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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Geometric interpretation of analyticity?

Suppose the real valued functions $u(x,y)$ and $v(x,y)$ are continuous and have continuous first order partial derivatives in a domain $D$. If $u$ and $v$ satisfy the Cauchy Riemann equations at ...
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37 views
+50

Geometric Interpretation of Antiderivative?

Could someone please give me a geometric interpretation of the above theorem?
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0answers
50 views
+100

Quick Question on a Proof of Artin-Wedderburn Theorem

Question [Edited]: [See below.] Are the isomorphisms in $(1)$ and $(2)$ (additive) group homomorphisms? If I'm right, $\text{End}_R(M)$ is a ring, but ...
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12answers
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Intuitive explanation of Cauchy's Integral Formula in Complex Analysis

There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive ...
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1answer
44 views

Proof and interpretation of $\mathbb{E}[\mathbb{E}[X \mid Y, Z] \mid Z] = \mathbb{E}[X \mid Z]$

First, I understand that $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$, but how to prove that $$\mathbb{E}[\mathbb{E}[X \mid Y, Z] \mid Z] = \mathbb{E}[X \mid Z]?$$ Second, for ...
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36 views

Geometric interpretation of Cauchy-Goursat Theorem?

This theorem seems almost magical. The algebraic derivation doesn't really provide any insight into why it works. So could someone give me a geometric interpretation of it? This: Geometrical ...
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2answers
131 views

Significance of homology groups of a topological space

I am studying homology groups of topological spaces. In books I have found that the $n$th homology group counts the number of "$n$-dimensional holes" which exist in that space. If I consider homology ...
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2answers
69 views

Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
7
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1answer
186 views

Example of a flat manifold with non-zero (global) holonomy group.

I'm having some trouble coming to terms with there being non-zero global holonomy but zero local holonomy. Is there an easy to visualize example of a manifold whose curvature is zero but has non-zero ...
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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 ...
3
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3answers
99 views

Self-studying Russell's Paradox

I'm self-studying and having trouble wrapping my head around Russell's paradox, even after looking here. I'd really appreciate a more intuitive explanation of the concept before I move on to ...
0
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4answers
138 views

Two plus two equals four when earth has one moon?

As is well known, we have the least intuitive of basic operations, the 'implication' or '=>'. Consider 'A => B'. Most beginners get stumped on the vacuous truth, that implication could be true even ...
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2answers
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Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
5
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3answers
55 views

Intuitive understanding of path integral formula

I have learned a general formula for a path/line integral $$ \int_a^b f\left(\mathbf{r}(t)\right) \|\mathbf{r}'(t)\|\ dt \tag{1} $$ and I'm trying to better understand it. Specifically, I'm ...
32
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6answers
5k views

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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1answer
73 views

Why is conic isomorphic to $\mathbb{P(C^2)}$?

Given a quadratic form $C(x)=x_1 ^2+x_2 ^2 + x_3^2$ in $\mathbb{C}[x_1,x_2,x_3],$ we have a conic $$C=\{C(x)=0\} = \{[x_1:x_2:x_3]: x_1 ^2+x_2 ^2 + x_3^2 = 0\}$$ in $\mathbb{P(C^3)}$, given in ...
2
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1answer
30 views

Why is $x$ restricted this way? (limits of functions)

Here is a corollary from Ross' Elementary Analysis: Why is $x$ restricted this way?
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4answers
356 views

What does continuity *in general* mean?

I am looking from : http://en.wikipedia.org/wiki/Lipschitz_continuity Continuously differentiable $\subseteq$ Lipschitz continuous $\subseteq$ α-Hölder continuous $\subseteq$ uniformly continuous ...
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0answers
17 views

Why is an open interval needed in this definition? (definition of a limit of a function)

Here's a part of the definition Ross' Elementary Analysis states for limits of a function: In both parts of the definition, why are open intervals needed? Would it fail if it were a closed interval ...
6
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2answers
293 views

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 ...
5
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2answers
57 views

Intuition behind sum of multiplication arithmetic sequence

Maybe this is a stupid question but please guide and enlighten me patiently. I have just known something fact that quite shocking me. Let start from this simple fact $$\sum_{k=1}^n ...
0
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1answer
39 views

Gambling interpretation of conditional probability

In Billingsley, when defining conditional probability the following property has been given a gambling interpretation : $$ \int_G P[A||\mathscr{G}]dP = P(A \cap G), G \in \mathscr{G} $$ where at ...
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1answer
49 views

How to visualize $ \mathcal P \ ( \ \mathcal P \ ( \ \mathbb R ^2 \ ) ) $?

