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 is the intuition behind the definition of the differential of a function?

What is the intuition behind the definition of a differential of a function in differential geometry? i.e. $$df(p)(v_{p}) =v_{p} (f)(p) $$ where $v_{p} \in T_{p} M$ is a vector in the tangent space to ...
3
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2answers
133 views

Why General Leibniz rule and Newton's Binomial are so similar?

The binomial expansion: $$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$ The General Leibniz rule (used as a generalization of the product rule for derivatives): $$(fg)^{(n)} = ...
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3answers
44 views

Cosine and Sine Angle Addition Intuition [duplicate]

I am lacking in understanding in the cosine and sine angle addition formulas. I have seen several questions similar to this but I have not seen an answer that explains how this conclusion can be ...
0
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0answers
32 views

Definition of Mapping

Some formal characterization is seems very abstract for me. For instance: Let $X$ be a finite set of alternatives. We denote by $\chi$ (respectively, by $B$) the collection of all non-empty ...
3
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1answer
82 views

Help with intuition on Cardinal Arithmetic Problems

It happens a lot to me that when I find an intuitive model (picture) of a mathematical entity, the proofs left as exercises in books are very easy to solve. For example when dealing with filters and ...
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2answers
57 views

how to understand the definition of continuity in analysis?

Please have a look at the picture above. This is about the continuity in analysis. I don't really understand how to utilize this definition? It says that is statement is equivalent to f is ...
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11 views

Visualization of the fact that the integers defining lens spaces must be coprime

This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @Earthliŋ: Why are the integers appearing in lens spaces coprime? I understand the ...
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1answer
29 views

Is there an interpretation for writing a polynomial in $x$ as a polynomial in $(x-b)$?

Let $Q(x)$ be a polynomial in $x$ of order $n$. The Taylor polynomial of $Q(x)$ of order $n$ developed around $x=b$ (denoted by $P_{n,b}(x)$ ) corresponds to $Q(x)$ written in $(x-b)$. This can be ...
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2answers
156 views

Given a drawing of a parabola is there any geometric construction one can make to find its focus?

This question was inspired by another one I asked myself these days Given a drawing of an ellipse is there any geometric construction we can do to find it's foci? I think this is harder, I can't ...
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3answers
465 views

Geometric intuition behind pullback?

I am having hard time with forming a geometric intuition of pullback and pushforward. The definition the book gives is like this: There are two open sets, $A$ and $B$. There is a dual transformation ...
7
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1answer
189 views

Ellipse inscribed on a quadrilateral

The problem is: Given that an ellipse is inscribed on a convex quadrilateral and each one of it's diagonals pass through one foci of the ellipse show that the product of the opposite sides ...
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59 views

Why Delta Complex structure?

I am a bit confused about delta complex structure and CW complex structrue. I was wondering is n-dimensional disks homeomorphic two n dimensional triangles? I mean if that is the case then I can make ...
7
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1answer
164 views

Visualization of Lens Spaces

I am trying to visualize lens spaces geometrically. While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious ...
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0answers
34 views

What does the Dedekind Rule `say'?

In Relation Algebra, the modal law or dedekind rule $$R;S \,\cap\, T \;\subseteq\; (R \cap T;S^\circ);S$$ appears often and I wonder what is the motivation behind it. Moreoever, what does it "say". I ...
6
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95 views

Intuition behind variation of parameters method for solving differential equations

I have used the variation of parameters method (and have been taught it, although not hugely in depth) and I was wondering if I've understood the intuition behind it. In particular I've been thinking ...
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1answer
50 views

What is the different between a product of gaussian random variables and their PDFs

Multiplying two univariate Gaussian PDFs $$ X \sim \mathcal{N}(\mu_X,\sigma_X) \\ Y \sim \mathcal{N}(\mu_Y,\sigma_Y) \\ Z = X*Y $$ results in closed form equations for $\mu_Z$ and $\sigma_Z^2$: $$ ...
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1answer
79 views

Puzzle: Players A,B,C,D are in a line

Players A,B,C,D stands in a line. Players A, D do not move. round 1: player B moves one distance closer to the midpoint of A,C round 2: player c moves one distance closer to the midpoint of B,D ...
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1answer
69 views

What is the idea behind “representability” in a first order theory?

