I am a German mathematics student. I have two questions:
First question: When writing several equivalent equations, we usually write this as:
$x^2 = 4$
$\Leftrightarrow x = \pm 2$
I have never seen this in English, i.e. the usage of $\Leftrightarrow$ to indicate equivalence in a sequence of equations. Is it considered bad style? How would you write this?
Second question: In German, we often declare variables within a sentence similar to:
"The output $x$ depends on the time of default $t$ and on the sum $S$ of all payments made prior to $t$."
Is this correct in English? I am not sure whether or not I have seen this before. If not, what would be an alternative formulation?
Thank you very much in advance
Your first statement would be acceptable, in English, for lectures or for work on a board. In written English, people are encouraged to use fewer symbols; for example the $\Leftrightarrow$ might be avoided. Someone might write this statement as "$x^2=4$ is equivalent to $x\in\{-2,2\}$."
Note that there are many alternatives to the phrase "is equivalent to" which would be acceptable, but few are shorter.
The style of your second statement is common in English mathematics.
Note that this is for American English. I'll leave other forms to native speakers in those languages.