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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Why are polynomials defined to be “formal”?

Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial. ...
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28 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
83 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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260 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
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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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Find the area of the curved shape

How to find area of this curved shape?
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1answer
12 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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46 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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Forming equation of a plane by solving linear equation set

Given three points on the plane: $ A(x_1, y_1, z_1) $, $ B(x_2, y_2, z_2) $ and $ C(x_3, y_3, z_3) $. I'm trying to obtain the equation of the plane in this format: $ ax + by + cz + d = 0 $ I ...
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Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
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66 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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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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49 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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185 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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Why is it so hard to find the roots of polynomial equations?

The question that follows was inspired by this question: When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic ...
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Nine plus ten really does equal 21… [on hold]

Well I was watching some vines and this kid responded 21 to "What is 9+10?"Well if you add it normally it's 19 but I tried a few other ways and it was 21 BUT I added a few modifications... Either the ...
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1answer
37 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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27 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
20 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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40 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
149 views

integrate $\frac{x}{\cos x}\sin(\tan x)$ [on hold]

I can't solve it. please help! I tried everything. Integration by parts doesn't work. but maybe I didn't do it right. I tried to substitute , but I'm stuck. $$\int \frac{x}{\cos x}\sin(\tan x)\,dx$$ ...
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1answer
41 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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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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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
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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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Questions on Proofs - Equivalent Conditions of Normal Subgroup - Fraleigh p. 141 Theorem 14.13

(1.) Why did Fraleigh shirk the proof for $(2) \implies (1)$? By dint of Arthur's comment, $(2) \iff \color{crimson}{gHg^{-1} \subseteq H} \quad \wedge \quad gHg^{-1} \supseteq H \implies ...
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Intuitive Aproach of 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
25 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 Mobius transforms

Let the Mobius 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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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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29 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 ...
4
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117 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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2answers
36 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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$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
0
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1answer
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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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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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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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1answer
85 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 ...
2
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1answer
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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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1answer
490 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
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Why “characteristic zero” and not “infinite characteristic”?

The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
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Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
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In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
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Elementary proof of the Prime Number Theorem - Need?

Although I am very much new to "Analytic Number Theory", there are some non mathematical questions which puzzle me. First of all, why was G.H.Hardy so much keen to have an elementary proof of the ...
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Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
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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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What are some examples of a mathematical result being counterintuitive?

As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition. My first and favorite experience of this is Gabriel's Horn that you see in ...
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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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112 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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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 ...