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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58 views
Moment, spheroid, charge redistribution
Let $$I_k:= c \int_{\mathbb R^3} (3x_k'^2-r'^2) \,\,\,d^3 x'$$ where ${r'}^2={x'}_1^2+{x'}_2^2+{x'}_3^2$ and $c$ is a constant = density of charge (uniform) in the body.
Suppose this integral is ...
8
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3answers
203 views
What does an outer automorphism look like?
I am working on a project in my group theory class to find an outer automorphism of $S_6$, which has already been addressed at length on this site and others. I have a prescription for how to go about ...
4
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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 ...
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1answer
38 views
How to choose substitution to make the difference equation linear with fixed coefficients?
I am going over some lecture notes and there is the following exercise:
Solve $$(k+1)^{2}y(k+1)-k^{2}y(k)=1$$ with the initial condition
$$y(1)=0$$
where $k$ it for the time, hence not ...
3
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1answer
167 views
Are Legendre transforms of non-convex functions useful?
Do Legendre transforms have any applications that do not appeal to convexity?
What is the intuitive interpretation of the Legendre transform of a non-convex function?
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3answers
99 views
How to think about one-point schemes?
As topological spaces, all of $\text{Spec}(k), \text{Spec}(k(x)), \text{Spec}(k[x]/(x^2))$ and $\text{Spec}(k(x_1,\cdots,x_n))$ are all homeomorphic, since they are all one point-spaces.
However, as ...
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1answer
170 views
Why is the determinant invariant under row and column operations?
I know that we may add any row to any other in a determinant and its value remains the same.
This is clear enough since elementary matrices corresponding to row and column operations have determinant ...
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2answers
98 views
The Riemann Surface of Logarithmic Function
Who can tell me why the compactification of RS of Log is just Riemann Sphere, please?
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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 ...
2
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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 ...
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2answers
183 views
Is there an idea of “primeless isomorphism” studied somewhere in finite group theory?
What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't ...
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1answer
664 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 ...
2
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1answer
159 views
interpreting the power of adjacency matrix
Given a directed graph $G$, and let $A$ be $G$'s adjacency matrix, whose $(i,j)$-entry is 1 when there is an edge from $i$ to $j$.
Is there any interpretative meaning of the $(i,j)$-entry of the ...
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2answers
214 views
What is the role of Topology in mathematics?
What is the role of Topology in Mathematics? Is it like Logic that you need in every areas of math?
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4answers
313 views
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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1answer
99 views
Formal power series ring of a valuation ring of dimension $\geq 2$ is not integrally closed.
I recently tried exercise 10.4 in Matsumura's Commutative Ring Theory but got stuck. The question is:
"If R is a valuation ring of Krull dimension $\geq 2$, then the formal power series ring $R[[X]]$ ...
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6answers
715 views
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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2answers
71 views
Does the sum of the digits tell us anything about the relation between these two functions
The number $K$ is expressible as part of one of either of two quadratic functions with integer coefficients. One of the functions is quadratic in $\pi$ and one is quadratic in $e$:
...
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1answer
76 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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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 ...
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2answers
56 views
How this solution is working?
In the question asked here, task is to convert decimal numbers to alphabets, like
1: A
2: B
.
.
.
26: Z
27: AA
.
.
.
703: AAA
the method used to generate the ...
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1answer
95 views
Physical meaning of spline interpolation
I remember that when I took my Numerical Analysis class, the professor said the spline interpolation take its name from a kind of wood sticks used to draw curved lines. Also Wikipedia say that the ...
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1answer
81 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, ...
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3answers
185 views
Summary of Optimization Methods.
Context: So in a lot of my self-studies, I come across ways to solve problems that involve optimization of some objective function. (I am coming from signal processing background).
Anyway, I seem to ...
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2answers
72 views
visualisation of pointwise boundedness
A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$
...
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4answers
299 views
Intuitive significance open sets (and software for learning topology?)
I have just started to learn topology and I referred to some books and online lectures. The problem is that they all talk the same things and are missing the same things.
I want to know "what is the ...
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2answers
144 views
Intuition and applications for the p-Laplacian
Consider the p-Laplacian of a suitably nice function $u$:
$\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u)$
Are there useful ways of thinking about the p-Laplace operator, or of thinking about ...
