Rare Greek Variables

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: |