p$-adic numbers?

My current intuition about the p-adic numbers comes from the following three facts: You can describe $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with the $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ ...