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 is the Jordan Curve Theorem not “obvious”?

I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ...
2
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1answer
237 views

An intuitive proof for one of the fundamental property of a parallelogram

"The sum of the squares of the diagonals is equal to the sum of the squares of the four sides of a parallelogram." I find this property very useful while solving different problems on ...
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What is a physical “dimension” - in the sense of “dimensional” analysis?

Mathematically speaking, what does it mean to say that a physical quantity is some numerical value with a “dimension” associated with it? When we say that the velocity of light is some constant, c ...
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Definition of e

Possible Duplicate: Why is $1^{\infty}$ considered to be an indeterminate form Is $dy/dx$ not a ratio? I'm very eager to know and understand the definition of $e$. Textbooks define $e$ as ...
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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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What is the intuition behind the concept of Tate twists?

For any field $K$ we can define the cyclotomic character $\chi: \operatorname{Gal}(K)\rightarrow GL_1(\hat{\mathbb{Z}})$. For any representation $V$ (I will view this as a module over ...
33
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768 views

How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?

My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity. Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow ...
10
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611 views

What could be an intuitive explanation for $ \sum\limits_{k=1}^{\infty}\frac{1}{k2^k} = \log 2 $?

What could be an intuitive explanation for $\displaystyle \sum_{k=1}^{\infty}{\frac1{k\,2^k}} = \log 2$ ? $\displaystyle \sum_{k=1}^{\infty}{\frac{1}{2^k}} = 1$ has a simple intuitive explanation ...
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Complex graphs in 3D

Does anyone have red-green 3D software for plotting 4D graphs in 3D with 3D glasses? I've seen a 4D hypercube done this way and it's very revealing...
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492 views

I need explanation for this solution for the proof. (Perfect square ends with 0,1,4,5,6,9)

Give a proof to the sentence: "The final decimal digit of a perfect square is 0, 1, 4, 5, 6 or 9." Solution: A integer $n$ can be expressed as $10a+b$, where $a$ and $b$ are positive integers and $b$ ...
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Should one think of a network as a connected graph ? (Or: What is the right way to think of a network?)

In the definition of a network, are we only considering connected graphs ? Because I keep encountering definitions that don't assume explicitly that we deal with connected graphs, but which would be ...
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What are the Axiom of Choice and Axiom of Determinacy?

Would someone please explain: What does the Axiom of Choice mean, intuitively? What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice? as simple ...
3
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1answer
136 views

Looking for an article on general principles of discrete mathematics

In his article 2 cultures Timothy Gowers states that the structure in combinatorics is there in the form of somewhat vague general statements that allow proofs to be condensed in the mind, and ...
8
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3answers
644 views

Intuition regarding Chevalley-Warning Theorem

Three versions of the theorem are stated on pages 1-2 in these notes by Pete L. Clark: http://math.uga.edu/~pete/4400ChevalleyWarning.pdf Could anyone offer some intuitive way to think about this ...
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What is the intuition behind the “par” operator in linear logic?

I'm $\DeclareMathOperator{\par}{\unicode{8523}}$ trying to wrap my mind around the $\par$ ("par") operator of linear logic. The other connectives have simple resource interpretations ($A\otimes B$ ...
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5answers
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Intuitive interpretation of the Laplacian

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian (a.k.a. divergence of gradient)? ...
3
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3answers
150 views

“Contradiction-free” in logic vs. “Contradiction-free” in plain mathematics

In our course we have defined a theory $T$ to be contradiction-free, if there are no formulas $\alpha_1,\ldots \alpha_n\in T$ such that $\neg ( \alpha_1 \& \ldots \ \& \alpha_n )$ is provable ...
6
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451 views

Intuition behind the scaling property of Fourier Transforms

I had a course in PDE last year where we used fourier transforms extensively; I understand the rules of manipulation and can prove the scaling theorem directly from the definition using a ...
5
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1answer
641 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
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12answers
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I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]

I didn't know how to phrase the question properly so I am going to explain how this came about. I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
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4answers
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Why is it that Complex Numbers are algebraically closed?

I find it curious that Complex Numbers give enough flexibility to be algebraically closed, where the reals, rational numbers do not. For the reals it is easy to see that they cannot be used to solve ...
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Four men, hats and probability

I encountered the four men in hats puzzle for the first time today. My question is about a realisation I (think I) had while arriving at the solution, but I have no idea whether I've made a mistake ...
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2answers
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Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a - b|$

I came across the following problem in my self-study of real analysis: For any real numbers $a$ and $b$, show that $$\max \{a,b \} = \frac{1}{2}(a+b+|a-b|)$$ and $$\min\{a,b \} = ...
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Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
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Are cyclic groups always abelian?

If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is? Thanks in advance!
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One divided by Infinity?

Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me. I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - ...
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1answer
242 views

Intuition behind elliptic curves and $K$-rational points

I find myself becoming confused whenever I try to think about this. In the following, $K$ is a field. An elliptic curve $\mathcal{C}$ is defined to be a nonsingular projective cubic curve over $K$, ...
5
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1answer
447 views

A way to see that $\int_{0}^{\infty}\exp(-x)dx=1$?

