I scrape Arxiv to find underused Greek variables which can add some
diversity to math; the top 10 underused letters are ϰ, ς, υ, ϖ, Υ, Ξ, ι,
ϱ, ϑ, & Π. Avoid overused letters like λ, and spice up your next
paper with some memorable variables!
Some Greek alphabet variables are just plain overused. It seems like no paper is complete without a bunch of E or μ or α variables splattered across it—and they all mean different things in different papers, and that’s when they don’t mean different things in the same paper! In the spirit of offering constructive criticism, might I suggest that, based on Arxiv frequency of usage, you experiment with more recherché, even, outré variables?
Instead of reaching for that exhausted π, why not use… ϰ (variant kappa), which looks like a Hebrew escapee? Or how about ς (variant sigma), which is calculated to get your reader’s attention by making them go “ςςς” and exclaim “these letters are Greek to me!” If that is too blatant, then I can recommend the subtle use of Υ (capital upsilon) instead of ‘Y’, which few readers will notice—but the ones who do, the hard way, will be asking themselves, “Υ‽ would any jury in the world convict me…?”
The top 10 least-used Greek variables on Arxiv, rarest to more common:
Quickly skimming papers on Arxiv, the equations often run together. It doesn’t help that everyone seems to overuse the same handful of Greek letters for everything, which rather defeats the point. Wouldn’t it be better if authors used variables which were a little unique, and we could enjoy the wonders of expressions like dividing a variable by its estimate, or as Eric Neyman would write it (inspired by Serge Lang), Ξ¯¯¯Ξ?
First, though, we’d need to know which variables are underused—we can see which ones are overused, but it’s harder to see what ones are neglected. To estimate the frequency of variable use in scientific papers, I did some ad hoc parsing of Arxiv; this is a good proxy for STEM research in general, and because Arxiv is TeX-centric, it’s easy to parse out Greek variables with reasonable accuracy.
Dataset: The Pile. The Arxiv scrape I use was released as part of the large 2020 text dataset The Pile; the shadowy hacker group known as EleutherAI scraped Arxiv in 2020 and converted it to a Markdown+TeX format, to simplify away the enormous amounts of TeX boilerplate while leaving the equations/variables. The Arxiv subset is provided as a 17GB tarball, 59GB uncompressed; it has 1.26m files with 417.6m lines / 7.2b words / 60.2b characters:
# if The Eye is down, use the torrent:# https://academictorrents.com/details/0d366035664fdf51cfbe9f733953ba325776e667/techwget\'https://the-eye.eu/public/AI/pile_preliminary_components/2020-09-08-arxiv-extracts-nofallback-until-2007-068.tar.gz'tar xf '2020-09-08-arxiv-extracts-nofallback-until-2007-068.tar.gz'cd ./documents/find . -type f |wc--lines# 1264405find . -type f |xargs-n 9000 cat |wc# 417,626,335 7,819,609,303 60,267,810,641
To quickly visualize this dump, here’s 100 random lines from 1,000 random files:
