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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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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1answer
86 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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59 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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1answer
44 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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1answer
62 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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1answer
29 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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56 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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1answer
42 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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1answer
55 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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41 views

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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1answer
79 views

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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1answer
107 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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1answer
288 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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191 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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380 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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37 views

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 ...
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1answer
60 views

The set $\omega \times \omega$ is equinumerous with $\omega$.

Proposition: The set $\omega \times \omega$ is equinumerous with $\omega$, i.e. the set $\omega \times \omega$ is countable. "Intuitive Proof" $$\mathbb{N}^2=\{ (n,m): n,m \in \mathbb{N} \}$$ $$1 ...
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53 views

Is my intuition on projectivization correct?

Is my intuition on what a projectivization of an affine curve in $C^2$ is and why it is useful correct? From what I understand given an affine curve $C$ we are trying to find a projective curve ...
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55 views

Clifford Algebras for Projective and Conformal Geometry

According to Clifford Algebra: A Visual Introduction, A Clifford Algebra over $\mathbb{R}^3$ may describe the rigid motions in space (namely, conjugation acts as a reflection by a plane). A ...
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98 views

Meaning of a long exact sequence

Edit: The setting for the question is some abelian category. From this question I learned that one way to view a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is as ...
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4answers
158 views

Connections between the solution of simple ordinary equation, normal distribution and heat equation

The solution to the following simple first-order linear ordinary differential equation: $$x'=-tx, x(0)=\frac{1}{\sqrt{2\pi}}$$ is the Standard normal distribution! One solution to another famous ...
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1answer
47 views

Rotman, Algebraic Topology, Lemma $4.22$

Lemma 4.22. Let $X$ be a space and, for $i=0,1$, let $\lambda _i:X\rightarrow X\times I$ be defined by $x\mapsto (x,i)$. If $H_n (\lambda _0)=H_n(\lambda _1)$, then $H_n(f)=H_n(g)$ whenever ...
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52 views

Intuition behind definition of limit of sequences at infinity with example

The definition given is for every $c>0$, there exists an N such that $a_n$>c for all $n>N$ Please could someone explain this is really basic terms as im struggling to get my head around it. ...
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39 views

Homotopy Invariance: Cone Construction and Prisms Operators

I'm looking at different approaches to proving the homotopy invariance of homology. Rotman and Dieck both mention "the cone construction", but hatcher only introduces the prism operators and does not ...
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39 views

Substructure of $\omega$-catogorical theory $T$.

I need some help understanding part of my Model Theory notes: "Given that $T$ is $\omega$-categorical and $\mathfrak{A} \vDash T$, for $S \subseteq A$, let $\langle S\rangle$ denote the smallest ...
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118 views

How can I understand $\Bbb Z\times \Bbb Z/2\Bbb Z$

This may be stupid request, but I would like to have a intuition for the group $\Bbb Z\times \Bbb Z/2\Bbb Z$ in terms of 'real' objects. 'Real' could mean geometric but not necessarily. I perhaps what ...
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3answers
128 views

Why is $(AB)^{-1}=A^{-1}B^{-1}?$ [closed]

If we have two matrices $A$ and $B$ then the following property is true. $$(AB)^{-1}=A^{-1}B^{-1}.$$ I can't understand how the property is true. Can anyone give me a intuitive proof for the ...
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1answer
101 views

Modern treatment of Topology that focuses on intuition and is full of explanations and visual insights.

I'm interested in a modern treatment of Topology (point-set, and general topology at the undergraduate level) that focuses on intuition and is full of explanations and visual insights. This will be ...
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216 views

How are long proofs “planned”?

I just graduated with my bachelors in mathematics last year, so I have little experience in writing huge, very involved proofs. The longest proof I've ever written was about 10 pages, but it wasn't ...
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68 views

How to prove this using natural deduction

⊢ P ∨ ¬P I found this question on the net. I know the solution but i find it complicated. How should i approach to this sort of question? Or can you provide me another solution ?
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24 views

Parallel Programming the 1-D dam breakage problem

I am to write a parallel program to simulate the 1D dam break problem by using the Galerkin Equations with WENO limiter. The equations are on domain [0,2000]. At the beginning a dam divides the ...
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1answer
87 views

An elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system

Arnold in his essay On teaching mathematics made the following statement: The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only ...
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Why do repeated linear factors have to be dealt with in this way?

When dealing with partial fractions, and your denominator has a repeated linear factor, the way to solve is this: $\frac{2x+3}{(x-2)^2}=\frac{A}{(x-2)^2}+\frac{B}{(x-2)}$ $2x+3=A+B(x-2)$ and so on. ...
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7answers
259 views

How the cardinality of $\mathbb{R^+}$ and $\mathbb{R}$ same?

Let me first confirm you that this question is not a duplicate of either this, this or this or any other similar looking problem. Here in the current problem I'm asking to disprove me(most probably ...
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69 views

Let $z_1$, $z_2$ and $z_3$ be complex vertices of an equilateral triangle. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$.

Prompt: Let $z_1$, $z_2$ and $z_3$ represent vertices of an equilateral triangle in the complex plane. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$. Question: I hope the following ...
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1answer
76 views

Why is this intuitive method valid?

Problem. There are $2$ white and $3$ black balls in the urn. A person randomly picked $2$ balls and put $1$ white ball. What is the probability of the event that the next randomly-picked ball would be ...
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142 views

How to upgrade Category Theory skills for Algebraic Geometry?

I am doing a second advanced graduate course in Algebraic Geometry, with Hartshorne as a textbook. The skillset I am least satisfied with is the application of the Category Theory to Algebraic ...
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How do I close the gap between intuitively knowing something is true vs being able to prove it?

For example, one of my review problems is: Let $S_k$ be the kernel of $T^k$. Show there is a $K$ such that $S_K = S_{K+1} = \cdots$ Somewhere in the back of my brain there's an intuition that told ...
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2answers
64 views

How to visualize(inside ones brain) the Four-dimensional_space

Can the fourth dimension https://en.wikipedia.org/wiki/Four-dimensional_space be visualized intuitively by the humans. Does the professional mathematicians can do this ? If so what are the things to ...
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498 views

Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
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Intuition for using vectors in sale related problems

I am reading Linear Algebra from David Lay's book. He gives one example to showcase use of linear combination of vectors : I understand the solution, but I am completely clueless about how to ...
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247 views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
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260 views

Is there a notion in mathematics saying that, in a sense, all finite dimensions are actually infinite dimensional?

So then every ordered pair or triplet and so on would be actually represented by an infinite sequence of numbers, and what we think of as 3 dimensions would mean that the point has an infinite number ...
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22 views

Steady state state distributions.

I am looking for a less "proofy" explanation of how a finite, irreducible, aperiodic Markov chain has a unique steady state $\pi$. No need define terms or include proofs of Bezout's lemma or number ...
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28 views

Gradient points in the direction of greatest change

Can anyone provide me with an alternative, possibly more intuitive proof of this proposition? I'm confused with where $cos\theta$ has come from?
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1answer
56 views

Non- intuitive connected space.

There exist knowing examples of connected spaces such that its picture is a counter intuitive for us?. I mean a topology on a set who makes see the space as connected (no connected) but it is no ...
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103 views

Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
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285 views

Intuitive understanding of the uniqueness of the Fundamental Theorem of Arithmetic.

Basically I am trying to understand why Fundamental Theorem of Arithmetic (FTA) exists, i.e why a natural number cannot be factored primely in two or more different ways. There are two proofs given ...