One can easily find the integral $\int_{0}^{\infty}\exp(-x)dx$. It is equal to 1. But is there a way to understand this geometrically without integration? If i rotate the picture i see that ...
2
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1answer
205 views

What is the mathematical process behind fully homomorphic encryption?

I've often wondered about how to compute encrypted data, and appears that a "hack" to do so has been found: http://www.technologyreview.com/computing/37197/ Is anyone able to offer an better, yet ...
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practical uses of matrix multiplication

The use of matrix multiplication is usually given with graphics initially (scalings, translations, rotations, etc). Then there are more in-depth examples such as counting the number of walks between ...
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1answer
432 views

Intuition behind Sobolev norm

This morning I was thinking at the following (simple) fact. Let us consider $[0, 1] \to \mathbb{R}$ functions and define a linear functional $$F(u)=u(1)-u(0).$$ $F$ is not continuous on $L^2(0, 1)$ ...
9
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Signs in the tensor product and internal hom of chain complexes

Let $R$ be a commutative ring and $\text{Ch}(R)$ the category of chain complexes of $R$-modules. $\text{Ch}(R)$ is first of all an abelian category, but it can also be equipped with the structure of a ...
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How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
2
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3answers
206 views

A question on logic - where intuition can fail

Suppose I have two predicates $P(x)$ and $Q(x)$, such that $\overline{P(x)\wedge Q(x)}$ holds for all $x$. Now, if $\displaystyle \bigwedge_{x\in A}P(x)$ for a set $A$, it must be certainly true, ...
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The simplest $\Delta$-complex structure on $S^2$

I think my reasoning is correct, but I want to run through it here because having the right intuition will make similar problems easier in future. A 2-simplex is homeomorphic to a closed disc, and a ...
8
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1answer
438 views

The Mayer-Vietoris sequence

If $X$ is a space with a pair of subspaces $A, B \subset X$ such that $X$ is the union of the interiors of $A$ and $B$, then there is a long exact sequence of homology groups $\displaystyle \ldots\to ...
8
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Demonstrating the value of abstracting away from elements/subsets to maps

Given a set $S$, here are 5 ways of thinking about elements of $S$, in increasing abstraction: an actual element, e.g. $s\in S$ an inclusion map, e.g. $i_s:\{s\}\hookrightarrow S$ an ...
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In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
18
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613 views

Categorical description of algebraic structures

There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
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2answers
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Intuition behind euler's formula [duplicate]

Possible Duplicate: How to prove Euler's formula: $\\exp(i t)=\\cos(t)+i\\sin(t)$ ? Hi, I've been curious for quite a long time whether it is actually possible to have an intuitive ...
31
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Can someone explain the Yoneda Lemma to an applied mathematician?

I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood ...
8
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1answer
685 views

Intuition Building: Visualizing Complex Roots

I can graph the parabola $y = x^2 + 1$. I see that it does not intersect the $x$-axis, and therefore it must have complex roots, namely $+i$ and $-i$. I can plot these roots on an Argand diagram at ...
6
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2answers
209 views

Improper integrals question: why does this work?

So, I'm solving $$ \int_0^3{ \frac{dx}{ \sqrt{ 9 - x^2 } } } $$ The catch here is $f(x)$ is defined on ( -3, 3 ) only. And I know the way to solve this is to integrate (yielding $ \arcsin{ ...
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Intuitive meaning of Pearson Product-moment correlation coefficient Formula

I can't understand the intuition behind Pearson Product-moment correlation coefficient Formula for bivariate data. The formula is : $\rho$ = cov(X,Y)/($S_x$ * $S_y$) where cov is covariance. $S_x$ and ...
4
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1answer
178 views

Visualizing Operators on $\mathbb{C}^n$

I am trying to get some better intuition about operators on complex inner product spaces. When we identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$, is there a nice geometric interpretation for the ...
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3answers
275 views

Some basic problems of group theory

1.Prove the every group G of order 4 is isomorphic to either Z4 or 4-group V,that is {1 (1,2)(3,4) (1,3)(2,4) (1,4)(2,3)} 2.If G is a group with $|G|\leq 5$ then G is abelian. I have learned ...
4
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1answer
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Explain what is a linear transformation

My textbook says "unique linear transformations can be defined by a few values, if the given domain vectors form a basis." However, that is all it says. So can someone explain what a unique linear ...
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15answers
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List of Local to Global principles

What are some of the local to global principles in different areas of mathematics?
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1answer
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Why is the ratio test for $L=1$ inconclusive?

One of the often used tests for convergence ($L\lt 1$) and divergence ($L\gt 1$) of an infinite series is the ratio test. The idea behind it, why it works is the geometric series which dominates (or ...
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1answer
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What's the intuition behind and some illustrative applications of probability kernels?

Given measure spaces $(X, \mathcal{X})$ and $(Y, \mathcal{Y})$ we define measure kernel $\pi : \mathcal{X} \times Y \to [0,\infty]$ such that $\pi(\cdot|y)$ is a measure on $\mathcal{X}$ for every $y ...