formatting – Are there ways to make Google Docs page layout more stable across browser & platform?

When I open a Google Doc on different computers, or even different browsers on the same computer, the line, column, & page breaks often (but not always) change. When I use “download” to export the doc to another format, the line, column, & page breaks often (but not always) change.

I’ve tried a variety of different fonts: built-in system fonts, Google fonts installed onto the computer, Google fonts not installed onto the computer. They all seem to change size.

Are there tricks or techniques to reduce this problem?

reference request – Is Fourier dimension finitely stable?

Let $A,B$ be compact subsets of $mathbb R$. Let $a=mathrm{dim}_F(A)$, $b=mathrm{dim}_F(B)$ be their Fourier dimensions, respectively. My questions are as follows:

  1. Is it true that $mathrm{dim}_F(Acup B)=max{a,b}$?

  2. If $A$ and $B$ are positively separated, is it true that $mathrm{dim}_F(Acup B)=max{a,b}$?

  3. In addition to 2, if $Asubseteq (0,1)$ and $Bsubseteq (2,3)$, is it true that $mathrm{dim}_F(Acup B)=max{a,b}$?

  4. In addition to 3, if $a=0$, is it true that $mathrm{dim}_F(Acup B)=b$?

I have searched online but I cannot find any references for that.

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computer algebra – Numerically stable reverse automatic differentiation of power(x, y)?

I would like to compute the adjoints $bar x$ and $bar y$, from a reverse automatic differentiation perspective, of the expression $x^y$. The adjoint $bar{x^y}$ is already known; and we can assume $x geq 0$ and $y geq 1$.

The “easy” solution consists of rewriting the expression $x^y=e^{y ln x}$, and proceed piece-wise from there. However, this solution proves to be unstable numerically, it gives:

$$bar y = bar{x^y} x^y ln x$$

and

$$bar x = bar{x^y} x^{y-1} y$$

Is there any way to rewrite and/or approximate those expressions with something that is not going to numerically diverge when $x approx 0$?

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Stable sorting without using sort libraries – Python

I want to sort a list of words by word length but maintain the original order of words with same length (stable sorting).

For example:

words = ('dog', 'cat', 'hello', 'girl', 'py', 'book')

Should become

words = ('py', 'dog', 'cat', 'girl', 'book', 'hello')

I know this can be done easily with python’s sorted function, which is stable:

sorted(words, key = lambda l:len(l))

But how would one do this efficiently without access to the library? I’ve been trying to brainstorm a solution and I’m thinking

  1. Create a dict that will contain the index (key) and length (value) of each word in words
  2. Maybe use a Counter to track how many instances of each length word there is (for this ex., would look like Counter({3: 2, 5: 1, 4: 2, 2: 1}))
  3. Go through the dict looking for the minimum length of a word in counter (2 in my ex), and keep doing that for amt of times that length appears
  4. Step 3 but moving onto next minimum in Counter

But this seems super inefficient. Could someone help with a better implementation?

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