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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Visualizing the quotient of a torus and a circle

We were asked to compute the homology for the double torus, $X$, and a circle around one of the loops, $B$, of the torus (not a circle between the two halves of the torus) and were told that this ...
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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 ...
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Intuition probability of two pairs in poker dice

I need some intuition for an element of the following question: The answer starts with this: I would like to know how they get to 6 choose 2. If I write it out (1122, 1133, 1144, 1155, 1166, ...
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Areas under the graphs of $\frac{1}{x}$ and $\frac{1}{x^2}$ from $1$ to $\infty$

A simple evaluation of the definite integral tells us that the area under the graph of $[\frac{1}{x}]^2$ from $1$ to $\infty$ is finite whereas that of $\frac{1}{x}$ for the same limits is infinite. ...
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Trying to understand why circle area is not $2 \pi r^2$

I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) ...
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What comes after diagram chasing?

An early edition of Lang's algebra textbook gives the famous exercise to Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book. Here ...
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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 ...
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Absolute value of a random variable

I have never encountered this concept before. Is this equation valid for $y>0$? $$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$ What about this? $$\mathbb{P}(|X|>y) = ...
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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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What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

Given a set of points $x_1,x_2,\dots,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m\times m$ Euclidean Distance Matrix $D$ where $D_{ij}={\|x_i-x_j\|}^2$. We know a little bit about ...
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What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?
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Geometric Interpretation: Parallel forms are harmonic

Let $(M,g)$ be a Riemannian manifold. The canonical volume form $\mu=\sqrt{\det g_{ij}}\mathrm{d}x^1\wedge\dots\wedge\mathrm{d}x^m$ is parallel w.r.t. the induced Levi-Civita conection $\nabla$ ...
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38 views

Matrix --> Scalar Valued Function: Differentiation

In class, we called a real-valued function from the space of matrices to the reals $f: \mathbb{R}^{m \times n} \rightarrow \mathbb{R}$ differentiable at $\mathbf{X}$ if: $$\lim_{\mathbf{H} \to ...
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Is there some geometric intuition for the quotient $G/Z(G)$, where $G=GL_n(\mathbb{R})$?

Let $G=GL_n(\mathbb{R})$ be the $n$th general linear group. Its center $Z(G)$ is given by all scalar matrices $aI$ with nonzero determinant. How can I get an intuitive picture of $G/Z(G)$? I know that ...
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Elevator pitch for a (sub)field of maths?

When I first saw the title of this question, I forgot for a moment I was on meta, and thought it was asking about quick, catchy, attractive, informative one-or-two-liner summaries of various fields of ...
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Taking a derivative of a function with respect to another function

I read a set of notes recently (unfortunately I can't find the link) in which the author made a statement of the form "differentiation of a function with respect to a function doesn't make sense". By ...
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Is differentiation with respect to a vector always defined componentwise?

When one takes the derivative of a function $f$ along the direction of some vector $\mathbf{v}$, i.e. the directional derivative of $f$ along $\mathbf{v}$ this operation is defined componentwise, i.e. ...
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Intuition of weak solutions of elliptic equations in divergence form

Let $\Omega \subset \mathbb{R}^{n}$ a domain, and consider the following equation (1) $-D_{j}(a_{ij}D_{i}u) = 0$ (Einstein notation) The function $u \in H^{1}(\Omega)$ is a weak solution of (1) if ...
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Intuition for universal quotient maps [migrated]

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman, the two characterizations of ...
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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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Intuitive proof for a Combinatorial Problem

Given a set $S$ such that $|S|=N$ and $S$ contains exactly $K$ $0$s $(K >0)$ and $N-K$ $1$s, then exactly half of the subsets of $S$ contain an $odd$ number of 1s, $indepedent$ of the value of ...
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Tricks to remember Fatou's lemma

For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality $\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow ...
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Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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The frog puzzle

So here's the puzzle. You're poisoned in the jungle and the only way to save yourself is to lick a special kind of frog. To make matters worse, only the female of that species will do. Licking the ...
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Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
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Intuitively, why does $\dfrac{a}{c} = \dfrac{1}{\dfrac{c}{a}}$?

