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Questions tagged [intuition]

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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Aha moment regarding the Gauss map

Given a oriented regular surface $M$ with Gauss map $N$. Is the idea of mapping tangents at some point $p \in M$ by $dN$, motivated by the fact that one can model all possible motions of a tangent ...
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4answers
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What is the intuition behind why a rank deficient matrix does not have an inverse?

Suppose that we have a $p$ dimensional square matrix $A$ whose rank is less than $p$. We know that such a matrix cannot have an inverse and there are several different ways to prove that the $A$ does ...
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1answer
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+50

Building Intuition for Differential forms, exterior derivative, wedge

I think I understood 1-forms fairly well with the help of these two sources. They are dual to vectors, so they measure them which can be visualized with planes the vectors pierce. Gravitation 1973 ...
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1answer
16 views

Thought process behind vector valued functions and parameterisation

Can someone please confirm whether my intuitive notions behind what vector-valued functions and parameterisation is correct. Below are some questions. Are vector-valued functions like functions of a ...
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1answer
42 views

Expanding on the intuitive meaning of singular matrices

My question is based on "What is the geometric meaning of singular matrix" posted here some years ago. To make this a bit more intuitive I would like to add an example. A three-dimensional force ...
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1answer
36 views

Diffeomorphism between the unit ball to itself

Let $T:B\to B$ be a diffeomorphism , where $B\subset\mathbb{R}^n$ is the unit ball. I want to show there exists $x\in B$ such that $|J_T(x)|=1$ ($|J_T|$ is the absolute value of the Jacobian of $T$) ...
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2answers
362 views

The importance of the different method of proofs? [closed]

I am very familiar with the direct method of proof. Is this proof sufficient enough to be able to understand higher level proofs? Do you need to be fluent in every type of proof? And does knowing how ...
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1answer
16 views

Intuitive explanation of solutions to a linear diophantine equation

"Given a linear diophantine equation $ax+by=c$ with a particular solution $(x_0,y_0)$ the general solution is given by $$\biggl(x_0-\frac{b}{gcd(a,b)}t,y_0+\frac{a}{gcd(a,b)}t\biggr)$$ for all $t\in \...
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1answer
37 views

Intuition behind inequality for measure of $\liminf$ and $\limsup$

For a set $X$ with a $\sigma$-algebra $\xi \subseteq \mathcal{P}(X)$ and $\sigma$-additive $\mu: \xi \rightarrow [0, \infty]$. The following inequality holds for $(A_n)_{n \in \mathbb{N}} \in \xi^\...
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5answers
129 views

Proof explanation of $``\exists x\in\mathbb{R}$ with $x^2=2"$

Can someone please help me break down the proof below from $(*)$ onwards. I'm lost at what is going on and where the proceeding steps are coming from. Is this a proof by contradiction? Why are we ...
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3answers
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Intuitive explanation of Euler's formula $e^{it}=\cos(t)+i\sin(t)$ [duplicate]

I'm trying to understand $$e^{it}=\cos(t)+i\sin(t)$$ This comes from the definitions $$\cos(t)=\frac12(e^{it}+e^{-it}) \quad\text{and}\quad \sin(t)=\frac1{2i}(e^{it}-e^{-it})$$ and those are ...
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Problems understanding: $\Sigma \vdash \theta \text{ iff }\Sigma \vdash \forall x \theta$ [duplicate]

I'm trying to make sense of a theorem I came across recently: $$\Sigma \vdash \theta \text{ iff }\Sigma \vdash \forall x \theta$$ Say $\theta$ is $(x=1)$, then we have $\Sigma \vdash (x=1)\text{...
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Approximating Differential Equations Using Difference Equations/The $z$-Transform.

I'm aware that difference equations and the $z$-Transform method can be used to approximate differential equations but I'm wondering how exactly it does this as I don't have a clue? All I've come ...
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86 views

Probability of getting 6 heads in a row from 200 flips and intuition about this high value

A few days ago i had an argument with a friend about this question : What is the probability of getting 6 heads in a row from 200 flips ? I argued it is high probability (significantly bigger than ...
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49 views

Math books emphasizing geometrical aspects (for differential geometry).

I'm not very good at geometry, but I feel, most of the time, math concepts in calculus, linear algebra, complex analysis, abstract algebra etc. can only be intuitively understood when and if an appeal ...
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5answers
214 views

Intuition for why $f_{xy} = f_{yx}$

If we have a function $f(x,y)$, why is it that $f_{xy} = f_{yx}$? I'm looking for an intuitive, qualitative reason rather than a rigorous proof. $f_{yx}$ represents the rate of change of the gradient ...
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2answers
44 views

Why is the intersection empty for open nested intervals?