find . -type f |shuf|head-1000|xargs cat |shuf|head-100# \end{array}# Zeldovich Pancake# {\begin{pmatrix}# \kappa_{11}(I_r) - \kappa_{11}(\Omega \setminus I_r) \\# If $G$ is a ‘blown-up’ $C_7$, then it is clearly $3$-colourable (as false twins can use the same color). Thus, Claim \[cl:c7\] enables us to assume $G$ has a cycle $C$ of length $5$, say its vertices are $c_1, c_2, c_3, c_4, c_5$, in this order. From now on, all index operations will be done modulo $5$. Because $G$ has no triangles, the neighbourhood $N_C$ of $V(C)$ in $G$ is comprised of 10 sets (some of these possibly empty):## \end{aligned}$$# [^1]: Note that in contrast to Logic Programming conventions, queries are not denoted by denials.# )}{\bar{n}_{m}(\omega)}e^{-i\left( \nu-\omega\right) \left( \tau-t\right)# \mathcal{M}_\beta\mathcal{K}^1_{-c}(\xi)&# &=# 512 3.50e$-$02 1.91 1.01e$-$01 1.11 2.57e$-$02 1.93 1.52e+02 1.04 1.65e$-$02 1.95## \mathbf{Q}\left(\alpha\right)^{-1} \otimes \boldsymbol{\Sigma}_{jk} & \mathbf{Q}\left(\alpha\right)^{-1} \otimes \boldsymbol{\Sigma}_{jj} \notag# $ 58243.6877674 $ & $ 32.9314 $ & $ 0.0070 $ & $ -0.005 $ & $ 0.010 $ & FEROS\# So our restricted Schubert variety is in the affine space $\operatorname{Spec}(A')$ where $$A'=\operatorname{Sym}({\mathfrak{g}}_1) = \operatorname{Sym}(\bigwedge^3 F\oplus\bigwedge^2F\otimes G\oplus F\otimes\bigwedge^2 G).$$# ✱✱Proof✱✱ First note that $$Q({S}_l({\mathcal A})) = \sum_{u \in \partial S_l({\mathcal A})} \pi(u)$$ and $$Q(\Omega \setminus {S}_l({\mathcal A})) = \sum_{u \in \partial S_l({\mathcal A})} \pi(u).$$ Further, $$\begin{aligned}#### ✱✱Keywords✱✱: Dilute suspensions, Eulerian models, direct and large eddy simulations, slightly compressible flows, dam-break (lock-exchange) problem.\# We have the following corollary.# Acknowledgements {#acknowledgements .unnumbered}### \frac{\Xi_c^{(',✱)+}}{\sqrt{2}} &\frac{\Xi_c^{(',✱)0}}{\sqrt{2}} &\Omega_c^{(✱)0}## (-1)^{p(i)} \lambda_{i} -\sum_{k=1}^{i-1}(-1)^{p(k)}# (N,\varrho_L,\varrho_R)$, and acting on the morphisms as the identity map. The identities $G_L \circ G_R^{-1}=U$ and $G_R \circ G_L^{-1}=V$ prove that $U$ and $V$ are mutually inverse isomorphisms, as stated.## 0 \mathop{\to}^\alpha - \eqspace + \mathop{\to}^\beta 0\eqspace (i=N)\;.$$ Note that all dynamical rules, conserving and non-conserving, are CP-symmetric, namely symmetric under the exchange of positive-negative charges and left-right directions. Generalizations of this model to the case were both types of particles can move in both directions, and when the dynamical rules break the CP symmetry, will be considered elsewhere.# {#section-19}# \[alg machine\]# \bar{u}_j \pa_j \, \l \nonumber \\## title: 'The Synthetic-Oversampling Method: Using Photometric Colors to Discover Extremely Metal-Poor Stars'# 9 5 (1764,567,525,266,150,132,49,27,8,1) (1,1,2,2,1,1,2,2,1,1) 39204## \nonumber\end{aligned}$$ Only $u$ and $du$ must be stored, resulting in two storage units for each variable, instead of three storage units for equation (\[s1\]). The third order nonlinearly stable version we use, Gottlieb & Shu (1998), has $m=3$ in (\[s2\]) with $$\begin{aligned}# We inherit the notation of §\[SectionDModules\]. We begin with proving an explicit version of \[thm:mainDresult\]. Let $Y$ be a smooth product variety $Y=X\times Z$, $X$ be Affine and let $M=(E,\nabla)$ be a (meromorphic) integrable connection on $Y$. We denote $\pi_Z:Y\rightarrow Z$ the canonical projection. We revise the explicit ${\mathcal{D}}_Z$-module structure of $\int_{\pi_Z}M$. We can assume that $Z$ is Affine since the argument is local. From the product structure of $Y$, we can naturally define a decomposition $\Omega_{Y}^1(E)=\Omega^1_{Y/X}(E)\oplus\Omega^1_{Y/Z}(E)$. Here, $\Omega^1_{Y/X}(E)$ and $\Omega^1_{Y/Z}(E)$ are the sheaves of relative differential forms with values in $E$. By taking a local frame of $E$, we see that $\nabla$ can locally be expressed as $\nabla=d+\Omega\wedge$ where $\Omega\in\Omega^1(\operatorname{End}(E))$. We see that $\Omega$ can be decomposed into $\Omega=\Omega_x+\Omega_z$ with $\Omega_x\in \Omega^1_{Y/Z}(\operatorname{End}(E))$ and $\Omega_z\in \Omega^1_{Y/X}(\operatorname{End}(E))$. Then, $\nabla_{Y/Z}=d_x+\Omega_x\wedge$ and $\nabla_{Y/X}=d_z+\Omega_z\wedge$ are both globally well-defined and we have $\nabla=\nabla_{Y/X}+\nabla_{Y/Z}$. Here, $\nabla_{Y/X}:\mathcal{O}_{Y}(E)\rightarrow\Omega^1_{Y/X}(E)$ and $\nabla_{Y/Z}:\mathcal{O}_{Y}(E)\rightarrow\Omega^1_{Y/Z}(E)$. Note that the integrability condition $\nabla^2=0$ is equivalent to three conditions $\nabla_{Y/X}^2=0, \nabla_{Y/Z}^2=0,$ and $\nabla_{Y/X}\circ\nabla_{Y/Z}+\nabla_{Y/Z}\circ\nabla_{Y/X}=0$. For any (local algebraic) vector field $\theta$ on $Z$ and any form $\omega\in\Omega_{Y/Z}^✱(E)$, we define the action $\theta\cdot \omega$ by $\theta\cdot \omega=\iota_\theta(\nabla_{Y/X}\omega)$, where $\iota_\theta$ is the interior derivative. In this way, ${\rm DR}_{Y/Z}(E,\nabla)=(\Omega^{\dim X+✱}_{Y/Z}(E),\nabla_{Y/Z})$ is a complex of ${\mathcal{D}}_Z$-modules. It can be shown that ${\rm DR}_{Y/Z}(E,\nabla)$ represents $\int_{\pi_Z}M$ ([@HTT pp.45-46]).## \## \widehat{m}_{a}$$ or, using the rescaled $u_{B}$ to the conventional Bondi $u_{c}$ coordinate\[convention\][@coorTransf], by $u_{B}=\frac{u_{c}}{\sqrt{2}}$.# Let $G=GL(m,n)$ and $B$ be the subgroup of upper triangular matrices. Then $\Lambda^+-\rho$ coincides with the set of dominant weights. Moreover, it is well-known (see for example [@P]) that for any $\lambda\in\Lambda^+$, $\Gamma_i(G/B,C_\lambda)=0$ if $i>0$. Moreover, $$\Gamma_0(G/B,C_\lambda)\simeq K_\lambda:=U({\EuFrak{g}})\otimes_{U({\EuFrak{g}}^+)}L_\lambda^0,$$ where ${\EuFrak{g}}^+={\EuFrak{g}}_0+{\EuFrak{b}}$ and $L_\lambda^0$ is the irreducible ${\EuFrak{g}}_0$-module of highest weight $\lambda$ with trivial action of ${\EuFrak{b}}_1$. The module $K_\lambda$ was first considered in [@Krep] and is usually called a Kac module. It was proven in [@Z] that every indecomposable projective module $P_\lambda$ has a filtration by Kac modules $K_\mu$ and that the multiplicity of $K_\mu$ in $P_\lambda$ equals the multiplicity of $L_\lambda$ in $K_\mu$. A combinatorial algorithm for calculating $a(\lambda,\mu)$ in this case was obtained by Brundan, [@B]. We will explain it in Section \[wd\] after introducing weight diagrams.# Let $r = 1 + \frac{q}{p'}$. These two corollaries also imply that the A${}_r$ characteristic of each $|W^{-\frac{1}{q}} {\ensuremath{\vec{e}}}|^{p'}$ is bounded by ${\ensuremath [W]_{\text{A}_{p,q}}}^{r' - 1} $.## The photocatalytic material is immersed in a solution of hydrogen peroxide fuel H$_2$O$_2$. Under activation by light, it decomposes the hydrogen peroxide fuel and the concentration profile of hydrogen peroxide, $[H_2O_2]_{(r, t)}$, is given by the solution of the diffusion-reaction equation: $$\partial_t [H_2O_2]_{(r, t)}= D^{\star}\Delta [H_2O_2]_{(r, t) } - \alpha [H_2O_2]_{(0, t) }## 2.1. Single-machine scheduling with a common due window assignment [@jana04; @mos10b; @yeu01]# \ln\mathcal{Z}=\frac{S}{2}\left[\frac{1}{Da_z}\ln\left## G. Paltoglou, M. Theunis, A. Kappas, and M. Thelwall. Prediction of valence and arousal in forum discussions. submitted to [✱Journal of IEEE Transactions on Affective Computing✱]{}.# M. Stoll. Implementing 2-descent for [J]{}acobians of hyperelliptic curves. , 98(3):245--277, 2001.# $. In terms of the new variable $x$, Eq. (\[schrodinger1\]) is recast in the “Schrödinger-like” form $$\label{schrodinger2}# --------------------# \sum_{\beta=1}^{\alpha} \frac{1}{\omega(r_\beta)}.