# 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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### What is meant in the quotation of Terry Tao?

Terrence Tao commented of internalizing [here: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/] "It is true that some mathematicians can be vastly more ...
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### Why are Unique Factorization Domains (UFD's) geometrically significant?

We know that for $A$ a UFD, it's class group is trivial. More generally, for a factorial (stalks are UFD's) scheme $X$ (that is also noetherian and normal), we have an isomorphism between it's Picard ...
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### Intuition for minimum spanning tree result

Let K$_ n$ denote the complete graph on n vertices. To each edge in K$_n$ independently assign a weight drawn from the uniform distribution on [0,1]$\,$. Finally, define MST(n) to be the expected ...
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### What is functional analysis in simple words?

To begin with , I am only a secondary school student (17yo) but I am very interested in higher mathematics. However we only learn so little in my school (only single variable calculus and basic linear ...
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### Techniques to find a subset of possible values for the rank of a matrix

The rank of a matrix is the number of linearly independent rows or columns of that matrix. In some exercises I need to find the rank of a matrix, but for some (lol unknown) reason I'm always stuck ...
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### Explaining Newton Polygon in elementary way

In this question of mine Proving irreduciblity over $\mathbb{Z}$ I was recommended to read Newton Polygon. Also this appears to be an interesting topic. Also I am currently studying irreducibility of ...
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### Fuzzy logic vs probability

In reading about fuzzy logic it says that fuzzy logic is different from probability. Can some one please explain how these two differ. How can this be explained to a person with no mathematical ...
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### Intuition behind the “infinite velocity” of a falling ladder

In Calculus there is a "classic" related rates problem involving a falling ladder. Say the ladder is $25$ ft tall and is leaning against a wall. The bottom edge of the ladder is pulled away from the ...
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### Abstraction and/or concreteness - What should be emphasized

Alexandar Grothendieck was probably a mathematician focusing on theory developement and abstraction much much more than focusing on concrete examples and/or problems. In his biography, he wrote: ...
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### Intuition: groups, quotient groups, cosets, homomorphisms.

If we start with the group of rational numbers $\mathbb{Q}$ and the subgroup of $\mathbb{Q}$; $\mathbb{Z}$ the integers, and then form the quotient group $\mathbb{Q}$/$\mathbb{Z}$ we have that this ...
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### Concerning Approximations, Multiplicative Order, and Residue Number Systems…

I'm interested in particular representations of reals in residue number systems. Specifically, if we are given a real $0 \le n \le 1$, we wish to represent that number as a fraction in a residue ...
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### Approximating $x=\sqrt{2}+1$

Suppose $y>1$ is some approximation to $x=\sqrt{2}+1$. Give a brief reason (not a proof) why one should expect $(1/y)+2$ to be a closer approximation to $x$ than $y$ is. After testing this out ...
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### Is there a geometric meaning associated with the condition “dot product equals $1$?”

Consider $x,y \in \mathbb{R}^n$. Then the condition $x \bullet y = 0$ is easy to understand; it just means that $x$ and $y$ are orthogonal. Question. Does the condition $x \bullet y = 1$ have an ...
R-processes and L-processes R-process on $[0,\infty)$ to signify a process all of whose paths are right-continuous on $[0,\infty)$ with limits from the left on $(0,\infty)$. R-function or R- ...