Is this correct? The length of, for example $(0, 1/n)$ as n appoaches infinity, doesn't reach 0 because it's not in the set. But, for the closed set it does because it is in the set? Apologies for ...
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1answer
19 views

Recursive proof of expectation of geometric distribution?

If $T \sim \mathsf{Geo}(p)$ ($0 < p < 1$) then $\mathbf{E} T = 1/p$ is well known. One "story" that captures this probablistic statement is the following: Question. Suppose that I toss a ...
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1answer
30 views

how to understand sigmoid(x+y)- sigmoid(x-y)

As shown in the following graph, why does function sigmoid(x+y)- sigmoid(x-y) has smooth instead of sharp edges around (0,1) in the contour plot? Could you please explain it both intuitive and ...
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0answers
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Intuition behind calculating expected value of log(q(x)) based on the distribution p(x)

I know that lots of divergences between two distributions are based on the expected value of one distribution with another one. Assume that for two distributions $p(x)$ and $q(x)$, we want to ...
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2answers
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How is it that a third line segment doesn't always divide the first two?

How can it be shown by the pythagorean theorem that it's not always possible to find a third line segment that evenly divides into the first two? I'm using the unit square as an example. Does this ...
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1answer
37 views

Explanation - prove that SupB=InfA

Why is it stated in the solution that $SupB\in{B}$? By the completeness axiom the set B has a supremum, but that does not imply that it's the maximal element of the set. My intuition is telling me ...
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5answers
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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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1answer
33 views

Polynomial Estimation proof explanation

Can someone please help me understand and break down this estimation lemma for polynomials. I don't understand what the conclusion is saying and what it means. I'm really confused with the logical ...
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1answer
42 views

Visualizing double points.

I was trying to visualize by drawing a curve / figure to get a double point on a curve. As per the Wolfram article, a double point is a point traced out twice as a closed curve is traversed. Any ...
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1answer
71 views

Proof explanation that the square root of 2 exists [closed]

Theorem. There exists $x\in\mathbb {R}: $ with ${x^2=2} $. Can someone please share a simple proof of the above theorem by defining, $$\{y\in\mathbb R: \mbox{$y^2\leq {2}$}\}.$$ How do we proceed ...
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2answers
30 views

Help on the derivation of Abel's Theorem using the wronskian

I'm finding it difficult to solve the first order differential equation in the proof to obtain the conclusion. I've solved 1st order ODE's before and not had problems and I feel really silly for not ...
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2answers
52 views

Why doesn't a 20 degree rotation change the slopes of $y=x$ and $y=\frac{x}{2}$ by the same amount?

It seems that if I rotate different lines (lying in the same quadrant) the same number of degrees they move different amounts (in terms of their slope). (where the rotation is such that all the lines ...
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1answer
31 views

Confusion regarding the general solution theoerm

Theorem 1.13 (General Solutions) If $y_1$ and $y_2$ are linearly independent solutions of the equation $L[y]=0$ on the interval $I\subset\mathbb R$ where $L[y]=y''+p(t)y'+q(t)y$ and $p(t),\ q(t)$ ...
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117 views

Intuition behind Unprovable Truths: Godel

Godel's Incompleteness Theorem says there are statements in an axiomatic system which are TRUE but UNPROVABLE. I have read other answers here, but none of them captures the essence of such statements. ...
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0answers
41 views

Intuition on Stokes' Theorem

Stokes' Theorem says that $$\int_C\vec{F} \cdot d\vec{r} = \int \int_S (curl \space\vec{F}) \space\cdot \vec{n} \space dS$$ I understand that $curl \space\vec{F}$ is the "spin" or "circulation" on a ...
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0answers
21 views

Intuition of sub sigma-algebra definition

I am having trouble understanding the sub σ-algebra definition on Wikipedia. I understand the following: Let $X$ be a set, and let $A,B$ be σ-algebras on $X$. Then $B$ is said to be a sub-σ-algebra ...
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2answers
119 views

For all $n>1$ there are positive $a+b=n$ such that $a+ab+b\in\mathbb P$

For all integers $n>1$ there are positive integers $a,b$ such that $a+b=n$ and such that $a+ab+b\in\mathbb P$. Tested for all $n\leq 1,000,000$. Hopefully, someone can explore and explain the ...
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1answer
34 views

