text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
Huancabamba é um distrito do Peru, departamento de Pasco, localizada na província de Oxapampa.
Transporte
O distrito de Huancabamba é servido pela seguinte rodovia:
PE-18D, que liga o distrito de Pozuzo (Região de Pasco) à cidade de San Rafael (Região de Huánuco)
PE-5NA, que liga a cidade de Yuyapichis (Região de Huánuco) ao distrito de San Luis de Shuaro (Região de Junín)
Distritos da província de Oxapampa | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 450 |
{"url":"https:\/\/ancient.darcynorman.net\/darcynorman.net\/2021\/2005\/12\/19\/mediawiki-spam_blacklist-extension\/index.html","text":"# MediaWiki Spam_blacklist Extension\n\nI'd installed Spam_blacklist back in October, but we'd been getting the occasional spam attack since then, lately every single weekend. So I just dug into the Mediawiki config, and realized that Spam_blacklist never got properly configured on my server, meaning it was essentially running wide open. Crap. What a waste of time that was. Paul and I have removed dozens\/hundreds of spams over the last few weeks, and I was assuming Spam_blacklist was enabled properly. Oops.\n\nSo, if you're running Mediawiki, be sure to carefully (re)follow the instructions, especially Morbus Iff's patch for 1.5 compatibility (I had to manually patch the file, since patch was barfing on an unmatched }).\n\nThe wiki is now configured to pull the blacklist from meta.wikimedia.org every hour, and will check edits against both that blacklist and the one we're growing ourselves.\n\nThings to watch out for:\n\n\u2022 If you're running the latest MediaWiki (1.5.x), be sure you've applied Morbus Iff's patch.\n\u2022 Also, be sure you've moved your local Spam_blacklist page to Mediawiki:Spam_blacklist so the plugin can see it. It apparently must be in the Mediawiki: namespace.\n\u2022 Fix your copy of load_lists.sh to use the proper URL for updating the blacklist - bots have to use http:\/\/meta.wikimedia.org\/w\/index.php?title=Spam_blacklist&action=raw\n\u2022 If you want, add the load_lists.sh file to your crontab so it runs automatically. I think the plugin tries to do that, but better safe than sorry ;-)\n\u2022 Test your wiki's now-hopefully-working antispam measures. Try to edit a page with a URL from the shared blacklist, then one from your own. Hopefully both attempts fail.\n\u2022 (hopefully) relax a bit, and with luck the spammers will move on to greener pastures...\n\nWhy on earth isn't this plugin included with the core Mediawiki distro? And why hasn't the download been updated for 1.5 compatibility yet? It would be great to keep it up to date, and to make it easier to get it running properly. There are likely a LOT of Mediawiki instances that aren't configured properly, making all Mediawiki instances nice, juicy targets.","date":"2023-03-24 16:47:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4577193260192871, \"perplexity\": 6265.454505252571}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945287.43\/warc\/CC-MAIN-20230324144746-20230324174746-00115.warc.gz\"}"} | null | null |
Leslie Gold was an IILJ Scholar from 2005-2008. After receiving her J.D. from NYU Law, Leslie began her career at Paul, Weiss, Rifkind, Wharton & Garrison LLP as an associate in the Corporate Department, focusing on transactions in the areas of Financing and Mergers and Acquisitions. She was subsequently an associate in the Corporate and Intellectual Property Departments of Morrison Cohen LLP where, in addition to her transactional practice, she worked in the areas of trademark licensing and prosecution. Leslie joined Deborah A. Nilson & Associates as an Associate in 2014. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,371 |
{"url":"https:\/\/mathoverflow.net\/questions\/154600\/separating-infinite-words-sharing-factors-by-automata\/157911#157911","text":"# Separating infinite words sharing factors by automata\n\nTwo infinite words $\\xi, \\eta \\in X^{\\omega}$ are separated by an (B\u00fcchi-)automaton if it accepts one but not the other.\n\nDenote by $F_n(\\xi)$ the factors of length $n$ of an infinite word $\\xi$ and also by $F_n^{\\infty}(\\xi)$ the factors of length $n$ that occur infinitely often. Define two equivalence relations on words: $$\\xi \\sim_k \\eta \\mbox{ iff } \\xi[1\\ldots k] = \\eta[1\\ldots k] \\land F_k(\\xi) = F_k(\\eta)$$ and $$\\xi \\sim_k^{\\infty} \\eta \\mbox{ iff } \\xi[1\\ldots k] = \\eta[1\\ldots k] \\land F_k^{\\infty}(\\xi) = F_k^{\\infty}(\\eta).