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.
4,107
questions
1
vote
5
answers
105
views
Density of $\mathbb{Q}$ in $\mathbb{R}$ seemingly contradictory to infinite set cardinality [duplicate]
In my real analysis module we proved that both $\mathbb{Q}$ and $\mathbb{R \setminus Q}$ are dense in $\mathbb{R}$, meaning that between any two real numbers there exists a rational and an irrational ...
-1
votes
0
answers
29
views
How does the elements of a formula explain the shape of an object. [closed]
Multiplication seems to add an extra dimension to explain the shapes of objects.
The area of a rectangle is given by (L*W), its length and its width put the rectangle in the second dimension, ...
0
votes
2
answers
57
views
How does hyperplane work?
The definition of a plane in $R^3$ is intuitive : $Ax + By + Cz = D$ , where $(A, B, C)$ is a normal vector to the plane and $D$ is some bias.
I can visualize it dividing the space into two halves (...
0
votes
0
answers
33
views
How to prove inequalities in definite integrals or find the integrals' boundaries?
$$ \int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} \frac{e^x+ e^{\frac{3}{2x}}}{x} dx$$
is the definite integral and the question is how to calculate its upper/lower bound.
I know we can differentiate some ...
2
votes
0
answers
20
views
Understanding Probability Distributions [closed]
Is there some way of understanding more deeply why exactly it is that the probability mass function or probability density functions of various random variables have the shapes that they have when you ...
12
votes
2
answers
325
views
+50
The vertices of a triangle are three random points on a unit circle. The side lengths are $a,b,c$. Show that $P(ab>c)=\frac12$.
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
Show that $P(ab>c)=\frac12$.
The result is strongly suggested by ...
0
votes
1
answer
34
views
What does it mean for the variance to be squared?
Suppose we are measuring the heights of different people. Our measures will be in $cm$, but the variance will be in $cm^2$. If we were measuring the time to complete a race course, our measures would ...
1
vote
0
answers
34
views
Motivation and intuition behind the geometric product integral
According to the relevant wiki article, there are three types of product integrals. Type II, the geometric integral, is named as the continuous analogue of the discrete product operator.
The geometric ...
1
vote
0
answers
83
views
Random Walk Intuition: why do we multiply probabilities?
In the response to this question (question and response below), Did mentions that the probability that we eventually reach $0$ from $1$ is labeled as $r$. This is distinct from the probability that ...
1
vote
1
answer
24
views
Proof of the Cycloid Parametric Equation
One of the steps of deriving the equations for the parametric curve of a cycloid is the following:
Here we establish that the distance PT is equal to the distance OT, which then (alongside other ...
0
votes
1
answer
64
views
What is the intuition behind picture in Fulton and Harris about $sl_3(\mathbb C)$
We are seeing at representations $sl_3(\mathbb C)$ here from Fulton and Harris. The space $\mathfrak h$ is collection of diagonal matrices and we have written $\mathfrak{sl_3}(\mathbb C) = \mathfrak h ...
1
vote
0
answers
82
views
Intuition why pulling out $\mathcal{G}$-measurable $X$ of $E[f(X,Y)|\mathcal{G}]$ requires $Y$ indep of $\mathcal{G}$
Given $\mathcal{G}$ a sigma-field and $X$ a $\mathcal{G}$-measurable random variable, on an intuitive level, why do we need $Y$ to be independent of $\mathcal{G}$ to pull out $X$ from the conditional ...
5
votes
4
answers
1k
views
Tautologies in classical logic [duplicate]
According to its truth table, $P \lor \neg P$ is a tautology, i.e. it is true for all truth values of its constituent propositions. But how come that is true in classical logic? $\lor$ is the ...
3
votes
2
answers
280
views
What's the intuition for reflexive pairs?
A reflexive pair is a pair of morphisms $f, g: A \to B$ such that there exists a common section $s: B \to A$, i.e. $fs = gs = 1_B$.
What's the intuition behind this concept?
I tried looking at this in ...
1
vote
0
answers
44
views
Why is the demonstration of the "sample" standard deviation somehow more exact than the classic standard deviation, if $\frac{x}{n} ≠ \frac{x}{n-1}$?
I cannot see why to use the "sample standard deviation" instead of the classic "standard deviation" (the "population") ever, most explanations I find are just prescribing ...
2
votes
1
answer
40
views
Is my interpretation of Shannon entropy correct?
We all know the formula. All the interpretations don't make sense to me, however. What does it mean that I get $0.81$ information after knowing X? Or what does it mean that there is $2.33$ uncertainty?...
0
votes
0
answers
45
views
Intuition for conditoning a random variable on a related random variable
I'm confused by the reasoning for the answer to subproblem (c) from Blitzstein and Hwang,chapter 3, problem 27:
A company with n women and m men as employees is deciding which employees to promote.
(...