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1answer
126 views
When do modifiers denote sub or super? Pseudo-, quasi-, ultra-, strong-, well-, pre-, c0- …
One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- ...
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3answers
107 views
Seemingly similar but different probability games
Burger King is currently running its "family food" game in which each piece can be modeled as a scratch off game where exactly one of three slots is a winner and you may only scratch one slot as your ...
3
votes
3answers
104 views
Intuition on the sum of first (n-1) numbers is equal to the number of ways of picking 2 items out of n.
While going through an equation today i realized that sum of first (n-1) numbers is [n*(n-1)/2] which is equal to combinations of two items out of n i.e [n!/((n-2)! * 2!)]. I need some intuition on ...
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175 views
Graduate level research [closed]
This is a very vague question but I would like to do research in mathematics, specifically quantum computation and I'm not really too sure how one goes about developing a topic to investigate. I've ...
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1answer
202 views
Are there ways to describe the Martin Axiom intuitively?
I'm looking for some way of understanding the Martin Axiom for some $\kappa$ in an intuitive way. I know that the motivation is the Baire Category Theorem, however is there any other way of thinking ...
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3answers
216 views
Understanding algebraic method of successive approximations to solve quadratic equations
I have straight away copied and pasted a worked example from john birds higher engineering mathematics {page 80 problem 4} which reads as follows.
Use an algebraic method of successive
approximations ...
17
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1answer
185 views
Understanding what $\sqrt{p}$ means for an event of probability $p$
Say I have a random event $E$ with probability $p$. There is a natural interpretation in terms of $E$ for the probability $p^2$: it's the probability that $E$ occurs twice if I perform two independent ...
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3answers
508 views
Intuition behind Snake Lemma
I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
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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 ...
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3answers
192 views
Why gradient descent works?
On Wikipedia, this is the following description of gradient descent:
Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable ...
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2answers
229 views
What exactly is a manifold?
Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth.
Does this mean that in mathematics a manifold is essentially a representation of something that ...
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2answers
171 views
Why does the $2$'s and $1$'s complement subtraction works?
The algorithm for $2$'s complement and $1$'s complement subtraction is tad simple:
$1.$ Find the $1$'s or $2$'s complement of the subtrahend.
$2.$ Add it with minuend.
$3.$ If there is ...
2
votes
2answers
355 views
Intuition for iterated function for log log log n.
Intuitively, $\log n$ (base 2) is the number of times you have to divide $n$ by 2 before reaching a number around 2. (Waving our hands a little to gloss over floor vs ceiling and $\pm$ 1 errors.) ...
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5answers
430 views
The definition of metric space,topological space
I have read some books in analysis,all of them define metric space,topological space or vector space directly,without any reason. Therefore, I want to know the background of the definition, the ...
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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 ...
11
votes
4answers
390 views
What makes elementary row operations “special”?
This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" ...
5
votes
2answers
213 views
Is there any graphical explanation of the derivative of $\sin x$?
I'm trying to understand in a practical/graphical view the derivative of $\sin(x)$ (that results in $\cos(x)$).
Is there any animation or illustration explaining that?
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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 ...
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2answers
113 views
Average run lengths for large numbers of trials: Intuition and proof
This article states that the formula for the average run lengths for large numbers of trials is:$$\frac{1}{1-Pr(event\ in\ one\ trial)}.$$
My questions
What is the intuition behind this formula?
Do ...
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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 ...
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1answer
152 views
Showing that an integral can not be expressed in terms of elementary functions
I recently encountered an integral of the form:
$$\int{\frac{\log(a+bx+\sqrt{x^2+c})}{x}}dx$$
The result involves the dilogarithm function, but I was wondering if there is a fast way of showing that ...
5
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2answers
164 views
Is symbolically solving $a(x)=f(x)g'(x)+f'(x)g(x)$, given $a$, ever easier than integrating $a$?
If we are given a function of $x$, $a(x)$, how hard is it to find an $f(x)$ and $g(x)$ such that $$a(x)=f(x)g'(x)+f'(x)g(x)$$ For comparison, I'd like to know when this is easier than symbolically or ...
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1answer
206 views
How can I intuit the role of the central limit theorem in breaking the curse of dimensionality for Monte Carlo integration
I would like to more intuitively understand where the power of Monte Carlo integration comes from for large-dimensional domains of integration.
Other questions on this site have referenced the proof ...
