# 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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### An intuitive explanation of how the mathematical definition of ergodicity implies the layman's interpretation 'all microstates are equally likely'.

I'm self-studying Statistical Mechanics; in it I got Fundamental Postulate of Statistical Mechanics and that took me to ergodic hypothesis. In the most layman's language, it says: In an isolated ...
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### Intuition behind exponential sum convergence

My textbook states without proof that the summation: $$\sum_{x=0}^{\infty} \frac{1}{x!} e^{ax}$$ converges for all real $a$. I am trying to understand this. I assume the reasoning is that the ...
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### What is the intuition for adding vs multiplying probabilities?

Caution: I modified this original answer to simplify the examples. You add probabilities when the events you are thinking about are alternatives (eg: A soccer team scores 0 goals or 1 goal or 2 ...
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### What is meant by finite (infinite) bernoullis?

I came across the following quote while searching for intuition behind Poisson distribution - think about a Poisson process. It really is, in a sense, looking at very, very small intervals of ...
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### Why do natural transformations express the fact that a vector space is canonically embedded in its double-dual but not in its dual?

I've been struggling for quite a while to understand why a vector space is considered to be "canonically embedded" into its double dual, but not its dual. As has been remarked in many other places, ...
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### Question about geometric interpretation of modules

I would like to understand the accepted answer to this MO question about the geometric interpretation of modules. In particular, I would like clarification on the following excerpt. Let $R$ be the ...
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### Geometric meaning of vanishing of higher cohomology of quasi-coherent modules over affine schemes

One of the basic vanishing results about quasicoherent (sheaves of) modules over affine schemes is that their non-zero cohomology vanishes. My only geometric intuition for sheaf cohomology is via ...
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### More intuition on the curl formula

I have a question regarding this quesiton. It says that $3$ simple fields that describe rotations around $x,y,z$ axis are: $$H_1(x,y,z)=(0,−z,y)\\ H_2(x,y,z)=(z,0,−x)\\ H_3(x,y,z)=(−y,x,0)$$ but why? ...
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### What is the geometric meaning of representability?

Representable functors play a large role in algebraic geometry when developed through the 'functor of points' approach. One finds schemes represent Zariski sheaves and this gives access to the great ...
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### Intuitively, why is compounding percentages not expressed as adding percentages?

I pursue only intuition; please do not answer with formal proofs. I already know the theoretical reason: because each percentage expresses a different base. $1.$ But why not intuitively? My problem:...
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### What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...
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### Is a series of successive derivatives known/useful?

So, while trying to find something else, it looks like I've found, for many $f(x)$: $$f(x) + f'(x) + f''(x) + f^{(3)}(x) + \dots + f^{(n)}(x)$$ Assuming that there is an easy way to find this sum ...
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### How to understand intuition behind compactness? [duplicate]

I have taken a course in general topology this semester.while solving problems,i find it difficult to go by the definition which says that a space is compact if every open cover of it has a finite ...
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### Intuitive proof that $U(n)$ isn't isomorphic to $SU(n) \times S^1$

One way to prove this is by comparing their centers. However, I do not feel that this proof gives me much insight into the structures of the groups. (It would make me very happy if I were to be ...
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### Can anyone help to explain Benfords Law [duplicate]

Okay so recently I have heard of what is known as Benford's law, and this is the first time I absolutely cannot think of my own inutuion about how this is true. Just trying to think about it makes my ...
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### How to get intuition in topology concerning the definitions?

Most topology texts go on directly to give definition of topology, then they give some examples and that's it, like they directly tell you right Let $X$ be a set and let $τ$ be a family of subsets ...
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### Difference between “real functions” and “real-valued functions”

According to my textbook: A function which has either $\mathbb R$ or one of its subsets as its range is called a real valued function. Further, if its domain is also either $\mathbb R$ or a ...
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### Proper solution of the limit of $\sin(x)/\tan(x)$ as $x \to 0$

In one of my math book, I have a problem where I need to compute $\lim_{x\to0}{\frac{\sin(x)}{\tan(x)}}$ I came up with a solution that I am not able to write formally. The reasoning is the following ...
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### Basic question on the infinitely many solutions of a linear system Ax=b,

I just want to verify the geometry of solutions to $Ax=b$, for the case when we have infinitely many solutions: If say for a $3\times 3$ matrix, after Gaussian Elimination, I have two pivot variables ...
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### Intuition for Kuratowski-Mrowka characterization of compactness

Fact. A space $X$ is compact iff for every space $Y$, the projection $X\times Y\rightarrow Y$ is a closed map. The finite subcover definition of compactness seems reasonably intuitive: finite covers ...
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### What is the Lagrange remainder in a Taylor series expansion

I know what a Taylor series expansion is and I know how to find the Lagrange remainder but what does it mean intuitively? I need an explanation of what the Lagrange remainder represents in terms of ...
### Motivating $y'=y \implies y=Ce^x$
Is there some intuitive reason why one should think that a function which is its own derivative should be of the form $Ca^x$ for some number $a$? Of course I can prove that the unique solution set to ...