# 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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### 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 ...
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### What are the examples to understand L-process and R-process?

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- ...
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### How can you picture Conditional Probability in a 2D Venn Diagram?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I pursue only intuition; do not answer with formal proofs. Which ...
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### How to intuitively understand that an open subset of the reals can contain the rationals and have finite measure?

A question that one could ask is the following: if $U \subset \mathbb{R}$ is an open subset such that $\mathbb{Q} \subset U$, then is the measure of $U$ infinite? The answer is no, as the (relatively)...
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### The significance of failure of uniqueness in differential equations

The nonlinear ODE: $y'(t)=y(t)^{1/2}$ with initial condition $y(0)=1$ has two solutions. Non-uniqueness is not surprising because of the failure of Lipschitz continuity in the $y$ term. While this ...
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### Limit point Compactness does not imply compactness counter-example

I think that I understand why compactness implies limit point compactness: Suppose $A \subseteq X$ has no limit points. Then $A^{\prime} \subseteq A$. Thus, $A$ is closed. Then for all $a \in A$, ...
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### Intuition of LHS in Green's thorem for a velocity field

I am trying to get an intuition for the LHS part of Greens theorem. For a potential field like gravity the LHS part is work, but if the vector field is a velocity field then what does the LHS ...
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### compactness requirement for the tube lemma of a product space.

The tube lemma: Let $X,Y$ be topological spaces s.t $Y$ is compact. Let $X_0 \in X$ and let $N$ be an open set in $X \times Y$ so that $x_0 \times Y$ is contained in $N$. Then there exits a ...
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### How to picture a first countable space?

I find myself forgetting what it means for a space to be first countable on a frequent basis. This is unlike say other terminologies such as "Hausdorff space", where you can picture balls separating ...
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### integrals with no analytic answer - intuition and proof

the following integral has no analytic solution it appears: $$\int_0^\pi e^{\sin(x)} \, dx$$ intuitively, what is the reason for this integral having no analytic answer? (is there a way to prove it ...
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### How was Zeno's paradox solved using the limits of infinite series?

This is a not necessarily the exact paradox Zeno is thought to have come up with, but it's similar enough: A man (In this photo, a dot 1) is to walk a distance of one unit from where he's standing to ...
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### Degrees of Freedom in Covariance: Intuition?

If we say $Var(x)$ has $n-1$ degrees of freedom which are lost after we estimate $Var(x)$, this matches how $n-1$ observations are now constrained to be sufficiently close to the remaining observation ...
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### Elements of bounded distributive lattice belonging to same prime ideals are equal?

I have read in a paper that by an easy application of Zorn's lemma one may show that two elements of a bounded distributive lattice are equal iff they are contained in exactly the same prime ideals of ...
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### Intuition for the construction of the product topology and its equivalence to the euclidian metric

While I have been provided a proof for the previous statement, I still cannot fully grasp why the euclidian metric [ $d(x,y)=((x_1-y_1)^2+...(x_{n}-y_{n})^2)^{1/2}$] generates the same topology as the ...
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### Moving from sheaves over spaces to sheaves over sites

The first example of a sheaf that I have consciously come across is the sheaf of continuous (real) functions on some topological space. The fact it is a sheaf is equivalent to the pasting lemma, which ...