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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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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34 views

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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12 views

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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8answers
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Why is the area under a curve the integral?

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ...
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1answer
25 views

Elements of bounded distributive lattice belonging to same prime ideals are equal?

I have read in a paper that by an easy application of Zorn's lemma one may show that two elements of a bounded distributive lattice are equal iff they are contained in exactly the same prime ideals of ...
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40 views

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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7answers
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Intuitive Explanation of Bessel's Correction

When calculating a sample variance a factor of (N-1) appears instead of N (see http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance ). Does anybody have an intuitive ...
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2answers
41 views

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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How to come up with the gamma function?

It always puzzles me, how the Gamma function's inventor came up with its definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of ...
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2answers
65 views

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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0answers
42 views

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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3answers
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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16 views

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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3answers
95 views

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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0answers
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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1answer
28 views

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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1answer
31 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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10answers
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What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
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2answers
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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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1answer
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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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12answers
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Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and ...
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2answers
467 views

Characteristic function of union of two sets formula and intuition

From http://topologicalmusings.wordpress.com/2008/03/20/inclusion-exclusion-principle-counting-all-the-objects-outside-the-oval-regions-2/ Is there an easier proof or way to calculate $1[A \cup B]$?...
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0answers
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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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Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
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5answers
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What is the difference between necessary condition & sufficient condition?

My book says : For having extreme point $a$ of function $f$, the necessary condition is that $f'(a) = 0$. However, it isn't a sufficient condition. Now, what is the difference between necessary ...
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Explain for students: Why does 0 mod n equal 0 (zero)?

I told my students that the mod operator basically gives the remainder of division, so upon seeing: 0 mod 10 Some students (apparently) reasoned that, "10 goes ...
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1answer
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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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Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=...
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2answers
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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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0answers
122 views

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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Intuition on the sum of first (n-1) numbers is equal to the number of ways of picking 2 items out of n.

While going through an equation today i realized that sum of first (n-1) numbers is [n*(n-1)/2] which is equal to combinations of two items out of n i.e [n!/((n-2)! * 2!)]. I need some intuition on ...
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1answer
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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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128 views

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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1answer
27 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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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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1answer
297 views

Intuition of projective plane and space

What is the geometric intuition of projective plane and space? I can understand affine plane and 3 dimension affine space, for higher dimension, at least I can imagine it similarly as the 2,3 ...
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Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if $f$ is a group homomorphism, is there a way to see that $\ker(f)=1\...
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Intuitive meaning of immersion and submersion

What is immersion and submersion at the intuitive level. What can be visually done in each case?
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120 views

Intuitive idea of immersions?

So I understand the definition of immersions and submersions, as well as the motivation for defining such ideas. Not only are they important in understanding properties of mappings of tangent spaces, ...
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2answers
151 views

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...
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7answers
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Dominoes and induction, or how does induction work?

I've never really understood why math induction is supposed to work. You have these 3 steps: Prove true for base case (n=0 or 1 or whatever) Assume true for n=k. Call this the induction hypothesis....
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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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1answer
186 views

Intuitive motivation to try to factor an ideal

In $\mathbb{Z}[\sqrt{- 5}]$, $2$ is irreducible, but the ideal $(2)$ factors into non-units: $$(2) = (2, 1 + \sqrt{- 5})(2, 1 - \sqrt{- 5}).$$ In general, what gives one the intuitive motivation (...
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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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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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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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Unique Representation and The Fundamental Theorem of Arithmetic [closed]

While reading this thread Why 1 is not considered to be a prime number?, I recalled that The Fundamental Theorem of Arithmetic (FTA) which says that every positive integer greater than $1$ can get ...
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4answers
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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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6answers
13k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
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1answer
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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. ...