So,$ \ \mathcal P \ ( \ \mathbb R ^2 \ ) $ , the power set of the set of all ordered pairs of real numbers, contains every imaginable (2D) function, black and white image and text as per its ...
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2answers
25 views

Smallest open, dense, G-invariant subset of a metric space

Let $X$ be a metric space and $G$ be a topological group acting continuously on $X$. Let $ \mathcal S $ be the set of open, dense and $G$-invariant subsets of $X$. I need to take inverse limit (of ...
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4answers
455 views

KAM theory and the Ergodic hypothesis

I have seen several authors mentioning that KAM theory contradicts the Ergodic hypothesis. Unfortunately, the authors do not elaborate on this. I have some background in KAM theory but very little in ...
3
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1answer
61 views

Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
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4answers
558 views

Intuition for $\omega^\omega$

I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
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0answers
9 views

Developing intuition for the modulus of continuity $\omega$?

I am currently in a course which is going through rigorous definitions of continuity, convergence, integrals, etc. I am trying to develop an intuition in understanding the modulus of continuity, ...
6
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2answers
73 views

If f(x)dx is a rectangle with height f(x) and width dx, what is f(z)dz in complex analysis

I am trying to intuitively understand the multiplication $f(z)dz$ in complex analysis. For instance, $f(x)dx$, we are all aware, is a rectangle with height $f(x)$ and width $dx$ so its multiplication ...
2
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1answer
271 views

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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1answer
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Find $T_\mathrm{id}\left(\mathrm{Diff}(S^1)\right)$

We established on last tutorial that $T_\mathrm{id}(\mathrm{Diff}(S^1))$ are vector fields on $S^1$. I'd be grateful for any explanation (formal or intuitive) standing behind this answer.
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1answer
588 views

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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F(A ∩ B) ⊆ F(A) ∩ F(B) laymen translation?

I am suppose to prove the above statement but i have got diffculty understanding it in the first place. Could anyone help me translate it into laymen language?
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Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
11
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3answers
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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 ...
2
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3answers
33 views

Extensions Fields and Zeros of Polynomials

I've just digested the proof for Kronecker's theorem that states every polynomial $f(x)$ over a field $F$ has a zero in an extension field of $F$, mainly, the factor ring of polynomials ...
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2answers
29 views

Intuitive understanding of relationship between unit vectors and position vector

On Wolfram Mathworld they give a unit vector in the $\textbf{x}_n $ direction as: $$ \hat{\textbf{x}_n} \equiv \dfrac{\frac{\partial \textbf{r}}{\partial x_n}}{\lvert \frac{\partial ...
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1answer
67 views

Building intuition in group theory

I'm finding it hard to translate abstract results of group theory into something that intuitively makes sense. Putting this into a concrete example: if $f:G\to H$, $Im(G)$, is a subgroup of $H$? Is ...
5
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4answers
163 views

rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
2
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1answer
397 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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15answers
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Intuitive Understanding of the constant “e”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say ...
7
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3answers
229 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
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1answer
53 views

Intuition behind a certain limit.

We want to find $\displaystyle\lim_{\theta\to\frac{\pi}{2}} b_1-a_1$, we are given $c=1$ and that $\cdot=90^{\circ}$ This is my solution; $$\begin{equation}\sin \theta=\frac{b_1}{a_1} \iff b_1=a_1 ...
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28 views

Pairwise Independent Events that are not Independent

Let a ball be drawn from an urn containing 4 balls, numbered 1,2,3,4 . let E = {1,2}, F = {1.3}, G = {1,4} . If all outcomes are assumed equally likely, then P(EF) = P(E)*P(F) = 1/4 P(EG) = ...
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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 ($A\otimes B$ ...
3
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1answer
110 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
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2answers
46 views

Is it possible to gain intuition into these trig compound angle formulas - and in general, final year high school math?

Does anyone have any insight into the trig sum and difference formulas? The formulas in themselves are very elegant, but I don't really like the proofs that have been given, even the geometric proofs. ...
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4answers
212 views

How can one visualize a homomorphic mapping.

It has been a year or so studying Group theory and Ring theory. Funnily enough, this is the part where i am able to solve most of the questions of the book quite easily, yet not fully understanding ...
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Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
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6answers
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The Intuition behind l'Hopitals Rule

I understand perfectly well how to apply l'Hopital's rule, and how to prove it, but I've never grokked the theorem. Why is it that we should expect that $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x \to ...