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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Intuition about heat equation with Neuman boundary data

On a bounded domain, consider the heat equation $u_t - \Delta u = 0$ with $\partial_\nu u = c$ (constant) and initial data $u_0$ which is non-negative. As usual $\nu$ is the outward normal vector. ...
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Intuition behind the derivative of are of a square? How to properly use the derivative ?

If I derive the formula $$S=16t^2$$, where S denotes the distance and t denotes time I get $$ds/dt= 32t$$. This in return give me a formula for the speed of the object at any time t. However if we ...
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What is the intutition behind the negative exponential ? in linear logic?

The positive exponential ! has a very satisfying interpretation in terms of the standard resource interpretation of linear logic. Given a resource $a$, we know that $!a$ means an infinite supply of $a$...
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Is my intuition of dense sets correct?

I am working with the usual definition of a dense set, which is Let $U$ be any non-empty open subset of $X$. A set $A$ is dense in $X$ iff $A \cap U \neq \emptyset$. My highly informal and ...
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67 views

Is this symbolic statement impossible?

Is this statement logically impossible if x is a single real number (i.e. not a set)? $$(x<5) \land(x>7)$$ it seems to me that x cannot both be greater than 7 and less than 5 if ...
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Systems with Principle Roots of Unity

Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying: $$\sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, ...
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Intuitive reason why the Euler characteristic is an alternating sum?

The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups. Since homeomorphic spaces have isomorphic homology groups, however, even the non-...
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47 views

Neumann vs Dirichlet eigenvalue problem - Intuition

What is the fundamental different between a Neumann eigenvalue problem and Dirichlet eigenvalue problem? I know that for DEP, we just fix the boundary (e.g. a drum), but what about the NEP. Now, ...
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Intuition behind the number of roots of a polynomial and dimension of the solutions of a linear scalar differential equation.

I'm sorry if my question is ill-posed but it's something I've been wondering about for quite a while. I understand that a polynomial of degree n must have less than n real roots (because each power ...
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30 views

Intuition behind the finite geometric series formula?

Can anyone give some intuition or insight on why $S_n = a(\frac{1-r^n}{1-r})$ works? (I've seen the proof but I like being able to visualize to think about formulas in different ways.)
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Deriving the Airy functions from first principles

I have just started reading about the Airy functions and am stuck on a particular step of their derivation. But first here is some background information to give this question some meaning, more ...
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Counter-examples for these T/F probability questions?

Claim 1: If $f$ is a pdf of a random variable $X$, then $0 \leq f(x) \leq 1 \quad \forall x\in\mathbb{R}$ Why is this False? Is it because $f(x)$ does not need to be defined on the whole real line? ...
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Given $C \subset A \subset X$, why is that $C$ is closed in $X$ if $A$ is closed, $C$ is open in $X$ if $A$ is open?

I want to understand a result discussed here : Subspace of a normal space Let $(X, \mathfrak{T})$ be a topological space. Given $C \subset A \subset X$, let $C$ be a closed set in $A$, then claim ...
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Graphical explanation of the difference between $C^1$ and $C^2$ function?

We are all aware of the intuitive (graphical) explanation of the concepts of continuous and differentiable function. Whenever these two concepts are formally defined, the following elementary ...
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How quickly can we find a value that has large multiplicative order modulo $n$?

If we're trying to find an element modulo $n$ that has multiplicative order at least $\sqrt{n}$, how quickly can we do this? We don't know if $n$ is prime or composite, only that $n$ definitely has a ...
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What are permutable equivalence relations intuitively?

What are permutable equivalence relations, and what are they used for? What is the idea behind them? Could someone give me an example and a counterexample for finite sets? I have encountered the ...
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Legitimacy of drawing a complex curve like a plane curve

In algebraic geometry, we often consider a complex algebraic curve, and in order to get some intuition, we often draw it on the plane as if it were a plane curve. In most cases it turns out that the ...
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72 views

Finding 8 co-primes $\le 2^n$

We can find 8 co-prime integers $\le 2^n$ for sufficiently large $n$. I'm looking for asymptotic bounds for the minimum distance away from $2^n$ we have to go before finding 8 co-primes. In other ...
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26 views

Good explanation on generators of intersection of two cyclic subgroups?

If $\langle a \rangle$ is a cyclic group and $\langle a^n \rangle$ and $\langle a^m \rangle$ are two subgroups then a generator for $\langle a^n \rangle \cap \langle a^m\rangle$ is $a^{\text{lcm}(n,m)}...
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Intuitive explanation of Tao's work on Navier-Stokes equations needed [closed]

In February 2014 mathematician Terence Tao posted his work on a partial solution to the Navier-Stokes existence and smoothness problem. He proved that a "finite time blowup" exist "for an averaged ...
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I need an intuitive explanation of eigenvalues and eigenvectors

Been browsing around here for quite a while, and finally took the plunge and signed up. I've started my mathematics major, and am taking a course in Linear Algebra. While I seem to be doing rather ...
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42 views

Dirichlet conditions - Explanation of the proof of theorem $4$

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...
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Bayes' rule application proving my intuition wrong

This is the example I am looking at. Question: Suppose a family has $2$ children and one is a boy, and that the probability of having a child of either sex is equal and independent across ...
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4answers
75 views

Intuitive explanation of $L^2$-norm

I have to play a lot with the $L^2$-norm defined as $\|w\|=\sqrt{\int_a^b <f,f>}$. However, I don't understand the interpretation of that norm. We know that the euclidean norm measure the length ...
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How do curves consist of points?

