Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Lagrange's theorem intuition

I cannot grasp the intuition behind |G|/|H|=[G:H]. Starting from the equivalence relation x~y if and only if x^(-1)*y is in H, I can see a sort of division, but in my mind, the equivalence relation ...
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35 views

Motivation behind Arithmetic Mean

I know that the arithmetic mean $(x_1+x_2+...+x_n)/n$ is the value that minimizes $f(x)=\sum_{k=1}^n (x_k-x)^2$; however, I'm looking for an intuitive relationship between the mean and ...
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Understanding limits and how to interpret the meaning of “arbitrarily close”

I have read several introductory notes on limits of functions, and in all of them they introduce the notion of a limit of a function $f(x)$ by discussing what happens to the value of $f$ as $x$ ...
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87 views

Why this intuition about natural transformations corresponds to its formal definition?

Almost everywhere people introduce the notion of natural transformations between two functors $ F$, $ G$ : $ \textbf C \Rightarrow \textbf D$ by examples like what follows: This is the intuition ...
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Covering maps as bundles

One geometric way to see a continuous map (or any set function really) is as a "fiber bundle" with the usual picture of a comb - the base space indexes the fibers of the map and there's a nice picture ...
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Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...
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49 views

Intuitive understanding of the “Multiplication Rule”?

I apologize in advance that this question has a long set-up. In the set up I am presenting how I currently understand the material, and the actual question is if my understanding is correct and ...
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27 views

Dividing a plane with lines

A while back, one of my friends challenged me to find out how many regions I can divide a plane into given $n$ lines. For instance: He also told me that the formula to find the maximum number of ...
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Fourier Analysis and its applications [duplicate]

My question has two parts: $1)$ Could anyone explain in simple terms what a Fourier Transform is? $2)$ What are some of the applications of Fourier Analysis in the field of high school mathematics? ...
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26 views

How does the multiplication theorem correspond to the concept of intersection?

Given two events A and B defined on a sample space S. S : Rolling a six-sided dice A : Getting an even number B : Getting a number ≥ 4 In an elementary sense (the experiment being carried out once), ...
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35 views

Integral of a function defined on dense subset

[Edited] For a real-valued continuous function $f$ defined on a Lebesgue measurable dense subset of $[0,2]$, consider an integral $$ \int_{[0,1]}\frac{f(s)}{\sqrt{1-s}}ds. $$ My question is whether ...
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3answers
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What does the derivative of a function at a point describe? [duplicate]

I understand that the derivative of a function $f$ at a point $x=x_{0}$ is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where $\Delta x$ ...
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123 views

Intuition behind Fourier and Hilbert transform

In these days, I am studying a little bit of Fourier analysis and in particular Fourier series and Fourier/Hilbert transforms. Now, I am confident with the mathematical definitions and all the ...
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1answer
42 views

Landau Notation - Practical explanations

Someone told me this week that the Landau notation is very pratical in general in analysis. Definition : Let the function $\phi$ defined on an open set containing $x_0$.We want to compare $f$ à ...
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1answer
47 views

If $f$ is holomorphic, what is the meaning/intuition behind $f_z=f'(z)$ and $f_{\overline z}=0$?

If $f$ is holomorphic then we know the derivative of $f$ with respect to $z$ is defined, i.e., $f'(z)$ exists. But $\overline{z}$ is a different variable, so if we take the derivative of $f$ with ...
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2answers
37 views

What is geometrical interptetation of a set being measurable.

What is geometrical interptetation of a set being measurable. I mean what does it mean geometrically by a set is measurable...
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3answers
66 views

Does it make geometric sense to say that open rectangles and open balls generate the same open sets

I have always been bothered by when people say: The open ball (i.e. $L_2$ ball) and the open rectangle (i.e. $L_\infty$ ball) generates the same open sets (topology) on $\mathbb{R}^2$ The proof is ...
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How, intuitively, does commuting with filtered colimits capture “smallness”?

Definition. A compact object is an object representing a copresheaf which commutes with filtered colimits. In algebraic categories, the compact objects are the finitely presented ones, so commuting ...
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Why do isotropic spaces deserve their name?

Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions? From my experience so far, ...
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The role of the Zariski topology in algebraic geometry

I am having trouble understading the relevance of the Zariski topology being a topology. Every time I see the proof that sets of the form $V(I)=\{p\in\mathbb{A}^n\mid f(p)=0 \ \forall f\in I\}$ ...
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4answers
354 views

Intuition about the semidirect product of groups

If we have two groups $G,H$ the construction of the direct product is quite natural. If we think about the most natural way to make the cartesian product $G\times H$ into a group it is certainly by ...
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How does the “arc tangent metric” $d(x,y) = | \arctan(x) - \arctan(y)| $ work?

I see there are some counterexamples and so forth in metric spaces regarding the metric $$d(x,y) = | \arctan(x) - \arctan(y)| $$ But honestly I have no intuition as to how it works For example, in ...
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What is meant in the quotation of Terry Tao?

Terrence Tao commented of internalizing [here: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/] "It is true that some mathematicians can be vastly more ...
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Why are Unique Factorization Domains (UFD's) geometrically significant?

We know that for $A$ a UFD, it's class group is trivial. More generally, for a factorial (stalks are UFD's) scheme $X$ (that is also noetherian and normal), we have an isomorphism between it's Picard ...
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Intuition about the first isomorphism theorem

I'm currently studying group theory and recently I've read about the first isomorphism theorem which can be stated as follows: Let $G$ and $H$ be groups and $\varphi :G\to H$ a homomorphism, then ...
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43 views

Homotopy Equivalence intuition

Can somebody tell me intuitively what does it mean geometrically when we say two spaces are homotopy equivalent ? I understand the technincal definition.
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62 views

Why should I expect the product of sum of four squares to be a sum of four squares? How did Euler come up with it?

