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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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What exactly is the significance of $a_{\phi}$ component in cylindrical co-ordinate system?

{ N.B: I have already looked for the answer in the following question but didn't quite get what I am looking for: What does phi component of a cylindrical coordinate signifies? } Let us say we have a ...
MSKB's user avatar
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Describing Frequentist vs Bayesian probability using casual dialogue

I am trying to presented Frequentist and Bayesian probability in casual dialogue format (i.e. a discussion between friends) for my younger sister, who is unfamiliar. The goal of this exercise is to ...
Abhishek Divekar's user avatar
1 vote
1 answer
51 views

Help developing intuition behind sufficient statistics (Casella & Berger)

Migrated to Cross Validated I am trying to understand the following intuition for sufficient statistics in Casella & Berger (2nd edition, pg. 272): A sufficient statistic captures all of the ...
Aaron Hendrickson's user avatar
1 vote
1 answer
168 views

What does ordinary differential equation mean, based on the word itself? [closed]

What does Ordinary Differential Equation mean? I know the mathematical definition, but I want to understand the meaning of the word "ordinary" here. I know that an ordinary derivative is not ...
Θάνος Κ.'s user avatar
5 votes
0 answers
121 views

Are creation and annihilation operators special?

In Weinberg's The Quantum Theory of Fields,volume I, the author quotes a theorem that left me a bit mystified. He states Any operator $O: \mathscr{H} \rightarrow \mathscr{H}$ may be written $$O=\sum_{...
Lourenco Entrudo's user avatar
1 vote
2 answers
40 views

Alternate Proof for Sum of Sides of a Triangle Inequality

I recently stumbled upon an idea for a proof for the sum of two sides of a triangle inequality. Note that I am just a high school student and feel free to correct me wherever if I am wrong. Statement/...
Rishwanth's user avatar
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0 answers
64 views

$(p, q)$ tensors and multidimensional arrays

I am trying to understand connections between different interpretations of tensors. In many contexts, tensors are treated simply as multidimensional arrays. Let us consider the following example. Let $...
mathslover's user avatar
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28 votes
8 answers
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Where is the pentagon in the Fibonacci sequence?

It is common wisdom that "When you see $\pi$, there is a circle close at hand". For example: The periods of sine and cosine equal $2\pi$? Properly constructed, the right triangles that ...
No Name's user avatar
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3 votes
1 answer
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Intuition for working with p-names.

So I have been trying to learn about forcing in set theory from kunen. There are these things called "p-names" which have a complicated definition in forcing. if $M$ is a countable ...
Kripke Platek's user avatar
2 votes
2 answers
91 views

If I have a sequence $a_0, a_1, a_2, \cdots$ , then is expressing the limit of this sequence as $a_\omega$ sensible?

If I have a sequence created by some rule which comes to a limit , then I can express it as $a_0, a_1,a_2,\cdots$. If I said $\lim_{n \to \infty} a_n = a_{\omega} $ , is that a sensible thing to do ? ...
Q the Platypus's user avatar
2 votes
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Countable system of differential equations

I'm interested in solving the following countably infinite system of ODEs \begin{equation} \frac{d}{dt}H_{n,k}\left(t\right)=\left(-p\right)nH_{n,k}\left(t\right)+\left(-q\right)nH_{n,k+1}\left(t\...
zokomoko's user avatar
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2 votes
1 answer
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Intuitively, why does $I(\lambda)$ decay as $\lambda \to \infty$ if $\Phi$ is not constant?

I'm quoting a few lines from Sogge's Fourier Integrals in Classical Analysis. Stationary phase is of central importance in classical analysis since integrals of the form \begin{equation} I(\lambda) = ...
stoic-santiago's user avatar
1 vote
3 answers
94 views

Solve an integral, e.g, $I = \int_{p}^{p+\delta p} f(x) dx=\int_{p}^{p+\delta p}\frac{e^{ax}(1-e^{-ax})}{x(1-x)}dx$ [duplicate]

I am trying to solve an integral over an infinitesimal interval, such as: $\begin{align}I = \int_{p}^{p+\delta p} f(x) dx=\int_{p}^{p+\delta p}\frac{e^{ax}(1-e^{-ax})}{x(1-x)}dx&\tag{1}\end{align}...
CafféSospeso's user avatar
2 votes
1 answer
110 views

Why does subgroup definition imply closure of binary operation?

