# Tagged Questions

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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### 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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### 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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### 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 ...
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### 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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### 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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### Is there a proof of Benford's Law? [duplicate]

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 ...
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### 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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### 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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### 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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### Formal power series ring over 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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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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### 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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### 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 ...
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### 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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### 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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### 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" ...
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### Is there any graphical explanation of the derivative of $\sin x$? [duplicate]

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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### Can we possibly combine $\int_a^b{g(x)dx}$ plus $\int_c^d{h(x)dx}$ into $\int_e^f{j(x)dx}$?

I'm wondering if this is possible for the general case. In other words, I'd like to take $$\int_a^b{g(x)dx} + \int_c^d{h(x)dx} = \int_e^f{j(x)dx}$$ and determine $e$, $f$, and $j(x)$ from the other ...
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### Why should coordinate transformations be reversible?

Intuitively I understand why coordinate transformation should be reversible. New coordinates should cover the same area covered by the initial coordinates, i.e. there should be one-to-one mapping. ...
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### 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 ...
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### The Duality Functor in Linear Algebra

I'm trying to gain an intuitive understanding of the following construction: For any vector space $M$ over a field $R$, one can define the algebraic dual of $M$ as $M^* := \mathsf{Hom}(M, R)$ and ...