$$\n\nNow I am interested in two questions: 1) If for two infinite words $\\xi \\sim_k \\eta$, i.e. their prefix and factors of length $k$ coincide, what is the size of the minimal automata separating them, and 2) if for two infinite words $\\xi \\sim_k^{\\infty} \\eta$, what is the minimum size of an automata separating them?\n\nFor i) I believe there is no interesting relationship, cause consider $\\xi_i = 0^i 1 0^{\\omega}$ and $\\eta_i = (0^i1)^{\\omega}$ then $\\xi_i \\sim_i \\eta_i$ for all $i$ and they could always be separated by a two-state automata which stays in the first state as long as it read $0$'s, switches to the second state on the first $1$, and then stays there as long as just $0$'s are read, so accepting $\\xi_i$ but not $\\eta_i$. This lead me to consider the second equivalence relation. Here for example $\\eta_i = (0^i1)^{\\omega}$ and $\\xi_i = 0^i100^i1000^i1\\ldots$, then $\\xi_i \\sim^{\\infty}_i \\eta$ and I guess the minimal automata needed to separate them has $i+2$ states, reading $0$'s in the first state, switch to second state on first $1$, and then a loop which counts the $0$'s (need $i$ states for the loop) and goes back to the second state if $i$ $0$'s are followed by a $1$, so passing the second state an infinite number of times. This automata accepts $\\eta_i$ but not $\\xi_i$.\n\nAre there any lower bounds on the size of an automata separating two words with $\\xi \\sim_k^{\\infty} \\eta$?\n\nAdded: A finite automata accepts an infinite word according to the B\u00fcchi-condition if there is a prescribes set of states such that the infinite words enters some state of this set an infinite number of times, see Wikipedia.\n\nConsider the word $\\xi_i=(0^{2i}1)^\\omega$ and $\\eta_i=(0^{2i+1}1)^\\omega$. Then we have $\\xi_i\\sim_i^\\infty \\eta_i$, but they are separated by an automaton of size $3$ , which accepts as soon as there is a block of $0$'s of even length. Notice that this is a Reachability automaton, there is no need for infinitary condition like B\u00fcchi to separate these words.","date":"2022-05-22 10:48:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.894956648349762, \"perplexity\": 222.078749166861}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662545326.51\/warc\/CC-MAIN-20220522094818-20220522124818-00271.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:Introduction}
The discovery of the Higgs particle, announced on
4 July 2012 by the LHC experiments ATLAS \cite{Aad:2012tfa} and CMS
\cite{Chatrchyan:2012xdj} marked a milestone for
particle physics. It structurally completed the Standard Model (SM)
providing us with a theory that remains weakly interacting all the
way up to the Planck scale. While the SM can successfully describe
numerous particle physics phenomena at the quantum level at highest
precision, it leaves open several questions. Among these are {\it
e.g.}~the one for the nature of Dark Matter (DM), the baryon
asymmetry of the universe or the hierarchy problem. This calls for
physics beyond the SM (BSM). Models beyond the SM usually entail
enlarged Higgs sectors that can provide candidates for Dark Matter or
guarantee successful baryogenesis. Since the discovered Higgs boson
with a mass of 125.09~GeV \cite{Aad:2015zhl} behaves SM-like any BSM extension
has to make sure to contain a Higgs boson in its spectrum that is in
accordance with the LHC Higgs data. Moreover, the models have to be
tested against theoretical and further experimental
constraints from electroweak precision tests, $B$-physics, low-energy
observables and the negative searches for new particles that may be
predicted by some of the BSM theories.
The lack of any direct sign of
new physics renders the investigation of the Higgs sector more and more
important. The precise investigation of the discovered Higgs boson may reveal
indirect signs of new physics through mixing with other Higgs bosons
in the spectrum, loop effects due to the additional Higgs bosons
and/or further new states predicted by the model, or decays into non-SM
states or Higgs bosons, including the possibility of invisible
decays. Due to the SM-like nature of the 125~GeV Higgs boson indirect
new physics effects on its properties are expected to be small. Moreover, different BSM
theories can lead to similar effects in the Higgs sector. In order not
to miss any indirect sign of new physics and to be able to identify the
underlying theory in case of discovery, highest precision in the prediction of the
observables and sophisticated experimental techniques are therefore
indispensable. The former calls for the inclusion of higher-order
corrections at highest possible level, and theorists all over the
world have spent enormous efforts to improve the predictions
for Higgs observables \cite{deFlorian:2016spz}.