# Наконец, если символ Сегре равен $[(1, 1), (1, 1), 1, 1]$, то $X$ содержит 4 особые точки. Они не лежат на одной плоскости, что видно из уравнений $X$, см. следствие \[sled1\]. Рассмотрим проекцию из трёхмерного проективного пространства, порождённого этими точками. Мы получаем $G$-эквивариантное расслоение над $\mathbb{P}^{1}$ на рациональные поверхности, являющиеся пересечиями двух квадрик. Применив эквивариантное разрешение особенностей расслоения, а затем эквивариантную относительную программу минимальных моделей, мы получим $G$-расслоение Мори на коники или поверхности дель Пеццо.# Using independence statistic for Two-Sample Testing {#using-independence-statistic-for-two-sample-testing .unnumbered}# \frac12 \bigl\| T_{\underline{\cZ}} P'_{\underline{\cB}|\underline{\cZ}} - P_{\underline{\cB}\underline{\cZ}} \bigr\|_1 \leq \epsilon_0 + \sum_{j=1}^m 2\epsilon_j.$$## This paper reports that a simple particle filter applied to data on a time sequence has the ability of real-time decision making in the real world. This particle filter is named PFoE (particle filter on episode). Particle filters[@gordon1993] have been successfully applied to real-time mobile robot localization as the name [✱Monte Carlo localization✱]{} (MCL)[@dellaert1999; @fox2003]. MCLs are mainly applied in the state space of a robot, while PFoE is applied in the time axis. This paper shows some experimental results. The robot performs some teach-and-replay tasks with PFoE in the experiments. The tasks are simple and could also be handled by some deep neural network (DNN) approaches[@hirose2018; @pierson2017]. However, unlike DNN, the particle filter generates motions of robots without any learning phase for function approximation. Moreover, unlike MCL, it never estimates state variables directly. This phenomenon has never been reported.# [^12]: This result is roughly consistent with the subsequent, similar analysis of @Choudhury15; however, here we only take the results from the former work as the latter does not provide statistically-quantitative constraints nor does it account for uncertainties in the ionising background.## Define $m_{\alpha,n}:=m_{\alpha,n,w}= \max\{1,\lceil \alpha - w - 2bn\rceil\}$, in particular $\alpha \le m_{\alpha,n}+w+2bn$. We split (\[eq:add2\]) into two subsums: $\sum_{m_{\alpha,n}\le m\le M}$ and $\sum_{1 \le m< m_{\alpha,n}}$ (the splitting of the sum is because the general function $h(z,w):=\int_0^1 t^z\exp(wt)\,dt$ behaves essentially differently according to whether $|w|<|z|$ or $|z|<|w|$). Each term in the subsum $\sum_{m_{\alpha,n} \le m\le M}$ can be calculated explicitly as### \right)^2$ of the flat direction. For $v \sim 10^{16}$ GeV, we expect this to be comparable to the (mass)$^2$ due to the inverted hierarchy which is $\sim -F_v^2 /v^2 \; d^2 Z(v)/ d (\ln v)^2$. It turns out that in this case the SUGRA contribution is smaller (by a factor of $\sim 4$) than the (mass)$^2$ due to the inverted hierarchy. This results in a shift of the minimum of $v$ by $\sim O(1/4) \;v$.### eno.x & 999 & 999 & 98486.1688\# bibliography:# When we switch on the dephasing, $R_3$ stays a good entanglement measure unlike the von Neumann and Rényi entropies. In the open system, $R_3$ undergoes a characteristic stretched exponential decay starting at time scales $\gtrsim 1/\Gamma$ as shown in Fig. \[fig:sc\_exp\]. Such a decay can be understood as a superposition of local exponential decays, and has also been observed in the imbalance in Refs. [@Fischer2016; @Levi2016; @Everest2017] and is also experimentally confirmed [@Lueschen2017]. We observed such a decay as well in our exact simulations for other entanglement measures like the negativity and the Fisher information as we show in Appendix \[app:qfi\].