1
vote
0
answers
44
views
Intuition behind statistical approach for Hypothesis testing
I have studied Hypothesis testing from thinkstats2 which does simulation to estimate p value.
For e.g. In coin toss example, we run the simulation, to generate the test statistic i.e. Difference ...
0
votes
0
answers
112
views
In real life, we can have a pencil of length 2 cm. Can we have pencil of length $\sqrt{2}$ cm? [migrated]
In real life, we can have a pencil of length 2 cm. Can we have pencil of length $\sqrt{2}$ cm?
My answer to that was no , we cannot even make 2 cm pencil.
My argument was that when are working ...
1
vote
1
answer
82
views
I wonder why the author wrote $\phi_i:\mathbb{R}^n\to\mathbb{R}$ instead of $\phi_i:A\to\mathbb{R}$. ("Analysis on Manifolds" by James R. Munkres.)
I am reading "Analysis on Manifolds" by James R. Munkres.
Theorem 16.3 (Existence of a partition of unity). Let $\mathcal{A}$ be a collection of open sets in $\mathbb{R}^n$; let $A$ be ...
0
votes
0
answers
28
views
Question Intuition behind mathematics of activation function in a neural network. [migrated]
Does this intuition behind why an activation function is used in a neural network make sense mathematically :
For this example lets consider a fully connected (NOT CONVOLUTIONAL) network that ...
1
vote
1
answer
132
views
Interpretation of "If A, then B" as "A coupled with B": Context and Applicability
It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. In certain fuzzy logic books, I have come across the interpretation that &...
1
vote
1
answer
83
views
What Does It Mean to Say that Betti Numbers Count Holes? [closed]
There seem to be two distinct notions of "hole" in homology. The first is cycles mod boundaries, which I'm fairly comfortable with. The second is Betti numbers, which are more confusing to ...
1
vote
1
answer
104
views
Geometrically, what does the linearity of an operator imply?
In linear algebra, I learned that we define linearity as
A map/mapping is linear if it preserved additivity and scaling.
If you prefer symbols:
$L(\vec{\mathbf{v}})$ is linear if:
$$ L(\vec{\mathbf{v}...
0
votes
0
answers
95
views
Intuition for vector calculus
In my statistics class, I was introduced to Fisher Information. As it comes from the Taylor Expansion in vector form, I wanted to know terms were ordered in a certain way - whether it was just to make ...
0
votes
1
answer
45
views
Developing visual intuition for proofs involving cartesian product and sets
I am beginning to learn set theory proofs. It has been extremely useful to draw Venn diagrams for proofs just involving union, intersection, complement, e.t.c. However with cross product involved, how ...
2
votes
0
answers
104
views
What's the motivation or intuition of defining the regularity of measures in Measure Theory?
In Measure Theory, the regularity of measures is defined as:
In any topological space $ \left(\Omega,\tau\right) $ , $ \mu $ is a measure on it; $ \mu $ is inner regular:
$$ \forall A\subseteq\Omega, \...
1
vote
1
answer
55
views
Intuition behind dot product
How exactly does a dot product give the similarity between two vectors ? I have just been doing the calculation without every actually giving it any thought. For example if Vector A is <0.5,2,1> ...
1
vote
1
answer
46
views
Is it possible to give an ordering of concave sets?
I don't know how to express the question in a "mathematically correct" way.
If I have objects (or sets) these can locally be concave, convex or flat.
Example:
a plate is less concave than a ...
7
votes
3
answers
304
views
Projective spaces: why adding points to make linear intersections work make everything else work too?
The (real) projective plane is often motivated by the issue of lines in $\mathbb R^2$ having exactly one intersection, except in the case of parallel lines. The solution is to mimic what we see when ...
4
votes
1
answer
122
views
A square contains many random points. From each point, a disc grows until it hits the nearest neighboring point. What is the total area of the discs?
A unit square lamina contains $n$ independent uniformly random points. Each point is the centre of a disc whose perimeter touches the nearest neighboring point. Here is an example with $n=20$.
In ...
2
votes
0
answers
65
views
Derivation and intuition of relative entropy
I am trying to understand the derivation of the relative entropy formula, but have trouble with the reasoning and intuition of the modification made to the differential entropy formula.
My current ...
1
vote
2
answers
267
views
How to explain that area under the curve is real number?
As is known, one can consider area under the curve as area that consists of infinitesimally small areas bounded by $dx$ & $dy$ , in general terms.
Why doesn't there exist an area that equal to $r+\...
4
votes
2
answers
149
views
Why does $x^{\log_a y} = y^{\log_a x}$ ? (intuitive reason)
So I have seen mathematical proofs for this. They used other logarithmic identities to get this result. I am looking for a more intuitive approach to this and not just equations which lead to this.
...
0
votes
1
answer
50
views
What are some subanalytic sets which are not semianalytic?
