Tootfinder

from my link log —
Faster remainder by multiplication, with applications to compilers and software libraries.
https://arxiv.org/abs/1902.01961
saved 2021-05-23

Faster Remainder by Direct Computation: Applications to Compilers and Software Libraries
On common processors, integer multiplication is many times faster than integer division. Dividing a numerator n by a divisor d is mathematically equivalent to multiplication by the inverse of the divisor (n / d = n x 1/d). If the divisor is known in advance---or if repeated integer divisions will be performed with the same divisor---it can be beneficial to substitute a less costly multiplication for an expensive division. Currently, the remainder of the division by a constant is computed from…

When my wife and I are watching things, we're always amused at how many characters in TV shows and movies are named Tommy or Billy. These are obviously fairly common names, but the amount of times we've noticed it in all kinds of different shows like X-Files, SVU etc is so often that it's comical-- it's bizarre. And sometimes in episodes right next to each other in a season. Just a ton of fictional guys out there with these two names across age groups, all over popular cultur…

This https://arxiv.org/abs/2505.11172 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the holomorphic foliations admitting a common invariant algebraic curve
In this paper, we prove the following result using an entirely elementary method: for any algebraic curve $C$ given by $f(x,y)=0$, where $f(x,y)$ is a weakly tame polynomial in $ \mathbb{K}[x,y]$, the $\mathbb{K}[x,y]$-module of polynomial 1-forms admitting $C$ as an invariant curve is generated by two polynomail 1-forms, one of which is $\mathrm{d}f$, if and only if, $f$ belongs to the Jacobian ideal $\langle f_x, f_y\rangle$ in $\mathbb{K}[x,y]$, where $\mathbb{K}\subseteq\mathbb{C}$ is an ar…

This https://arxiv.org/abs/2411.18486 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_…

A new potential method for the $X_{\rm max}$ measurement of extensive air showers based on backtracking radio signals
{Measurements of cosmic-ray composition based on air-shower measurements rely mostly on the determination of the position of the shower maximum ($X_\mathrm{max}$). One efficient technique is to image the development of the air shower using fluorescence telescopes. An alternative technique that has made significant advances in the recent years is to measure the radio emission from air shower. Common methods for $X_\mathrm{max}$ determination in the radio detection technique include fitting a two…

Moments of the shifted prime divisor function
Mikhail R. Gabdullin
https://arxiv.org/abs/2505.24050 https://arxiv.org/pdf/2505.24050

Moments of the shifted prime divisor function
Let $ω^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}ω^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent conjecture of Fan and Pomerance. Our proof uses a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, along with analytic tools from number theory.

I'm in a pickle! I have been giving mathematics tuition to this boy (say X) for 2 years now. His mother introduced herself as X's mother since the beginning. Sometimes, we talk to each other regarding X's studies, and during these conversations I refer to her as "you".
In this situation it's common to call someone like her as "দিদি (elder sister)" or "aunty". But, calling her "aunty" might be inappropriate as this will suggest …