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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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If $a,b>0$ and $a>b$ then $\sqrt[k]a-\sqrt[k]b<\sqrt[k]{a-b}$ geometric interpretation

Is there a way to understand this Formula Right away in a geometric way? Sorry for brevity. If I am Always using this Formula I have to prove it myself again it would help me if I would have a ...
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5answers
140 views

Intuitive explanation of $y' = y \Rightarrow y = Ce^x$

I understand why if $f(x): \mathbb{R} \to \mathbb{R}$ with $f'(x)= f(x)$ and $f(0) = 1$ then it must by $f (x) = e^x$, but I don't really feel it super intuitive. Intuitively, why would you expect ...
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Optimization problem produces trivial results under Gaussian assumption

I have two Gaussian's, $Y_1$ and $Y_2$ with parameters $(\mu_1, \sigma_1)$ and $(\mu_2,\sigma_2)$. These predict the amount of demand for some quantity. Let's assume the variances are much lesser than ...
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1answer
32 views

Intuition behind existence of mixed volumes?

Consider "Volume" as a function from set of $d$-dimensional convex bodies to real numbers. This function is homogeneous of degree $d$ (under rescalings of the convex bodies). Minkowski's theorem ...
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0answers
22 views

Probablity theory, sigma-algebra, rolling two dices and having partial information about outcome.

I was looking at this example to understand sigma-algebra better and why they are a good tool for probabilities. So my basic understanding about sigma-fields, is that the set in them are the events, ...
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1answer
41 views

Is the following ring isomorphism true?

In recent few days, I have been trying to develop an intuitive idea for quotient rings and what I've learnt from my explorations is that if you want to quotient a ring by a principal ideal, you can ...
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2answers
58 views

What is the link between topology and graphs, if one exists?

In spirit, topology and graph theory seems fairly similar - you have points/vertices, and a notion of "how they are connected", loosely. However, it's not obvious how these fields relate, despite ...
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2answers
63 views

Why is $H_0(X,x_0) \ne H_0(X)$

We know $\tilde H_n(X) \cong \tilde H_n(X,x_0)=H_n(X,x_0)$ for all $n\ge 0$. Also, when $n>0$, we have $H_n(X)\cong \tilde H_n(X) \cong H_n(X,x_0)$. However, when $n=0$, we have $H_0(X)\cong \...
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0answers
72 views

Wild visualization of higher dimensions [closed]

I have a very sophisticated mental picture of higher dimensions and I really need some guidance in correcting my wild imagination. Is it ok to visualize $ \mathbb{R}^4 $ like a regular 3D space ...
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2answers
96 views

Intuition behind these statements being equivalent

I have encountered a strange equivalence, and I am trying to make intuitive sense of it, which I have been able to do in many similar cases. Here it is: $$[(p\Rightarrow q)\Rightarrow r] \...
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3answers
95 views

How can I define an infinitely small positive value?

I have a question about infinity. (I'm in 8th grade so please just let me know if this is a stupid question, and please ask if you want any clarification). How do you define an infinitely small value ...
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Intuition re: Optimal matching distance

The optimal matching distance between the unordered sets $\lambda=\{\lambda_1,\ldots,\lambda_n\}$ and $\mu=\{\mu_1,\ldots,\mu_n\}$ is defined as $d(\lambda,\mu)=\min_{\sigma\in S_n}\max_{1\leq i\leq n}...
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31 views

Independence of Intersection of Chords in a cirle

$3$ chords are uniformly selected in a circle. We need to find the distribution of number of points of intersection of chords. To solve this, I first considered an easier problem, one containing ...
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How do I interpret an isomorphism of planar graphs?

Can you explain how isomorphisms in planar graphs are related to projected edges?
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1answer
85 views

Intuition behind a structure of a language in mathematical logic [long read but simple]

I do not get the intuition behind the structure (I think it's called "interpretation" in other texts) of a first order language. Here I use the following definition, given in Shoenfield's Introduction ...
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transpose of a doubly stochastic matrix intuition

Here are my observations regarding the Stochastic Matrix: If we have a Markov chain $(X_n)_{n \ge 0}$ defined by $(\lambda, P)$ we can see that the i-th row shows the probabilities of leaving the i-...
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2answers
40 views

How to prove that a function has a maximum and a minimum

Consider the function $f(x, y) = \sqrt{xy}$ on the domain: $$D = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 2, x \geq 0, y \geq 0\}$$ How would you explain that this function has a maximum and ...
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1answer
50 views

Remembering the golden ratio

I always forget how one can deduce the golden Ratio and its property. I hope somebody can explain me the chain of thoughts of its introduction in the book. By property I mean that the reciprocal of ...
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2answers
33 views

Conditional expectation should be $\cal F$ a measurable.