I've been reading through Enderton's logic, this notion is introduced and is given special attention as it's said that they are crucial in the proof of incompletness theorems. I grasp the formal ...
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6answers
126 views

Why is $P \to Q \equiv \neg P \vee Q$?

By truth table, we know that $P \to Q$ is equivalent to $\neg P \vee Q$. But I'm trying to understand why this work? How can connective "or" be implication. I tried some examples but I still can't ...
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1answer
52 views

definition of multiplication intuition (Set theory)

We define how to multiply two integers using set theory as follows [(a,b)][(c,d)] = [(ac + bd,ad + bc)] what is the intuition behind defining multiplication like this ?
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4answers
136 views

Why do graphs sometimes have unevenly spaced intervals in axis e.g. logarithmic

Why do some graphs not have evenly spaced out units on the x or y-axis? For example they have log on one or both of the axis? Here's an example from this page I guess the page does give a bit of an ...
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2answers
53 views

Finding angles plane geometry

$\Delta ABC$ is obtuse on $B$ with $\angle ABC = 90 + \frac{\angle BAC}2$ and we have a point $D \in AC$ (in the segment, I mean D is in between A and C) such that $\angle BDA = \angle ABD + ...
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1answer
132 views

Problematic lack of intuition for number theory

I Have a problem. I mostly do mathematics because I find it fascinating and enjoy doing it. Now whenever I skim through a book a number theory I always find myself thinking 'I wish I would understand ...
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0answers
62 views

What is the variance of self-information (or surprisal)?

The self-information of an outcome $x_i$, or surprisal, is defined as: $$ I(x_i)=-\log P(x_i), $$ where $P$ means probability. This way, the Shannon entropy can be seen as the "average" or "expected" ...
3
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1answer
61 views

Intuitive difference between a continuous map and a homeomorphism

I know the formal definitions of both continuous map and a homeomorphism between two spaces. If two spaces are homeomorphic intuitively they can be thought of spaces which can converted to each other ...
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66 views

Is it correct to say that if $\lim\limits_{x \to a}f(x) = 0$ it is an Infinitesimal?

I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ...
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3answers
56 views

The support and the non-vanishing set of a function on a scheme

I have some confusion regarding the two concepts: Let $(X,\mathscr{O}_X)$ be a scheme, let $f\in \Gamma(\mathscr{O}_X,X)$ and define the support of $f$ to be $$\operatorname{Supp}(f) : = \{p\in X: ...
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2answers
76 views

How can one intuit complex numbers from quaternions?

I understand that quaternions are sort of an extension of complex numbers in higher dimensions. If that's really the case conceptually (is it?), it must be possible to get back from the higher ...
3
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0answers
97 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
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1answer
31 views

Multivariable differentiability: where does the concept come from?

I'm not sure if I posed the right question, but this is my curiosity: That a function is differentiable in $P\in\mathbb{R}^n$ means that given $F:\mathbb{R}^n\rightarrow\mathbb{R}^m$ $$ \lim_{X\to ...
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1answer
22 views

Improving the proof by contraposition / why it works

This is the problem Prove that if n is an integer and 3n+2 is odd, then n is odd So for this I should take $3n+2$ to be true and assume $\lnot q$, therefore I ...
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1answer
37 views

Linear Transformation Between Different Dimension Vector Spaces

If there is a linear transformation from a smaller vector space to a larger one which is 1-1 and onto (can it be)? What will happen if the transformation is from a bigger vector space to a smaller ...
2
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1answer
38 views

For every $z\in \Bbb C$, the exponetial series converges uniformly on every bounded subset of the complex plane

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This series converges uniformly on every bounded subset of the complex plane. What does this mean in simple terms?
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1answer
31 views

Exponential of a complex number converges absolutely

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
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24 views

Visualising relations between inequalities and solution criteria.