Confusion regarding Kelvin functions

I am trying to implement the following equation from this paper and having some troubles in the interpretation of $bei'$ and $ber'$. I understand from the definition of Kelvin functions that for ...
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0answers
54 views

A visual subject in group theory

My math teacher gave us the following instruction: « Pick any subject of your choice (in math of course) and in 2 months present it to the class ». I really like this idea, and I really like group ...
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1answer
39 views

orthogonal group what does it represent

Let $A$ be an finite abelian group and $B$ be a subgroup of $A$. Then we defined the orthogonal of $B$ : $$B^{\perp} = \{f:(A,+) \to (\mathbb{Q}/\mathbb{Z},+) \mid \forall b \in B ,f(b) = 0 \}$$ I ...
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1answer
33 views

Is the derivative an approximation? (Cost function, marginal cost, deviation in costs)

Good evening everyone, I wonder if the derivative is an approximation. Because if I am calculating marginal costs there is always some small deviations regarding the marginal cost (additional cost ...
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0answers
36 views

Is it possible to give an intuitive idea of the notation at the base of the transition Kernel for a Markov Chain on a general state space?

Given $A \in \sigma(\mathcal{S})$, with $\mathcal{S}$ is the state space, the transition kernel is a function $K(\cdot , \cdot): \mathcal{S} \times \mathcal{B}(\mathcal{S}) \to [0,1]$ $\forall x \in ...
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Average distance from center of circle

Using calculus, we can show that the average distance of a point in a circle to the center is $2R/3$, where $R$ is the radius. However, I have a separate way of approaching this question through ...
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3answers
32 views

Interpreting an agreement function

Consider the following function: $$ P = \frac{1}{n(n - 1)} \sum_{j=1}^k n_{j} (n_{j} - 1) $$ where for $n = \sum_{j=1}^k n_{j}$. Intuitively, this function measure concentration of values the ...
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0answers
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Is MCMC (or any sampling for that matter) explainable?

Recently, at an interview, I was asked if you use MCMC to build Maximum a posteriori (MAP), and use it for an inference, will the system you create have an explainability? Now, explainability is ...
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3answers
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What are we really doing when we integrate a complex exponential?

Firstly, I have read this related question from this site, but it does not answer what I am asking here. What I like to know is why $$\int_{-n}^n e^{ix}dx \ne 0$$ for any finite $n \in \mathbb N$ My ...
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1answer
62 views

A book as complete as Hartshorne, but “better” on the intuition side

My question is a short one, but hopely I get long answers...I am looking for bibliography on Algebraic Geometry. A first but complete book, rigorous and also modern. I want to say that Hartshorne`s is ...
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1answer
54 views

An anomaly regarding the sum of all divisors that is square-free

Let $q(n)$ be the number of integers $m<n$ such that $m$ is square-free. Let $p(n)$ be the number of integers $m<n$ such that the sum of the prime factors of $m$ is square-free. And let $s(n)$ ...
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0answers
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What can we say about the Begaman transform of $f\ast g (t_2)- f\ast g(t_1)$?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$ Now we define $$ H(t)= H(...
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1answer
56 views

Why is the argument principle called the argument principle?

The Argument principle: If $f$ is meromorphic in an open connected set $\Omega$, with zeros $a_j$ and poles $b_k$ then $$\frac{1}{2 \pi i}\int_{\gamma}\frac{f'(z)}{f(z)} dz = \sum_j n(\gamma , a_j) - \...
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2answers
922 views

Am I the only one constantly forgetting the Eisenstein criterion? [closed]

Do you guys have tricks to remember the Eisenstein criterion? I constantly forget it and I am looking for some logic in it to never forget it again.
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1answer
43 views

Explanation of Cartesian formula for circumcenter

On Wikipedia there is a Cartesian formula for the circumcenter of a triangle. That is, given points $A$, $B$ and $C$ in $\mathbb{R}^2$, find point $U$ such that $d(A,U)=d(B,U)=d(C,U)$. The formula, as ...
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0answers
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Intuition about conjugate priors and parameter estimation

I have a problem that I am starting to work in a field where I need lots of non-rigorous probability theory for modelling.One large stumbling block for me is concept of conjugate Priors of random ...
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1answer
60 views

Soft question:Intuition for tangent space

I am trying to learn topology, from An introduction to manifolds of Loring W.Tu second edition to be precise, while I saw the definition of tangent space I don't think I understood what they are and ...