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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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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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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21 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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32 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 ...
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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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30 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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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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64 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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453 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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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: ...
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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 ...
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74 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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55 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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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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50 views

Intuition behind Gauss' Easter Sunday formula?

Easter Sunday is the first Sunday after the first full moon of spring. Given just the year, Gauss was able to derive a formula which gave you the month and day that easter sunday fell on. The formula ...
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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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23 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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39 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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263 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 ...
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133 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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49 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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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 ...
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52 views

How to relate algebraic properties of the fundamental group to the topological properties of a space

After just finally getting a somewhat intuitive grasp on just what the fundamental group of a topological is, I'm curious as to how one would relate the algebraic properties of the group to the ...
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36 views

why are these two intuitive ways of understanding division equivalent?

The elementary school example of division, say 12:4 is saying that you have to share out 12 cookies to 4 kids. However, another (only slightly less intuitive) way would be to ask how many times 4 ...
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Intuition for the Positive Real Number $\epsilon$ in Topology

Although this question might sound a little too simple, it is a problem that I must get addressed. In addition, there is no way for me to formally describe it. If you have something you can add, by ...
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52 views

Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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38 views

How (the graphic of) a $\mathcal C^1$ but not $\mathcal C^2$ function looks like

We know examples of functions (obviously we are in the context of real valued functions) which are continous but not derivable; the simplest is $x\mapsto|x|$. In particular we have a precise graphic ...
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1answer
109 views

What is an intuitive way to think of Cauchy's theorem?

I am looking at a problem which involves an understanding of why a finite group $G$ has an element with order $p$ if $p$ is a prime factor of $|G|$. I have looked at several resources and proofs ...
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1answer
69 views

Why are ill-conditioned systems of equations hard to solve iteratively?

Is there some intuition as to why ill conditioned system of equations hard to solve iteratively ( i.e. the convergence is slow) ? I've read convergence proofs of several methods, but still don't have ...
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Why does $\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$?

I was wondering why the following is true: $$\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$$ It is easy to obtain this result by doing a trig substitution but it's messy and not ...
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30 views

What separates the dot product from the scalar projection?

Just a little problem with geometric intuition here (or perhaps I just haven't slept in far too long!). I know that the scalar projection of vectors $ \vec{u} $ and $ \vec{v} $ is defined as $ ...
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92 views

Heuristics of the sum of squared naturals $(1^2 + 2^2 + 3^2 \cdots + n^2)$

I'm new and this is my first question (though I've been lurking). English is not my native language. Studying on my own. I'm really interested in deriving the formula $1^{2} + 2^{2} + 3^{2} + \cdots+ ...
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104 views

Intuituive reason why Fermats last theorem holds

I am unsure of whether it is normal, but to me, intuitively Fermats last theorem should not hold. If anyone intuitively believed it to be correct, why? Can someone explain so I understand ...
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29 views

Requirement That a Vector be Related to Itself Through Identity

If I have two vectors for which the relation can be written $$ \begin{bmatrix}\vec{I}_1\\\vec{I}_2\\\vec{I}_3\end{bmatrix} = \begin{bmatrix}A\end{bmatrix} ...
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62 views

Understanding why the public exponent $e$ is chosen the way it is in RSA

I am trying to get a better understanding of RSA. At the moment I am unable to understand the difference between the correctly chosen value of the public exponent $e$ and other possibilities ...
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45 views

Negation of uniform continuity

The definition of uniform continuity is: Given any $\varepsilon>0\ \exists\delta>0\ \forall x\in I \ \forall y\in I\ \left(\text{if }|x-y|<\delta\text{ then }\ ...
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89 views

Intiution behind the derivative of dirac delta function

Let me first begin what I mean by saying the intuition behind the " $\delta'(x)$ ". For example the smooth approximations of the delta function looks like the following: (Left:the smooth ...
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35 views

Motivation for the binary entropy function

What is the motivation for the definition of the binary entropy function $H(x) = -p\log_2(p) - (1-p)\log_2(1-p)$? I understand that we want the entropy to be zero at $p = 0$ and $p = 1$ (no ...
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62 views

Explicit formula for floor(x)?

In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. ( second Chebyshev ...
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53 views

Physically, what meaning have Taylor series which have their lower order terms equal to zero, but their higher order terms non zero?

Usually, when using a Taylor series to describe a function (which may itself be a model of some physical phenomenon), we often throw out the higher order terms, as they are quite small relative to the ...
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1answer
45 views

What does $H X H^T$ do?

I regularly encounter the use of: $$H X H^T$$ Where: $H$ is a $(n\times m)$ matrix, with $H^T$ it's transpose $X$ is a $(m \times m)$ matrix If we rephrased this as a function (unquely defined ...
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80 views

Product of two Gaussian PDFs is a Gaussain PDF, but Produt of two Gaussan Variables is not Gaussian

The Product of Two Gaussain Random Variables is not Gaussian distributed: Is the product of two Gaussian random variables also a Gaussian? Also Wolfram Mathworld So this is saying $X \sim N(\mu_1, ...
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On the matrix representation of a composition of Möbius transforms

Let the Möbius transform associated to the matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ be defined as $\mu_A:\mathbb C\to\mathbb C:z\mapsto\frac{az+b}{cz+d}$ provided $\det A\neq 0$. It is ...
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Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
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129 views

Intuitive Aproach to Dolbeault Cohomology

I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. All suggestions are welcome.
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427 views

Intuitive Approach to de Rham Cohomology

The intuition behind homology may be summarized in a sentence: to find objects without boundary which are not the boundary of an object. This has geometric meaning and explains the algebraic boundary ...
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203 views

Intuition behind functional dependence

What is the intuition behind functional independence ? (This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally ...
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489 views

Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors ...
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Is there an intuitive reason why hippopede, the intersection curve of a sphere and a cylinder, is traced by composing two rotational motions?

The hippopede is historically famous because Eudoxus used its properties in the first mathematical model of planetary motion. He nested concentric spheres rotating at different inclinations to each ...