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.
11
votes
1answer
663 views
Is there a proof of Benford's Law?
As stated by Wikipedia (here):
Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed ...
4
votes
1answer
84 views
Discreteness of eigenvalues for certain operators - can this approach be made rigorous?
I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
3
votes
1answer
49 views
Geometric intuition behind the Uniform Boundedness Principle
Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
2
votes
1answer
63 views
What is the intuition between 1-cocycles (group cohomology)?
This is, I'm sure, an incredibly naive question, but: is there a simple explanation for why one should be interested in 1-cocycles?
Let me explain a bit. Given an action of a group $G$ on another ...
2
votes
1answer
96 views
Hydra game and quantum superposition
Goodstein's theorem is not provable in Peano Arithmetic showed by Kirby and Harrington in 1982 [Wolfram Mathworld].
Any reference of a "quantum" hydra game where a head can remain in a state of ...
1
vote
1answer
443 views
Pattern-matching puzzle (shapes)
some puzzle that can be found online are typically a 3x3 grid where you've got nine spots filled with some images or symbols or geometric figures and you have to predict what the missing one should ...
0
votes
1answer
31 views
Intuition Behind Krantz Theorem
The theorem I'm referring to is as follows:
Let $z_0$ be a root of a nonzero holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$-th derivative of $f$ evaluated ...
0
votes
1answer
52 views
Euclidean Geometry in Classical Thought - Used for Realization or Representation?
I posted this in the Physics.SE Forum but I figured I'd ask this here as well since it's relevant to the forum subject :]
Taken from John J. Roche's "The Mathematics of Measurement: A Critical ...
34
votes
0answers
669 views
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 ...
27
votes
0answers
1k 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: ...
8
votes
0answers
278 views
What is a complex inner product space “really”?
To be clear on this, I know what is the definition of an inner product space and some properties and theorems about them. What I am asking for is an intuition for this definition in the complex case. ...
6
votes
0answers
184 views
Behaviour at infinity of a function in terms of first and second derivatives
In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$:
there exists a constant $C>0$ and a ...
5
votes
0answers
104 views
How should I think about homogeneous coordinates?
It's quite easy to make sense of ordinary coordinates on a space $X$: it is just an open embedding of an open subset of $X$ in some vector space. Along the same lines, one could say that homogeneous ...
5
votes
0answers
113 views
Why are injective $\mathscr{O}$-modules flasque?
Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
5
votes
0answers
153 views
Intuitive test of convergence
Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
4
votes
0answers
68 views
Geometrical Interpretetion of Half Derivative
How would you understand in a intuitive way the meaning of:
$$D^{\frac{1}{2}}x^2=\frac{\Gamma(3)}{\Gamma(\frac{5}{2})}x^{\frac{3}{2}}=\frac{8}{3\sqrt{\pi}}x^{\frac{3}{2}}$$
or
...
4
votes
0answers
90 views
Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)
I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form.
For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
4
votes
0answers
69 views
Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds
The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
4
votes
0answers
51 views
Quotient-lifting properties
I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies
Let $N\triangleleft G$. Then ...
4
votes
0answers
36 views
Something behind the substitution $h^0=\frac{1}{|G|}\sum_{t\in G}\rho^2_{t^{-1}}h\rho^2_{t}$?
I am quite new to representation theory and I reading Serre's Linear Representation of Finite Groups.
In the first and second chapter, one trick he uses quite often is the substitution ...
4
votes
0answers
92 views
Why is better to work with the spectrum of prime ideals than with the maximal one, for example in the definition of affine scheme.
When we have an algebraic variety we can identifie the points of the variety with maximal ideals of the coordinate ring.
I would like to know why is more natural to define the main structure of the ...
4
votes
0answers
74 views
Ways to think about one-relator groups
What are some intuitive ways to think about one-relator groups?
I am aware of the Freiheitsatz, and Bass-Serre theory. What I'm interested in are ways people who work extensively with one-relator ...
3
votes
0answers
59 views
Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.
There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
3
votes
0answers
65 views
A few questions about nonabelian cohomology of finite groups.
I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
3
votes
0answers
92 views
Do the $p$-norms in $\mathbb R^2$ have a nice geometric intuition behind them?
For $p\geq 1,$ the $p$-norm of a vector $(x,y)\in\Bbb R^2$ is the number $\|(x,y)\|_p=(|x|^p+|y|^p)^{1/p}.$
I learned this definition some time ago, but I never really understood it. Is there a ...
3
votes
0answers
56 views
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...
3
votes
0answers
309 views
Numerical Methods: Approximating an Integral with Exponentials
I'd like to know about the best numerical methods for approximating an integral. Unfortunately, I want to know about a fairly general case, so I cannot give a lot of information.
Essentially, I have ...
2
votes
0answers
30 views
Intuition behind criterion for an irreducible Markov chain to be transient
I have been looking over my notes for Markov chains, and I have come across the following:
Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
2
votes
0answers
88 views
Intuition behind the proof for Wiener's theorem?
