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.

learn more… | top users | synonyms (1)

2
votes
7answers
203 views

How the cardinality of $\mathbb{R^+}$ and $\mathbb{R}$ same?

Let me first confirm you that this question is not a duplicate of either this, this or this or any other similar looking problem. Here in the current problem I'm asking to disprove me(most probably ...
8
votes
4answers
2k views

Intuitive explanation of variance and moment in Probability

While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment. What is a good way to think of those two terms?
9
votes
2answers
1k views

Example: Function sequence uniformly converges, its derivatives don't.

Could anyone give an example of a sequence of differentiable (real) functions that uniformly converge to a differentiable function, but the derivatives of which don't converge to the derivative of the ...
7
votes
2answers
307 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 ...
1
vote
0answers
28 views

Why do repeated linear factors have to be dealt with in this way?

When dealing with partial fractions, and your denominator has a repeated linear factor, the way to solve is this: $\frac{2x+3}{(x-2)^2}=\frac{A}{(x-2)^2}+\frac{B}{(x-2)}$ $2x+3=A+B(x-2)$ and so on. ...
13
votes
3answers
591 views

Mathematical structures

Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
4
votes
3answers
112 views

Geometry of the dual numbers

A dual number is a number of the form $a+b\varepsilon$, where $a,b \in \mathbb{R}$ and $\varepsilon$ is a nonreal number with the property $\varepsilon^2=0$. Dual numbers are in some ways similar to ...
1
vote
1answer
172 views

How is percolation defined and measured in social networks?

From the wikipedia article on percolation it appears that the theory is applicable to graphs in general, and this presentation describes the theory nicely. This review article on complex networks ...
90
votes
2answers
3k views

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: ...
3
votes
3answers
51 views

Let $z_1$, $z_2$ and $z_3$ be complex vertices of an equilateral triangle. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$.

Prompt: Let $z_1$, $z_2$ and $z_3$ represent vertices of an equilateral triangle in the complex plane. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$. Question: I hope the following ...
3
votes
4answers
128 views

What is the motivation to build measure theory?

I started reading about measure theory on wikipedia, and downloaded some PDFs, but they all start defining things that I can understand, but can't imagine the motivation to define these things. ...
4
votes
1answer
63 views

Why is this intuitive method valid?

Problem. There are $2$ white and $3$ black balls in the urn. A person randomly picked $2$ balls and put $1$ white ball. What is the probability of the event that the next randomly-picked ball would be ...
7
votes
2answers
53 views

How do I close the gap between intuitively knowing something is true vs being able to prove it?

For example, one of my review problems is: Let $S_k$ be the kernel of $T^k$. Show there is a $K$ such that $S_K = S_{K+1} = \cdots$ Somewhere in the back of my brain there's an intuition that told ...
1
vote
2answers
99 views

How to upgrade Category Theory skills for Algebraic Geometry?

I am doing a second advanced graduate course in Algebraic Geometry, with Hartshorne as a textbook. The skillset I am least satisfied with is the application of the Category Theory to Algebraic ...
21
votes
10answers
310 views

Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
0
votes
2answers
50 views

How to visualize(inside ones brain) the Four-dimensional_space

Can the fourth dimension https://en.wikipedia.org/wiki/Four-dimensional_space be visualized intuitively by the humans. Does the professional mathematicians can do this ? If so what are the things to ...
13
votes
4answers
1k views

Why are polynomials defined to be “formal”?

Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial. ...
18
votes
6answers
3k views

Why do some mathematical ideas seem counter-intuitive?

Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the ...
9
votes
3answers
239 views

Intuitive understanding of the uniqueness of the Fundamental Theorem of Arithmetic.

Basically I am trying to understand why Fundamental Theorem of Arithmetic (FTA) exists, i.e why a natural number cannot be factored primely in two or more different ways. There are two proofs given ...
0
votes
2answers
16 views

Intuition for using vectors in sale related problems

I am reading Linear Algebra from David Lay's book. He gives one example to showcase use of linear combination of vectors : I understand the solution, but I am completely clueless about how to ...
0
votes
3answers
447 views

Intuition behind gradient VS curvature

In Newton's method, one computes the gradient of a cost function, (the 'slope') as well as its hessian matrix, (ie, second derivative of the cost function, or 'curvature'). I understand the intuition, ...
0
votes
5answers
145 views

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 ...
3
votes
3answers
247 views

Is there a notion in mathematics saying that, in a sense, all finite dimensions are actually infinite dimensional?

So then every ordered pair or triplet and so on would be actually represented by an infinite sequence of numbers, and what we think of as 3 dimensions would mean that the point has an infinite number ...
2
votes
1answer
448 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 ...
2
votes
1answer
275 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 ...
3
votes
1answer
649 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) ...
3
votes
3answers
65 views

Formal proof for $\lim_{x\to\infty}x\exp(-x) =0$

I intuitively understand that $$\lim_{x\rightarrow\infty} xe^{-x}=0$$ as the $e^{-\infty}$ approaches zero faster than $x$ approaches infinity. But this requires one to have a knowledge of the ...
3
votes
2answers
140 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 ...
0
votes
1answer
51 views

Non- intuitive connected space.

There exist knowing examples of connected spaces such that its picture is a counter intuitive for us?. I mean a topology on a set who makes see the space as connected (no connected) but it is no ...
0
votes
0answers
19 views

Steady state state distributions.

I am looking for a less "proofy" explanation of how a finite, irreducible, aperiodic Markov chain has a unique steady state $\pi$. No need define terms or include proofs of Bezout's lemma or number ...
2
votes
2answers
88 views

Which of these topological properties imply which?

I am going through the chapter on compactness and completeness from Sternberg's Advanced Calculus and trying to build an intuition for what many of this topological properties mean, and which imply ...
1
vote
0answers
21 views

Gradient points in the direction of greatest change

Can anyone provide me with an alternative, possibly more intuitive proof of this proposition? I'm confused with where $cos\theta$ has come from?
4
votes
2answers
134 views

Geometric Interpretation of Antiderivative?

Could someone please give me a geometric interpretation of the above theorem?
0
votes
1answer
19 views

How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
3
votes
2answers
80 views

Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
-1
votes
0answers
16 views

Linear programming and their geometry: cones

This is a snippet of a solution in my notes. Why can we conclude that from figure 1? Why can we say that the arrows are between and do not go all the way around the axis?
3
votes
1answer
94 views

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 ...
3
votes
1answer
54 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
6
votes
1answer
136 views

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 ...
7
votes
2answers
169 views

How would one arrive at the formulas for divergence and curl?

It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ...
52
votes
12answers
5k views

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 ...
0
votes
1answer
48 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 ...
5
votes
2answers
152 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 ...
0
votes
2answers
73 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
votes
1answer
189 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 ...
13
votes
7answers
5k views

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
votes
3answers
101 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
votes
4answers
144 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 ...
5
votes
3answers
57 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
votes
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 ...