I'm reading Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain, and S. R. Nagpaul, and struggling with their very basic definition of a subgroup, and why it implies that it must be ...
panto's user avatar
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Is the resemblance in the formula for distance between $2$ parallel planes and foot of perpendicular on a line a coincidence?

I was trying to derive the formula for the foot of the perpendicular of a point $(x_0,y_0,z_0)$ on the line $L_1:\dfrac{x-x_1}{l}=\dfrac{y-y_1}{m}=\dfrac{z-z_1}{n}$, where $\left<l,m,n \right>$ ...
Cognoscenti's user avatar
6 votes
3 answers
186 views

Prove that any sequence of five distinct integers must contain a 3-chain

This task is from MIT OpenCourse Mathematics for CS 2010 course, problem set 2, exercise 1(d). I am aware that this question has already been asked several times previously on this platform. Yet, the ...
Lina's user avatar
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0 votes
1 answer
99 views

Can someone explain when a limit is defined?

Why is it that the following limit is defined if $\lim_{x\to a}f(x) = \lim_{x\to a}g(x) = 0$? $$\lim_{x\to a}\frac{f(x)}{g(x)}$$ In contrast, the limit isn't defined if $\lim_{x\to a}f(x) \neq 0$ but $...
Aryaan's user avatar
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1 vote
2 answers
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Significance of tangent indicatrix

I'm reading a proof of the Four Vertex Theorem, and the proof introduces the notion of a tangent indicatrix. The precise text is as shown above. Can someone explain the intuition behind the tangent ...
DC2974's user avatar
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-2 votes
1 answer
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Image Problem: During a rough sailing trip Janne tried to sketch a map [closed]

Could someone explain the question, and how to go about solving it. Btw it is from a 2011 IKMC paper.
Travis's user avatar
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1 answer
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Matrix multiplication interactions with scalar multiplication

In my linear algebra textbook, it states the following theorem: Using this equation, it then derives the inverse matrix of $T$ (assuming $T$ is invertible): (in both definitions, the ⋅ represents ...
LateGameLank's user avatar
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0 answers
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Intuition behind the formula: $(ah, bk) = (a, b)(h,k) (\frac{a}{(a,b)}, \frac{k}{(h,k)}) (\frac{b}{(a,b)}, \frac{h}{(h,k)})$ [duplicate]

Recently, I've tried to derive $$(ah, bk) = (a, b) \cdot (h,k) \cdot \left(\frac{a}{(a,b)}, \frac{k}{(h,k)}\right) \cdot \left(\frac{b}{(a,b)}, \frac{h}{(h,k)}\right).$$ I think I succeeded but it got ...
Bryle Morga's user avatar
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1 vote
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32 views

The newton method is invariant to affine linear transformations. Why is this an important or useful property? [duplicate]

In my lecture notes this property is highlighted as an important one but it doesn't mention WHY it's important. Gradient descent in comparison doesn't have this property. What is a typical scenario ...
Sen90's user avatar
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0 answers
26 views

Intuition behind relation between Gamma and Standard Normal distribution [migrated]

I read if $Z$ is a random variable with a standard Normal distribution and $X=Z^2$ then $X \sim Gamma(1/2, 1/2)$. I understand the math (manipulations of formulas) behind it. What about the intuition? ...
Gabriele Bettineschi's user avatar
3 votes
2 answers
225 views

The notion of orientation in vector spaces

Do Carmo's Curves and Surfaces book states that "two ordered bases $e = \{e_i\}$ and $f = \{f_i\}$, $i = 1,\ldots,n$, of an $n$-dimensional vector space $V$ have the same orientation if the ...
DC2974's user avatar
  • 111
2 votes
0 answers
49 views

Parameterization by arc length: the concept

Can someone let me know if my understanding of parameterization by arc length is correct? If we have a regular parameterized differentiable curve $\alpha: I \rightarrow \mathbb{R}^3$, it is ...
DC2974's user avatar
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1 vote
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How to interpret equivalence relations graphically.