I was reading this answer that gives some intuition about what conditional expectation is. I can more or less understand it now but at some point in the response it is written : A single numerical ...
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2answers
268 views

Kernel and image of matrix: What are they? Why do they exist?

I've been trying to get an understanding of the Kernel of image of matrices. I'm studying them in college right now, but the problem is, while I can find a ton of resources on how to find them given a ...
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2answers
46 views

Understanding the definition of “proper maps” in Differential Topology by Guillemin and Pollack

I am trying to understand a piece of definition in Differential Topology by Gullemin and Pllack (GP). Before introducing the concept of "embedding", GP gives the following definition (page 17): A ...
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2answers
74 views

If $f$ and $g$ are one-to-one, then $gf$ is one-to-one.

I'm aware that there is a thread about this proof. However, I have a slightly different approach which I can not verify myself - hence, this thread. Proof. If $\,g(f(a_j))=g(f(a_k))$ and $g$ is ...
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0answers
16 views

Is $\mathbb Q(\sqrt 2) \times \mathbb Q(\sqrt 3)=\mathbb Q(\sqrt 2,\sqrt 3)$ if I prove $\sqrt 2,\sqrt 3$ are L.I. over $\mathbb Q$? [duplicate]

I proved that $\{1,\sqrt 2\}$ and $\{1,\sqrt 3\}$ are respective bases of $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$ over $\mathbb Q$. I want to show in some sense that since $\sqrt 2,\sqrt 3$ are ...
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2answers
55 views

How to see that this space with two vertices and four edges is homotopy equivalent to $S^1 \vee S^1 \vee S^1$?

I am having trouble seeing why this space has fundamental group $\mathbb Z * \mathbb Z * \mathbb Z$. I have read that this space is homotopy equivalent to $S^1 \vee S^1 \vee S^1$, from which we can ...
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2answers
181 views

Convincing yourself about choice of axioms for predicate calculus

Consider some Hilbert-type formal system of predicate calculus. I will use the one from Kleene's "Introduction to Metamathematics" 1971. While developing predicate calculus in this style we list set ...
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4answers
77 views

How is $\mathbb R \to S^1$ nullhomotopic?

Every map $\mathbb R \to S^1$ is nullhomotopic. The covering map $p: \mathbb R \to S^1$ is given by $p(x)=\cos(2\pi x, \sin 2\pi x)$. How is this map null-homotopic if we are wrapping $\mathbb R$ ...
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2answers
24 views

In computing $\partial: C_2(T) \to C_1(T)$ of the torus, why is $\partial(U)=a+b-c$? Why is the cell structure of the torus oriented this way?

In Example 2.3 from Hatcher, page 106, we are computing the homology groups of the torus. The $2$-chains is the free abelian group $\mathbb Z\{U,L\}$ and the $1$-chains is the free abelian group $\...
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0answers
64 views

Is my intuition of imaginary numbers correct?

From what I have been reading about imaginary numbers, I've arrived at this summarization: Numbers describe the existence and count of objects. Imaginary numbers describe the rotation of these object(...
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1answer
38 views

How is a standard $2$-simplex oriented? How is a standard $n$-simplex oriented?

If we have the standard $2$-simplex (pictured below from Hatcher), why is there an arrow from $v_2$ to $v_0$? Why not from $v_0$ to $v_2$? We have $\partial([v_0,v_2])=[v_2]-[v_0]$, so shouldn't ...
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1answer
77 views

Proof of a formula to calculate $1^p+2^p+…+n^p$ for arbitrary $n,p\in\mathbb{N}$

$1^p+..+n^p=\sum_{k=1}^{n}k^p$ Suppose I fix an $n$ and set $p=1$ Then one can prove by induction that $1+2+...+n=\frac{1}{2}(n)(n+1)$ Now there is an identity and I am looking for a proof for it ...
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2answers
159 views

Category with colimits but no limits

(I suspect this is a very easy question: I haven't spent much time thinking about category theory.) $\DeclareMathOperator{\colim}{colim}\DeclareMathOperator{\Dom}{Dom}\DeclareMathOperator{\im}{im}$ ...
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2answers
50 views

Prove that there exists no surjection between $X\rightarrow P(X)$ -Proof question

There is a hint that I shound take an arbitrary function $F:X\rightarrow P(X)$ and show that the subset $\{ x\in X|x\notin F(x) \}$ is not an element of $F(X)$. I don't know how one could came up with ...
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64 views

Zig Zag Product's Problem

Hi I read this paper about the Zig Zag Product's (page 73 and 74) http://www.cs.huji.ac.il/~nati/PAPERS/expander_survey.pdf and I encounter this equation (page 74 top). and also this equation (page ...
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2answers
162 views

U-Substitution Intuition

I've had a very hard time wrapping my mind around u-substitution. I understand how the chain rule applies with the following intuition: Say I have some car whose position function is defined as: $x=...
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0answers
115 views

Why are principal crossed homomorphisms coboundaries?