To discover the intuition, I refer to concrete objects: define $a$ as apples and $c$ as children. Question 1. How and why is $\dfrac{a}{c} \qquad (3) \quad = \quad\dfrac{\color{red}{1}}{\dfrac{c}{a}} ...
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Barendregt's Substitution Lemma (lambda calculus)

I am struggling to put words on an idea used in Barendregt's Substitution Lemma's proof. (available here) The lemma states that: If x≠y and x not free in L and M, L are $\lambda$-terms: then ...
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Intuition for Adeles and Ideles

I'm currently studying some class field theory and read about the notion of adeles and ideles. However, the object seems a bit arbitrary to me; is there a natural way to think about the adele-ring? ...
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How to figure out the “idea behind” proofs in analysis?

I'm taking a course in Real Analysis, and for the most part I can follow the rote mechanics of a proof (e.g. manipulation to produce a chain of inequalities as desired, etc.), but I have difficulty ...
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Volume of a cone [duplicate]

I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: ...
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Geometric intuition of tensor product

Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements. Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me ...
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What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$?

I'm trying to understand in an intuitive manner the relationship between the circular and hyperbolic functions in the complex plane, i.e.: $$\cos(z)= \cosh(iz)$$ $$\sin(z)=-i\sinh(iz)$$ where $z$ is ...
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Combinatorial Geometry explanation

I do not understand what is going on in $(4)$: for every flat $E \in \mathcal F$, $E \ne X$, the flats that cover $E$ in $\mathcal F$ partition the remaining parts. What is meant by "the flats ...
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Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
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Unique Representation and The Fundamental Theorem of Arithmetic

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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Is this inverted ANOVA possible?

If my understanding is correct, in an ANOVA usually you start with a null hypothesis that all groups have the same mean. You then calculate the within group and between group variance, and do an ...
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What is the appropriate ANOVA test for this situation?

So in this experiment I have have 7 devices. The response of each device (call it Y) are each measured ~20 at a 4 different levels of an independent variable (we'll call X). The Y response is known ...
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What does it mean to be going 40 mph (or 64 kph, etc.) at a given moment?

I was coming back from my Driver's Education class, and something mathsy really stuck out to me. One of the essential properties of a car is its current speed. Or speed at a current time. For ...
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What are the intuitions about matrix algebra operations?

In my current data analysis problem I am using models with complicated penalty structure that is a result of operations on some matrix $Q$. I do know definitions of basic matrix operations: $Q^T$ ...
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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 ...
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What is the intuitive meaning of a determinant? [duplicate]

I know how to calculate a determinant, but I wanted to know what the meaning of a determinant is? So how could I explain to a child, what a determinant actually is. Could I think of it as a measure ...
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Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding ...
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Intuition about the second isomorphism theorem

In group theory we have the second isomorphism theorem which can be stated as follows: Let $G$ be a group and let $S$ be a subgroup of $G$ and $N$ a normal subgroup of $G$, then: The ...
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Practical intuition for standard deviation

Simply: What is the intuition you get about the data if you are given standard deviation? More detailed: It is easy to imagine some information about the data if you are being told for example mean ...
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Lagrange's theorem intuition

I cannot grasp the intuition behind |G|/|H|=[G:H]. Starting from the equivalence relation x~y if and only if x^(-1)*y is in H, I can see a sort of division, but in my mind, the equivalence relation ...
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Understanding limits and how to interpret the meaning of “arbitrarily close”

I have read several introductory notes on limits of functions, and in all of them they introduce the notion of a limit of a function $f(x)$ by discussing what happens to the value of $f$ as $x$ ...
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Dot Product Intuition

I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). For ...
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Motivation behind Arithmetic Mean

I know that the arithmetic mean $(x_1+x_2+...+x_n)/n$ is the value that minimizes $f(x)=\sum_{k=1}^n (x_k-x)^2$; however, I'm looking for an intuitive relationship between the mean and ...
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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 ...