According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in ...
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Is counting roots with multiplicites at all a geometric concept?

It is well known that a polynomial of degree $n$ admits $n$ roots when the field is algebraically closed. However, this comes with a caveat, in particular that the roots be counted with multiplicity. ...
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63 views

Propositional Logic - Can you Derive $C \to A$ from $A$ alone, given the introduction rule?

Apparently, according to the Conditional Introduction rule, this is valid: Prove $C \to A$ Source: http://kpaprzycka.wdfiles.com/local--files/logic/W12R Page 5 So before this, the way I viewed ...
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26 views

How many complexes modulo a prime $p$ are of multiplicative order $p^2 - 1$?

If $i = \sqrt{(-1)} \bmod p$, $p$ prime, does not exist, then we can form numbers of the form $a+b i \bmod p$ with multiplicative order $p^2 - 1$. How often do these numbers occur modulo $p$? In ...
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57 views

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$?

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$? NOTE This $n$ is ...
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What is characteristic time?

What is characteristic time? Where is it useful? From this answer by Joriki: The characteristic time is usually defined to be the time in which a quantity decreases by $1/e$. Why is ...
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Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a "...
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How much can we compress a sequence of bits?

Suppose that we have a sequence of bits defined for each natural less than $n$. For example, if we have 3 bits (either 0 or 1), we can represent the sequence as a function of $x < 4$ in the ...
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Intuition of Immersed versus Embedded Submanifolds

The definitions I read in Lee's Smooth Manifolds is: Embedded Submanifold: $S\subset M$ is an embedded submanifold if $S \to M$ is an embedding. Immersed Submanifold: $S\subset M$ is an immersed ...
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Geometric Intuition about the relation between Clifford Algebra and Exterior Algebra

It is common to see a relation being established between the Clifford Algebra and the Exterior Algebra of a vector space. Recently reading some texts written by Physicists I've seem applications of ...
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Geometric meaning of the contact condition?

I am trying to understand contact structures. The definition of a contact manifold is this: Let $M$ be a $2n + 1$-manifold and let $\omega$ be a differential $1$-form such that $\omega \wedge (d\...
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I don't understand why $\oint_\gamma f\, dz=0$ holds for holomorphic functions.

I've recently learned a proof of Cauchy's Integral Theorem, i.e, If $U\subseteq \Bbb C$ is open and simply connected, $f:U\to\Bbb C$ is holomorphic and $\gamma$ is a closed curve, $\gamma\...
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How do you visualize $\mathcal{P}(1)$ in constructive mathematics?

If I understand correctly, constructive mathematics doesn't prove that the powerset $\mathcal{P}(X)$ of a set $X$ is a Boolean algebra; in general, all we can say is that its a Heyting algebra. This ...
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Intuitive explanation of $(1-x)^{-a-1}=\sum_{j=0}^{\infty}{{a+j} \choose j}x^j$

Could anyone please explain me the reasoning behind this formula? $(1-x)^{-a-1}=\sum_{j=0}^{\infty}{{a+j} \choose j}x^j$ Thanks so much!
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What is an intuitive way to understand Cayley's formula?

Is there any intuition behind Cayley's formula $n^{n-2}$ for the number of spanning trees of a graph?
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1answer
40 views

What is the intuition behind covering spaces?

I've come to study this definition and become interested on the intuition behind it mainly because of the study of spinors, motivated by Quantum Mechanics. The definition of covering space is as ...
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1answer
45 views

Limit of $n-1$ measure of the boundary of a sphere

The measure of a sphere of radius $R$ centered in $0_{\mathbb{R}^n}$ in $\mathbb{R}^n$ is \begin{array}{l l}\int_{B_0(R)}dx_1\ldots dx_n & =\int_0^R\rho^{n-1}d\rho \int_{-\frac{\pi}{2}}^{\frac{\pi}...
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66 views

Why are second order linear PDEs classified as either elliptic, hyperbolic or parabolic?

Is there a geometric interpretation of second order linear partial differential equations which explains why they are classified as either elliptic, hyperbolic or parabolic, or is this just a naming ...
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1answer
39 views

Every fiber bundle with Cantor set fiber is the suspension of a homeomorphism of the Cantor set.

I've heard that every fiber bundle (over $\mathbb S^1$?) with Cantor set fiber is the suspension of a homeomorphism of the Cantor set. Can someone explain the intuition behind the fact? Is there a ...
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28 views

Intuition of a k-connected graph?

The formal definition of a k-connected graph $G$ is: $\nexists x\subseteq V(G)$ with $|x| \le k - 1$ such that $G-x$ is disconnected. What is the intuition behind this? What does it mean to be ...
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Intuition behind negative radians in an interval [duplicate]

Say a function's domain is $[−\pi,\pi]$. How should I interpret this interval? It starts from where? To where? In what direction?
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How to understand the notion of a differential of a function

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal ...
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How does the Fourier transform of a “zero avoiding” function look?

Let $n$ be a very large positive integer. Let $f \in\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function, satisfying $0\leq f\leq1$, and supported on $[-n,-\frac{1}{n}]\cup[\frac{1}{n},n]$ such ...
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1answer
69 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point $x\...
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43 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
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