## \begin{array}{cc}## t^\gamma|\varphi(t)|^p\,dt\bigg]^{1/p},$$ then the symbol $a(\xi)$ is called a Mellin $\mathbb{L}_{p,\gamma}$--multiplier.# \tilde\omega^g_k\bigl(F_k(\phi_{tot})^2\bigr) &= \frac{1}{2},\end{aligned}$$ and the limit $k\to 0$ is trivial: $$F_0(n_{tot})=\lim_{k\to 0}F_k(n_{tot}) \quad\text{and}\quad### [^7]: Note that there are many different nomenclatures throughout the machine-learning literature for the terms defined in Eqn. \[eq:precision\_recall\]. is most commonly referred to as the true positive rate (TPR), though it can also be referred to as the sensitivity, hit rate, or completeness depending on the context. I adopt the convention of referring to this as the as this is only discussed relative to the . The of a model is sometimes referred to as the positive predictive value or purity.## \xi^{-1}(\hpart^\mu(\xi(f)\cdot_{U(\gg)}\xi(g))) =# The states in Figs. \[strangemoms\](a) and \[strangemoms\](b) can be understood as the extreme limits of the states in Figs. \[moms\](d) and \[moms\](g), respectively.# \sin v_{00}(t)# M. Bredel and M. Fidler, “A measurement study regarding quality of service and its impact on multiplayer online games”, in ✱Proc. NetGames✱, 2010.# Introduction and rational Arnoldi decompositions# Our paper is organized as follows; in the next Section we discuss the changes that must be made in the nucleosynthesis code when neutrino heating is taken into account and how we implemented them. In the final Section, we discuss our numerical results, compare them to previous estimates for the change in $^4$He production, and finish with some concluding remarks.# & & + \sum_{j} p(x_j^R) \ln p(x_j^R)# \left( {1 \over D-3} - 1.13459 \right) \Lambda^{4+2(D-3)} \,.# The Boroson and Green (1992, BG92) “Eigenvector 1” (EV1) is one of the reasons NLS1 are interesting, and why we are having a meeting about them. EV1 (also called Principal Component 1) is a linear combination of correlated optical and X-ray properties representing the greatest variation among a set of spectra. EV1 links narrower (BLR) H$\beta$ with stronger H$\beta$ blue wing, stronger FeII optical emission, and weaker \[OIII\]$\lambda 5007 emission from the NLR, with steeper soft X-ray spectra (Laor et al 1994, 1997). The narrower H$\beta$ defining NLS1s has been suggested to result from lower Black Hole masses, and therefore higher accretion rates relative to the Eddington limit in these luminous AGN. The NLS1s’ steep X-ray spectra are also suggested to tie in with high Eddington accretion rates (eg. Laor, these proceedings).# To get the theorem, a spin structure is not necessary on $M$: any second order operator of the Laplace type with values on a vector bundle $V$ over $M$ (see [@Vassilevich:2003xt eq. (2.1)]) can replace ${\mathcal{D}}^2$.## Following the notation of Eqn. (\[dl\]) we set up the following GP for $w$: $$w(u)\sim {\rm GP}(-1,K(u,u')).# &\propto \exp\bigg\{-\frac{1}{2}\Big[\boldsymbol{\beta}^T \left\{\mathbf{X}\left(\boldsymbol{\theta}\right)^T\boldsymbol{\Omega}^{-1}\mathbf{X}\left(\boldsymbol{\theta}\right) + \mathbb{C}_{k|j}^{-1}\right\}\boldsymbol{\beta}\\### title: |
We can see that the Greek variables are often run-together, not delimited by anything except backslashes or punctuation (which begin/end TeX commands), whitespace is chaotic, and so on. Parsing the TeX into an AST or pulling apart expressions like \sum/\prod is right out1, but fortunately, that’s enough structure for our purposes: gauging overall letter use.