Is there any intuitive, visual explanation of the following lemma: Lemma: Let $\{ \alpha_{ij} : i = 1, \ldots, m, j = 1,\ldots, n \}$ be an $m \times n$ matrix, $\alpha_i = (\alpha_{i1}, \ldots, ...
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1answer
67 views

How to formalize that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0 \implies$ $g$ “grows faster” than $f$?

I understand that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0$ implies that, for sufficiently large values of $x$, $f(x)<g(x)$, as a direct consequence of the definition of limit to ...
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0answers
570 views

Intuition for gradient descent with Nesterov momentum

A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ...
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2answers
51 views

Why is $\lim\limits_{x\to0+}x\cot x=1$?

Why is $\lim\limits_{x\to0+}x\cot x=1$? Since both $x$ and $\cot x$ are continuous at zero and both equal to zero at $x=0$ why is the limit of both of them $1$? i.e why isn't it: ...
0
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1answer
34 views

How do limit cycles explain curvilinear asymptotes?

I'm a 17 years old and I have no clue about a concept known as limit cycles. I looked it up and I understand it represents the orbit of functions approaching other A person told me that limit cycles ...
2
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1answer
76 views

Intuitively speaking, why was there a need to “eliminate” quantified variables in mathematical logic?

I'm trying to wrap my head around the understanding of lambda-calculus, from a math/computing/logic standpoint and am reading more about its very genesis. This has taken me to 1924 - Schonfinkel's ...
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1answer
56 views

About the function $f(x)=\sin x\ln x^2$ and derivative definition

$f(x)=\begin {cases}\sin x\ln x^2 & x\neq 0\\ 0 & x=0\end{cases}$ When I try to find the derivative on $x=0$ with the defintion I get: $\displaystyle\lim_{h\to 0}\frac ...
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0answers
46 views

subtracting mean of iid RVs increases mutual information?

I have a problem about intuition: substracting the mean of iid RVs seems to increase the mutual information. Say $X,Y$ are real iid RVs, then $\frac{X-Y}{2}$ and $\frac{Y-X}{2}$ are not independent ...
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1answer
30 views

How to “see” that this expression is $>0$.

$N \in \mathbb N$. $\displaystyle\int_{N-1}^N \left(\dfrac{1}{x} - \dfrac{1}{N}\right) dx>0$ This is the finish of a proof, a modification of $\log N-\log (N-1) -\frac{1}{N}$. Calculating it ...
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1answer
24 views

DiffEq: The correlation between the number of roots and the number of limiting behavior

So I understand that when dy/dt is 0, this means that $y(t)$ is a constant. But why does the number of limiting behavior depend on the roots of the differential equation. And why if $dy/dt ≠ 0$, the ...
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40 views

intuitive interpretation of the multiplicity

Although logically I can understand and use multiplicity (for defi􀀀nition see 4.1.5 of Bruns_Herzog), yet, the concept of multiplicity of a module is not completely clear for me. Is there an ...
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3answers
283 views

How to understand intuitively the Stolz-Cesaro Theorem for sequences?

I have to give a presentation on the theorem in Real Analysis with a fellow student. While I've looked over the proof and verified that, yes, step B does indeed follow logically from step A, etc. and ...
2
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1answer
149 views

Developing Examples for Basis $\mathcal{B}$ on Topology $\tau$

I put together my own example on the basis of topology. I wanted to know if it is a valid example displaying the properties of basis. Here is my example: Example: Since $X\subset\mathbb{R}$, let ...
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3answers
56 views

Confusion on how to calculate mean value

I've done this type of thing in multiple classes over multiple years since high school, and still, when it's presented to me, I fumble around like a dope. Consider a gas of $N_0$ non-interacting ...
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39 views

Protomodular categories

The axioms for abelian categories are nice and clear. The axioms for protomodular categories - and therefore semi-abelian categories - are beyond me entirely. I'm looking for a breakdown of the ...
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40 views

Derived Category in terms of Torsion Theory?

It is known that there's a bijection between hereditary torsion theories on, and localizations of, a fixed abelian category. Is this bijection natural? How/why not? How can I think of the derived ...