According to Wikipedia (and to many other sources): The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) $f : G \to M$ satisfying $f(ab)=f(a)+...
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1answer
96 views

Intuition behind a counterexample to $|A+A|\leq |A-A|$, where $A$ is a finite set

Define $$A+A=\{a+b:a,b \in A\}, A-A = \{a-b:a,b \in A\}$$ Then prove or disprove the following $$|A+A|\leq |A-A|$$ Intuitively, it should be true, as $$a+b=b+a$$ $$a-b \neq b-a$$ ...
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6answers
108 views

Intuition behind $ ||x|-|y|| \leq |x-y| $

In my math analysis course we proved that $ ||x|-|y|| \leq |x-y| $. However, unlike the triangle equality that I can visualise with the example of a triangle, I cannot visualize this one nor ...
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2answers
41 views

Why is $U ^tU $ an orthogonal projection on $\operatorname{Im}(U)$?

Let $U \in M_{n,k}(\mathbb{R})$ such that : $^t UU = I_k$. Then I would like to understand geometrically why $U ^t U$ is the orthogonal projection on $\operatorname{Im}(U)$ ? When $n = k$ we are ...
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2answers
47 views

Intuition for a limit found using L'Hôpital (geometry)

$OPR$ is a sector with central angle $\theta$. $A(\theta)$ is the area of the segment bounded by the line $PR$ and the arc $PR$ and $B(\theta)$ is the area of the triangle $PQR$. The ratio $$\frac{A(\...
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1answer
34 views

What does the set : $\{ (\lambda_1, …, \lambda_n) \in \mathbb{R}_+^n \mid \sum \lambda_i = 1 \}$ represent

What the set : $$S = \{ (\lambda_1, ..., \lambda_n) \in \mathbb{R}_+^n \mid \sum \lambda_i = 1 \}$$ represent geometrically ? I tried in dimension $2$ and it seems that I get a triangle. But I ...
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1answer
35 views

Can someone please explain Prufer's sequence?

I'm currently having a difficult time understanding the Prufers sequence. I'm a bit confused on understanding the concept of choosing the smallest node and how to know which nodes connect to which ...
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0answers
46 views

If $^{t\mkern-5mu}A A = I_k$ in which way should I think about : $A\, ^{t\mkern-5mu}A$

Let $A \in M_{n,k}(\mathbb{R})$ such that : $^{t\mkern-5mu}AA = I_k$ then what can be said about : $A\, ^{t\mkern-5mu}A$ ? In which way should I think about the transformation $A^tA$ ? First of when ...
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25 views

Lagrange’s method for PDEs

What is the intuition behind the Lagrange’s method for solving first order quasi linear PDEs? And what is its relation with the method of characteristics?
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1answer
35 views

Stable and unstable manifolds that are tangent to each other in a continuous dynamical system?

I am thinking of a scenario/ examples where the stable and unstable manifold of an equilibrium of a continuous dynamical system are tangent to each other? Any examples/ plots would be helpful?
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2answers
99 views

Intuition of Euclidean norm

I know the following : A norm is Euclidean iff it respects the parallelogram law. The problem is that the parallelogram law is not very intuitive geometrically. So I am wondering if there is a ...
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2answers
54 views

Intuition behind the $cl(cl(A))=cl(A)$ Kuratowski closure axiom in general topology?

If we use the real numbers as a model of topology (with the standard topology), then it is obvious why $cl(cl(A))=cl(A)$ should hold, given the definition of "closure of a set" in the context of the ...
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0answers
10 views

linear functions and physical constraints

I have a feeling I am missing something elementary with my question, but I can't seem to find where the problem is. In a certain physical problem, I have to evaluate an integral in order to obtain an ...
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2answers
32 views

What is an intuitive application of estimators?

So we're currently studying Estimators and we just proved Cramér-Rao's inequality and that when it is an equality, then whatever estimator we have is a unique MVUE. All of this to me just sounds like ...
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1answer
43 views

Intuition/clarification for product topology in Munkres' Topology

In Topology, the second edition by Munkres, in section 19, on page 113 he says the following (when talking about how to impose topologies on set): "Another way to proceed is to generalize the ...
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1answer
36 views

Is $\sup_{x\in \mathbb R} \sqrt{|x|} \sin x = + \infty$?

The graph of $\sqrt{|x|} \sin x$ seems like staying around the $x$-axis, but $\lim_{x\to \infty} \sqrt{|x|} = +\infty$ and $\sin x$ is bounded.