Split, sort, count, recount. The usual CLI strategy for counting histograms is to split into newline-delimited instances, sort, count, and resort by count. To look at the frequency of top-1000 ‘words’, I:
cat all the files in parallel
grep for math-related lines which contain a dollar sign or backslash, indicating TeX
Split lines at every backslash using sed (preserving the backslash)
Split lines at all whitespace and punctuation characters aside from backslash, using tr (deleting those delimiters)
I don’t do this in a single step using the [:punct:] regexp character class, because that would delete the backslashes, which we want to keep.
sort the resulting lines, each of which should be a single ‘word’
Count the runs of words using uniq, producing n/word pairs
sort again, to produce an ascending count
Take the 1,000 most common words at the tail of the output.
The results look plausible overall. We want to search that for the Greek alphabet specifically.2
Alphabet zoo. The TeX Greek alphabet is a little tricky. Several of the Greek letters come with variants which have the \var prefix. Who knew there’s a π variant, ϖ? (Not most Arxiv authors, turns out. At least the really obscure ones like ϡ & Ϙ aren’t supported at all and need not be considered.) And what’s up with digamma? Most letters are defined reasonably mnemonically with a lowercase and uppercase version, like \delta vs \Delta, but Knuth, in one of his overly-clever TeX design choices, leaves out cases where the English letter is ‘close enough’—eg there is no \Beta, you just have to know to type B instead (not to be confused with the Unicode version, which does distinguish, of course, and where ‘B’ ≠ ‘Β’). Epsilon, for example, both has a variant and an overloaded English uppercase, while omicron has neither lowercase nor uppercase and is just o/O!3
This complicates the search because we cannot simply match on backslash, and is one reason we split on punctuation and whitespace, which catches the English puns. We have to specify that we are matching the entire line: you cannot grep for ‘H’, you have to grep for the line being solely H, ^H$. And likewise for the others.
The distribution of usage looks like a smooth dropoff by rarity. Usually, these turn out to be something related to Zipf’s law; whether it’s a Pareto like natural language vocabulary, or something else (like a Poisson), this is probably too few items to test—it looks like one, but is too noisy to tell, perhaps due to inflation from my cheap error-prone parsing. Constantin et al 2024 analyzes physics equations more rigorously, and finds that the frequency of ‘operators’ (including functions like exponents or fundamental constants) seem Zipf-distributed.
‘Α’ anomalously popular. Most of the high-ranking variables are expected: of course \beta or \pi or \mu will be common. The spike/jump of Α is almost surely an unavoidable glitch from the ad hoc parsing: you can’t easily distinguish between ‘A’ which is being used in a normal English sentence (“… A good example…”), and ‘A’ which is in an TeX expression like “… + A + …”—that would require full parsing to understand whether it’s in a prose or formula context, and is not doable with shell scripting. (This applies less to the other capitals like E/B/X/O, which are generally not used isolated like Α is.)
‘Π’ anomalously unpopular. Perhaps unsurprisingly, the variants typically rank low, and there is enormous differences in frequency: the most common backslashed entity is 298× more common than the rarest. More puzzling are some of the low ones: given how incredibly common \pi is, why is its capital version \Pi so rare? (Are people shying away from it because Π—despite being much smaller—looks like the multiplication/product summation, which is itself usually written as \prod?) Is \iota really that rare?4 And what do people have against poor upsilon?
One could also argue that the use of sigma/pi that way is not a variable because the letter does not stand for any quantity—it merely denotes the function being called (summation or product) on the actual quantities of interest (specified by the rest of the expression).
Being so flexible, there are many other variables in common use which are made of compound commands or arguments (eg. \mathbb{\pi} (ℼ) or \not\eq (≠)), but these are hard to parse and there’s no master list, so I confine my attention here to just the Greek alphabet which are all single commands used consistently by authors.
Perhaps because it looks so much like an ‘o’, it turns out to be one of the least used letters. The 2021 SARS-CoV-2 Omicron variant may change that in future Arxiv scrapes.
Apparently it’s rare enough to be a Jeopardy! question! “The ninth letter of the Greek alphabet, it can also mean a very small amount.” (Epsilon is the fifth letter.)