I'm working with sheaf theory for the most part, and subanalytic sets are an integral part of the field. However, I'm struggling to get an intuition for subanalytic sets. I know, because I've seen it ...
1
vote
0
answers
44
views
Is the more general uncertainty principle related to bias-variance tradeoff?
Since I first saw this video on the more general uncertainty principle, I have taken the bias-variance tradeoff to be an example of it. They seem quite similar on the surface.
However, now that I'm ...
3
votes
1
answer
110
views
Infinitesimal Interpretation Of Exterior Derivative
I have a question regarding the conceptual understanding of the exterior derivative. I've read that one can view a $k$-form on an $n$-dimensional manifold as a collection of infinitesimal (oriented) ...
13
votes
4
answers
404
views
Break a stick at two random points. The probability that the longest piece is at least twice as long as each of the other pieces is $1/2$. Why?
Choose two independent uniformly random points on a stick, and break the stick at that those points. The probability that the longest piece is at least twice as long as each of the other pieces is $1/...
2
votes
2
answers
166
views
Quotient Rule: Why does Tao assume $g$ is non-zero on $X$?
Tao (Analysis I, 2022, p. 220):
Why can't Tao just assume $g(x_0)\neq 0$?
Wouldn't the conclusion still hold if we changed the assumption "$g$ is non-zero on $X$" to "$g(x_0)\neq 0$&...
2
votes
0
answers
47
views
What makes number rings special so that every nonzero ideal has finite index?
In a number ring, every non-zero ideal has finite index.
In order to build a better intuition on what makes number rings special to have this property, I'm trying to find some "minimal" ...
1
vote
0
answers
92
views
Intuitive explanation for why rotating through a great 2-sphere of a 3-sphere does not produce the whole 3-sphere
Why does rotating through a great $n$-sphere of a $(n+1)$-sphere not produce the whole sphere $(*)$ for $n \ge 2$?
More precisely: Let $X$ be the $(n+1)$-sphere. A great sphere is the intersection of ...
20
votes
1
answer
534
views
Conjecture: If $A,B,C$ are random points on a sphere, then $E\left(\frac{\text{Area}_{\triangle ABC}}{\text{Area}_{\bigcirc ABC}}\right)=\frac14$.
On (not in) a sphere, choose three independent uniformly random points $A,B,C$. Is the following conjecture true:
The expectation of the ratio of the area of (planar) $\triangle ABC$ to the area of ...
10
votes
2
answers
195
views
Cut a unit stick at $n-1$ random points. Expectation of product of fragment lengths is $\prod\limits_{k=n}^{2n-1}\frac1k$. Why?
On a straight stick of length $1$, choose $n-1$ independent uniformly random points. Cut the stick at those points, yielding $n$ fragments.
Let $\mathbb{E}_n$ be the expectation of the product of ...
0
votes
1
answer
59
views
Intuition for Well-Founded Relations
On page 11 of Holz's Introduction to Cardinal Arithmetic a well founded relation $R$ is defined as
that for which
$\forall x\in\text{fld}(R)\exists y\left(x\in y\land R^{-1}[y]\subseteq y\right)$, ...
1
vote
1
answer
102
views
Understanding the almost complex structure of a complex manifold
I start learning complex manifold by myself and hard to lift my previous intuition of differential geometry over the complex structure.
Let $M$ be a real $2m$-dimensional manifold. We define an ...
7
votes
1
answer
250
views
Why we Use Partitions of Unity. Specific Questions
While trying to better understand why we use partitions of unity I stumbled across this post. There are a few points I do not understand:
This is exactly how to compute integrals in practice. But if ...
0
votes
0
answers
132
views
Why do the higher order derivatives of $\frac 1 x$ grow extremely large?
The $n$th derivative of $f(x) = \frac 1 x$ is $$f^{(n)}(x) = (-1)^n n! \frac 1 {x^{n+1}}$$ so that e.g. $$f^{(10)}(x) = \frac {10!} {x^{11}}.$$
While this formula is easy to calculate, it's somewhat ...
2
votes
1
answer
95
views
Intuition of minimal surfaces in the class of sets of locally finite perimeter.
I'm reading the book Minimal Surfaces and Functions of Bounded Variation by Enrico Giusti. I'm wondering if anyone could help me understand the intuition of the theorem below.
We define
$$\...
10
votes
2
answers
209
views
Describing Mean Value Theorem as "Fundamental Theorem Of Differential Calculus"
Referring to "Introduction to Real Analysis" 4th Ed (2011) by Bartle and Sherbert. (ch 6, sec 2, page 174)
In fact the Mean Value Theorem is a wolf in sheep's clothing and is the ...
2
votes
5
answers
79
views
Does anyone have any general rules of thumb for when to use $e^x$ vs $2^x$ (or any other non-special exponential) during modeling?
Recently, I picked up a book about statistics and probability. I know $e^x$ is special because of its derivative and corresponding growth rate. However, I have a hard time connecting this academic ...