Imagine a relation $R$ as a subset of the cartesian product of the real numbers with itself. This relation can be interpreted as a graph. If R is an equivalence relation Reflexivity can be ...
19360254735168's user avatar
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64 views

What is the intuition behind all invertible complex matrices being congruent? [closed]

Despite the proof being quite straightforward i found the fact that two symetric matrices over $\mathbb{C}$ are congruent as long as thier ranks are the same surprising. In the language of bilinear ...
Michał Cieszyński's user avatar
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1 answer
22 views

The Frobenius Endomorphism is Surjective iff the field is perfect

I'm taking a Galois theory class right now. I've read and understood the proof that the Frobenius endomorphism is surjective iff the field is perfect (working in characteristic $p$). But it just feels ...
Boran Erol's user avatar
3 votes
0 answers
75 views

Intuition behind construction for Induced Representations.

I'm studying representation theory. I've come across induced representations. I really don't understand why the construction: $$ W = \bigoplus_{i = 1}^{[G:H]}g_iV $$ as shown in this page produces a ...
Apollonius's user avatar
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58 views

Question about Vector Field Geometry

I am confused about vector fields. Let $f_i:\mathbb R^3\to \mathbb R$ be real-valued functions for $i=1,2,3$. Let $\vec{f}=(f_1,f_2,f_3):\mathbb{R}^3\to\mathbb{R}^3$ and $\vec{g}=(f_2,f_1,f_3):\mathbb{...
佐武五郎's user avatar
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0 answers
22 views

Most natural definition of a product of smooth manifolds with a smooth manifold structure.

In this question,I want to ask and clarify(for myself) some points regarding the definition of product manifolds,so that I can appreciate the definition better. The definition of any product structure(...
Kishalay Sarkar's user avatar
5 votes
3 answers
161 views

Choosing between the 'and' and 'implies' connectives

$pol(x): x$ is a politician $liar(x): x $ is a liar All politicians are liars : $\forall x(pol(x) → liar(x))$ Some politicians are liars : $\exists x(pol(x) \land liar(x))$ No politicians are liars :...
user1327299's user avatar
3 votes
0 answers
63 views

Question About the Meaning of Notations: Big "O", $\leq$, $\lesssim$, $\approx$, $\lessapprox$, etc. in Combinatorics

I am completely new to combinatorics. I start to self-study some combinatorics but got confused in the very beginning. I came upon the following: $X\lesssim Y$ means that as $X$ and $Y$ grow large, ...
Beerus's user avatar
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1 vote
4 answers
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A possible error in a math question of 2022 Brazil's ENEM, a equivalent to US's SAT

This is a rather simple question, and it may seem like a physics question, but it actually belongs to a math test. I'll translate it into English first and then reproduce the Portuguese version. "...
Igor Paulino's user avatar
2 votes
3 answers
138 views

Why a vector function to $\mathbb{R}^n$ can be regarded as $n$ different functions?

I am taking advanced calculus, and the professor said the following: we can regard $f: U \rightarrow \mathbb{R}^n$ as $n$ different functions, and write $f = (f_1,...,f_n)$." I am not sure what ...
A24601's user avatar
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0 votes
1 answer
34 views

Why the orthogonal projection give minimal?

Given the data $ \vec{x}=(-2,-1,1,2) )$ and $( y=(1,1,-1,1) )$. Use an orthogonal projection to determine the coefficients $( a_{0}, a_{1}, a_{2} )$ of the quadratic polynomial function $ \begin{...
asdfgh jkl's user avatar
1 vote
1 answer
79 views

Circuit Probability Question from Mathematical Statistics

From Mathematical Statistics, 7th ed., Chapter 2, Supplementary Exercise no. 2.163: Relays used in the construction of electric circuits function properly with probability $0.9$. Assuming that the ...
Mailbox's user avatar
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2 votes
3 answers
161 views

A constructive proof of this innocent set theoretic proposition?

