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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Visualizing the quotient of a torus and a circle

We were asked to compute the homology for the double torus, $X$, and a circle around one of the loops, $B$, of the torus (not a circle between the two halves of the torus) and were told that this ...
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Intuition probability of two pairs in poker dice

I need some intuition for an element of the following question: The answer starts with this: I would like to know how they get to 6 choose 2. If I write it out (1122, 1133, 1144, 1155, 1166, 2233,...
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Areas under the graphs of $\frac{1}{x}$ and $\frac{1}{x^2}$ from $1$ to $\infty$

A simple evaluation of the definite integral tells us that the area under the graph of $[\frac{1}{x}]^2$ from $1$ to $\infty$ is finite whereas that of $\frac{1}{x}$ for the same limits is infinite. ...
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Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=...
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Absolute value of a random variable

I have never encountered this concept before. Is this equation valid for $y>0$? $$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$ What about this? $$\mathbb{P}(|X|>y) = \mathbb{P}(X>...
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What comes after diagram chasing?

An early edition of Lang's algebra textbook gives the famous exercise to Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book. Here ...
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45 views

Matrix --> Scalar Valued Function: Differentiation

In class, we called a real-valued function from the space of matrices to the reals $f: \mathbb{R}^{m \times n} \rightarrow \mathbb{R}$ differentiable at $\mathbf{X}$ if: $$\lim_{\mathbf{H} \to \...
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Is there some geometric intuition for the quotient $G/Z(G)$, where $G=GL_n(\mathbb{R})$?

Let $G=GL_n(\mathbb{R})$ be the $n$th general linear group. Its center $Z(G)$ is given by all scalar matrices $aI$ with nonzero determinant. How can I get an intuitive picture of $G/Z(G)$? I know that ...
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Is differentiation with respect to a vector always defined componentwise?

When one takes the derivative of a function $f$ along the direction of some vector $\mathbf{v}$, i.e. the directional derivative of $f$ along $\mathbf{v}$ this operation is defined componentwise, i.e. ...
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44 views

Taking a derivative of a function with respect to another function

I read a set of notes recently (unfortunately I can't find the link) in which the author made a statement of the form "differentiation of a function with respect to a function doesn't make sense". By ...
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Intuition of weak solutions of elliptic equations in divergence form

Let $\Omega \subset \mathbb{R}^{n}$ a domain, and consider the following equation (1) $-D_{j}(a_{ij}D_{i}u) = 0$ (Einstein notation) The function $u \in H^{1}(\Omega)$ is a weak solution of (1) if $...
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Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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Intuitively, why does $\dfrac{a}{c} = \dfrac{1}{\dfrac{c}{a}}$?

For intuition, I reference objects. Imagine making a dessert with: $a$ as apples and $c$ as chestnuts. Question. How and why is $\dfrac{a}{c} \qquad (3) \quad = \quad\dfrac{\color{red}{1}}{\dfrac{c}{...
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Intuition for Adeles and Ideles

I'm currently studying some class field theory and read about the notion of adeles and ideles. However, the object seems a bit arbitrary to me; is there a natural way to think about the adele-ring? ...
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Intuitive proof for a Combinatorial Problem

Given a set $S$ such that $|S|=N$ and $S$ contains exactly $K$ $0$s $(K >0)$ and $N-K$ $1$s, then exactly half of the subsets of $S$ contain an $odd$ number of 1s, $indepedent$ of the value of $K$....
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Volume of a cone [duplicate]

I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: $$\text{Vol}(C)=\...
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How to figure out the “idea behind” proofs in analysis?

I'm taking a course in Real Analysis, and for the most part I can follow the rote mechanics of a proof (e.g. manipulation to produce a chain of inequalities as desired, etc.), but I have difficulty ...
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Combinatorial Geometry explanation

I do not understand what is going on in $(4)$: for every flat $E \in \mathcal F$, $E \ne X$, the flats that cover $E$ in $\mathcal F$ partition the remaining parts. What is meant by "the flats ...
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Is this inverted ANOVA possible?

If my understanding is correct, in an ANOVA usually you start with a null hypothesis that all groups have the same mean. You then calculate the within group and between group variance, and do an F-...
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What is the appropriate ANOVA test for this situation?

So in this experiment I have have 7 devices. The response of each device (call it Y) are each measured ~20 at a 4 different levels of an independent variable (we'll call X). The Y response is known ...
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What are the intuitions about matrix algebra operations?

In my current data analysis problem I am using models with complicated penalty structure that is a result of operations on some matrix $Q$. I do know definitions of basic matrix operations: $Q^T$ $...
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What is the intuitive meaning of a determinant? [duplicate]

I know how to calculate a determinant, but I wanted to know what the meaning of a determinant is? So how could I explain to a child, what a determinant actually is. Could I think of it as a measure ...
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Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding ...
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Intuition about the second isomorphism theorem

In group theory we have the second isomorphism theorem which can be stated as follows: Let $G$ be a group and let $S$ be a subgroup of $G$ and $N$ a normal subgroup of $G$, then: The ...
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What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$?

I'm trying to understand in an intuitive manner the relationship between the circular and hyperbolic functions in the complex plane, i.e.: $$\cos(z)= \cosh(iz)$$ $$\sin(z)=-i\sinh(iz)$$ where $z$ is ...
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Practical intuition for standard deviation

Simply: What is the intuition you get about the data if you are given standard deviation? More detailed: It is easy to imagine some information about the data if you are being told for example mean ...
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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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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 $g(x)=\sum_{k=1}...
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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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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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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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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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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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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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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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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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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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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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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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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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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 ...