I am reading his proof for Wiener's theorem in Chp9 of Rudin's functional analysis. The theorems (9.4, 9.5 and 9.7) themselves are quite clear and Rudin did a good job explaining the intuition behind ...
2
votes
0answers
83 views
Is this calculus of variations intuition justifiable?
I'll preface this by saying that I haven't taken an in depth study of the calculus of variations and have only come across it recently in applications; in depth study is on my to do list.
I'm ...
2
votes
0answers
85 views
Intuition in permutations for Laplace Determinant Expansion
Starting with the Leibniz formula for the determinant, I wish to derive the Laplace (Cofactor) Expansion. At the risk of being overly verbose, please see the proof here. Now I understand the idea of ...
2
votes
0answers
127 views
Ways to think about the binomial coefficient
Just to sharpen my intuition in combinatorics, I ask you of ways to think about interesting combinatorical quantities and expressions like the binomial coefficient, for example, for the binomial ...
2
votes
0answers
46 views
Intuition about moment function derivation [OR] derivatives involving a time varying integration domain
$$
m_{{pq}}(t)=\iint\limits_{R(t)}h(x,y) dx dy
$$
where $ R(t)$ the domain of integration is time varying (In fact it is the only one which is time varying). And
$$
h(x,y) = x^p y^q f(x,y) dx dy ...
2
votes
0answers
201 views
Statements about logic (=“metalogic”(??))
Sometimes there are statements about logic e.g. "That's not logical" and I can neither prove nor disprove a statement about logic with no definition for logic itself. It's just a negation and it's ...
1
vote
0answers
47 views
What to take from representation of $S_d$?
I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
1
vote
0answers
31 views
Schonhage–Strassen algorithm
After brief intro to Fourier series, CFT, DFT and their basic properties I enjoyed implementing forward and backward FFT algorithm in complex numbers. I was happy to, at least, have an idea how is it ...
1
vote
0answers
86 views
Why Markov matrices always have 1 as an eigenvalue
Also called stochastic matrix. Let
$A=[a_{ij}]$ - matrix over $\mathbb{R}$
$0\le a_{ij} \le 1 \forall i,j$
$\sum_{j}a_{ij}=1 \forall i$
i.e the sum along each column of $A$ is 1. I ...
1
vote
0answers
87 views
Which space this space drawn in this picture is homeomorphic?
Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome.
[Context of Image: ...
1
vote
0answers
113 views
1
vote
0answers
57 views
A gratifying re-encounter with a piece of math that was out of my mind
A series of real numbers is said to be conditionally convergent if it is convergent but not absolutely convergent.
By rearranging the terms of a conditionally convergent series we can make the ...
1
vote
0answers
42 views
intuitive idea of deformations in topology
We know that when we prove that two topological spaces are homeomorphic to each other in fact we are proving that these spaces are in fact equal under deformations.
Why? this question is very ...
1
vote
0answers
88 views
Intuition for Calderon-Zygmund operator?
What is the best intuition for Calderon-Zygmund operators? Why are they so important in singular integrals, and complementary, which singular integrals don't they cover?
1
vote
0answers
275 views
How effective is this alternative to integration?
I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following:
$f(x) = ...
0
votes
0answers
24 views
expression that constrain the range of x to a positive interval
For any $x \in R$, I used the exponential $f(x)=e^x$ to constrain the value of $f$ to a positive interval. While serving this purpose, it happens that I cannot use the exponential for some other ...
0
votes
0answers
17 views
Understanding the topology of a variety concretely
My ultimate goal is to understand how to compute the cohomology groups of complex algebraic varieties, without having to know what a scheme is.
Therefore I want to be able to handle simple examples, ...
0
votes
0answers
64 views
Intuition for the Frobenius method
I'm teaching a differential equations class now and I am hoping to give a reason for the Frobenius series method beyond simply "we guess these solutions". Now, for the Euler equation
$$t^n x^{(n)}(t) ...
0
votes
0answers
137 views
Structure tensor of a function and the distribution of gradients
In computer vision, one often computes what's known as the structure tensor of an image. The structure tensor of a an image (i.e. a function) is a matrix that, I quote from Wikipedia.
"summarizes ...
0
votes
0answers
58 views
Lorentz reflection
What is a Lorentz reflection of $\mathbb R^3$? Is there a way to visualize it? Suppose I have a plane, P, what would (Lorentz) reflecting in it differ from (Euclid) reflecting in it?
I know that the ...
0
votes
0answers
192 views
What is “essentially equivalent”?
I thought that "essentially equivalent" for two sets means that there are a bijection between these sets.
But recently I thought that every two sets of the same cardinality are essentially equivalent ...
0
votes
0answers
144 views
Visualization of 2-dimensional function spaces
As a follow-up question to what is the norm measuring in function spaces
I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