I was reading Freiwald's An Introduction to Set Theory and Topology, and I came across the following exercise from Chapter 1: E8. Suppose $A$, $B$, $C$, and $D$ are sets with $A\ne\emptyset$ and $B\...
Atom's user avatar
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4 votes
1 answer
110 views

Making clear the definition of 'affine variety' in Mumford's book.

I am reading "The Red Book of Varieties and Schemes" by Mumford. In section 4 the author defines the term affine variety: An affine variety is a topological space $X$ plus a sheaf of $k$-...
Toni's user avatar
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3 votes
4 answers
73 views

If sets $X$ and $Y$ have at least two elements each, then $X \cup Y \preceq X \times Y$

This is a problem from Derek Goldrei's Classic Set Theory, a book that I am currently working through. This is in a chapter titled 'Cardinals (without the Axiom of Choice)' in case that context is ...
Jared Napier's user avatar
0 votes
1 answer
52 views

Absolute Scale and Distance Function in Hyperbolic Geometry

I have a question about intuition on key differences between axiomatic hyperbolic and Euklidean geometries. More precisely I'm pondering about last sentence of following explanations from wikipedia: [...
user267839's user avatar
  • 7,489
2 votes
0 answers
68 views

Understanding the rolling motion of a circle on $\mathbb{R}$ in terms of parallel transport and Ehresmann connections

I am trying to understand a toy example to help build my intuition about connections on fiber bundles and parallel transport. My main issue is trying to understand if and how the "no-slip" ...
Tob Ernack's user avatar
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0 votes
0 answers
48 views

What is the geometrical interpretation/intuition behind complex eigenvectors and eigenvalues?

As far as I know, matrices are transformations of the x-y coordinate systems i.e. the rotations and skews. As for eigenvectors, they are the vectors, which after transformation, remain oriented the ...
Soham P's user avatar
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0 votes
0 answers
49 views

Intuition for Exponential Sheaf Sequence via Winding Number

I have a question about the geometric intuition behind exponential sheaf sequence $$ 0\to 2\pi i\,\mathbb Z \to \mathcal O_M\to\mathcal O_M^*\to 0 $$ inducing long cohomological sequence. In wikipedia ...
user267839's user avatar
  • 7,489
1 vote
0 answers
60 views

What does this ratio of derivatives represent, geometrically?

Suppose I have an arbitrary smooth real-valued function $F(x,y,z)$ with the property that all the first-order partials of $F$ are strictly positive everywhere. The criterion of St-Robert says that $F$ ...
user326210's user avatar
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2 votes
0 answers
78 views

Intuition: Why is quadratic variation finite for martingales

(Disclaimer: I'm not well-read in this topic, so might be getting some details wrong. Hopefully not wrong enough to make my question for intuition moot) For any martingale $(X_t)_{t \geq 0}$ with $X_0=...
Bananach's user avatar
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0 votes
0 answers
46 views

Intuition of compactness in function spaces

I understand compactness in $\mathbb{R}^n$. However, I do not know much about what sets of function spaces are compact. For example, what are the compact sets of the spaces $B([0,1], \mathbb{R})$ of ...
Power's user avatar
  • 109
4 votes
3 answers
275 views

Can vacuously true statements be proved using proof by contradiction?

I always thought it was valid to prove vacuously true statements using proof by contradiction but now am not so sure. For example, consider the vacuously true statement $\forall x \in \emptyset, x + 1 ...
gfjfvhjk's user avatar
5 votes
4 answers
506 views

Why are Contour Integrals defined the way they are?

I have a question about Contour Integrals. Contour Integrals are defined as $$\int_{C} f(z)dz = \int_{a}^{b} f(z(t))z'(t)dt \tag{1}$$. But why define it this way? The textbook I'm using (Complex ...
Asterix's user avatar
  • 98
1 vote
4 answers
710 views

Why does set-theoretic union and intersection operate on reverse logic?

In set theory, $A \cup B$ is logically defined as $\{x : x \in A \lor x \in B\}$. In set theory, the result of unionizing A with B is a bigger set, but in logic, "or" is a softening ...
Fomalhaut's user avatar
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