Euler discovered the lovely identity shown here: https://en.wikipedia.org/wiki/Euler%27s_four-square_identity Is there a natural reason to assume a solution can be found? Any intuition? I saw that ...
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Extension theorem from Guillemin-Pollack, motivated sketch of proof?

Let $W$ be a compact, connected, oriented $k + 1$ dimensional manifold with boundary, and let $f: \partial W \to S^k$ be a smooth map. Could anybody sketch with good motivation that $f$ extends to a ...
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Log-concave function changes when scalar is added

A function $f$ is log-concave if $\log(f)$ is concave. Intuitively, one might guess that adding a scalar to a function would not affect properties like concavity, log-concavity, quasi-concavity etc., ...
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Basic probability. Is the textbook wrong?

A and B are independent events such that P(A)=0.7, P(B)=k, P(A U B)=0.8 Find the value of k. Solution given: P(A U B) = P(A) + P(B) - P(A ∩ B).....(i) [Addition Rule] P(A ∩ B) = ...
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Is it possible to tell whether a set of functions is equicontinuous from the graph of the function?

I never internalized why the set of functions $f_n(x) = \sin(nx)$ is not equicontinuous I know I can show that it is not equicontinuous by definition, by choice of appropriate $x,y$ for ...
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Intuition for minimum spanning tree result

Let K$_ n$ denote the complete graph on n vertices. To each edge in K$_n$ independently assign a weight drawn from the uniform distribution on [0,1]$\,$. Finally, define MST(n) to be the expected ...
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What is functional analysis in simple words?

To begin with , I am only a secondary school student (17yo) but I am very interested in higher mathematics. However we only learn so little in my school (only single variable calculus and basic linear ...
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Show that $\nabla\cdot\left(\dfrac{\mathbf{e}_r}{r^2}\right)=4\pi\delta(\mathbf{r})$ using the divergence theorem.

The book answer goes as follows: By the divergence theorem, in spherical coordinates we find ...
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How small can we make two numbers $a$ and $b$, with prime factorizations such that…?

Given a number $n$, I'd like to find it using either the sum or difference of two other numbers. The other two numbers, which we can call $a$ and $b$, must have a prime factorization with no primes > ...
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The frog puzzle

So here's the puzzle. You're poisoned in the jungle and the only way to save yourself is to lick a special kind of frog. To make matters worse, only the female of that species will do. Licking the ...
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Where can I found an explanation of group cohomology from the point of view of invariants?

I heard once that we can view group cohomology as the right derived functor quantifying precisely (i.e. by the usual long exact sequence) how much the functor of "taking the invariants" is not right ...
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Synthetic differential geometry and formally étale morphisms?

Upon looking throug Kostcki's synthetic differential geometry notes, I stumbled upon the following definition. (Here $R$ is the geometric line, $W$ is a Weil algebra, and $\operatorname{Spec}_RW$ is ...
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35 views

Geometric intuition for homotopy invariance of fiber bundles?

There's a nice result in algebraic topology saying that given a fiber bundle, its pullbacks along homotopic maps are isomorphic as bundles. Thinking of a bundle as a comb with the "teeth" as its ...
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Grasping ODE by intuition

Consider following differential equation $$\frac{dy}{dx} = y^2(1+x^2)$$ Its solution is: $$y(x) = \frac{-1}{x+\frac{1}{3}x^3+C}$$ As I read in one book, following must be satisfied: ...
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Techniques to find a subset of possible values for the rank of a matrix

The rank of a matrix is the number of linearly independent rows or columns of that matrix. In some exercises I need to find the rank of a matrix, but for some (lol unknown) reason I'm always stuck ...
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1answer
113 views

Explaining Newton Polygon in elementary way

In this question of mine Proving irreduciblity over $\mathbb{Z}$ I was recommended to read Newton Polygon. Also this appears to be an interesting topic. Also I am currently studying irreducibility of ...
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46 views

Fuzzy logic vs probability

In reading about fuzzy logic it says that fuzzy logic is different from probability. Can some one please explain how these two differ. How can this be explained to a person with no mathematical ...
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62 views

What do the properties of the dot product mean? [closed]

I am having trouble understanding the relevance or meaning of the properties of dot product. For example, the distribution property of dot product states: $$\vec a \cdot (\vec b+\vec c) = \vec a ...
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Intuition for power-set structure of finite Boolean rings

A course I am taking has started to introduce Boolean rings: rings where every element is idempotent. It was proved that every finite Boolean ring $R$ is isomorphic to a power set ring $\wp (S)$ for ...
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Motivation for the mapping cone complexes

I was reading some topics in Homological Algebra when I came across the concepts of cone of a map of complexes and cylinder. My knowledge of Algebraic Topology is pretty basic so I only used these ...
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Coordinate transformations and interpreting what the Jacobian determinant describes

Apologies for a perhaps rather trivial question, but I really want to get the concept cleared up in my head. I understand that when one changes from one coordinate system there is an appropriate ...
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68 views

“Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
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Simple Question about Interpretation of Poisson Distribution

So if you know for instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. Can the Poisson distribution ...
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Additional term in solution for wave equation

Given the wave equation: $u_{tt} = c^2u_{xx} \quad x \in \mathbb{R}, t> 0$ $ u(x,0) = \phi(x), x \in \mathbb{R}$ $ u_{t}(x,0) = \psi (x), x \in \mathbb{R}$ The solution is: $u(x,t) = ...