Among the new physics models supersymmetric (SUSY) extensions
\cite{Golfand:1971iw,Volkov:1973ix,Wess:1974tw,Fayet:1974pd,Fayet:1976cr,Fayet:1976et,Fayet:1977yc,Nilles:1983ge,Haber:1984rc,Sohnius:1985qm,Gunion:1984yn,Gunion:1986nh,Gunion:1989we}
certainly belong to the best motivated and most thoroughly
investigated models beyond the SM, and numerous higher-order
predictions exist for the production and decay cross sections as well
as the Higgs potential parameters, {\it i.e.}~the masses and Higgs
self-couplings \cite{deFlorian:2016spz}. The Higgs sector of the
minimal supersymmetric extension (MSSM) \cite{Gunion:1989we,Martin:1997ns,Dawson:1997tz,Djouadi:2005gj} is a 2-Higgs doublet model
(2HDM) of type II \cite{Lee:1973iz,Branco:2011iw}. While due to
supersymmetry the MSSM Higgs potential parameters are given in terms
of gauge couplings this is not the
case for general 2HDMs so that the 2HDM entails an interesting and
more diverse Higgs phenomenology and is also
affected differently by
higher-order electroweak (EW) corrections. Moreover, 2HDMs allow for
successful baryogenesis
\cite{McLerran:1990zh,Turok:1990zg,Cohen:1991iu,Turok:1991uc,Funakubo:1993jg,Davies:1994id,Cline:1995dg,Cline:1996mga,Fromme:2006cm,Cline:2011mm,Dorsch:2013wja,Fuyuto:2015vna,Dorsch:2014qja,Dorsch:2016tab,Dorsch:2016nrg,Basler:2016obg,Dorsch:2017nza,Basler:2017uxn,Basler:2018cwe} and in their inert version provide a Dark
Matter candidate
\cite{Deshpande:1977rw,Barbieri:2006dq,LopezHonorez:2006gr,Dolle:2009fn,Honorez:2010re,Gustafsson:2012aj,Swiezewska:2012ej,Swiezewska:2012eh,Arhrib:2013ela,Klasen:2013btp,Abe:2014gua,Krawczyk:2013jta,Goudelis:2013uca,Chakrabarty:2015yia,Ilnicka:2015jba}.
The situation with respect to EW corrections in non-SUSY models is not
as advanced as for SUSY extensions.
While the QCD corrections can be taken over to those
models with a minimum effort from the SM and the MSSM
by applying appropriate changes, this is not the case
for the EW corrections. Moreover, some issues arise with
respect to renormalization. Thus, only recently a renormalization
procedure has been proposed by authors of this paper for the
mixing angles of the 2HDM that ensures
explicitly gauge-independent decay amplitudes,
\cite{Krause:2016gkg,Krause:2016oke}. Subsequent groups have confirmed
this in different Higgs channels
\cite{Denner:2016etu,Altenkamp:2017ldc,Altenkamp:2017kxk,Denner2018,Fox:2017hbw}. Moreover, in Ref.~\cite{Denner2018} four schemes have been proposed based on on-shell and symmetry-inspired renormalization conditions for the mixing angles (and by applying the background field method \cite{Zuber:1975sa,Zuber:1975gh,Boulware:1981,Abbott:1981,Abbott:1982,Hart:1983,Denner:1994xt}) and on $\overline{\mbox{MS}}$ prescriptions for the remaining new quartic Higgs couplings, and their features have been
investigated in detail. The authors of Ref.~\cite{Kanemura:2017wtm} use an improved on-shell
scheme that is essentially equivalent to the mixing angle
renormalization scheme presented by our group in
\cite{Krause:2016gkg,Krause:2016oke}. It has been applied to compute
the renormalized one-loop Higgs boson couplings in the Higgs singlet
model and the 2HDM and to implement these in the program
{\texttt{H-COUP}} \cite{Kanemura:2017gbi}. In
\cite{Kanemura:2018yai} the authors present, for these models, the
one-loop electroweak and QCD corrections to the Higgs decays into
fermion and gauge boson pairs. The complete
phenomenological analysis, however, requires the corrections to all
Higgs decays, as we present them here for the first time in the
computer tool {\texttt{2HDECAY}}.
In \cite{Krause:2016xku} we completed the renormalization of the 2HDM and calculated
the higher-order corrections to Higgs-to-Higgs decays. We have applied
and extended this renormalization procedure in \cite{Krause:2017mal} to the
next-to-2HDM (N2HDM) which includes an additional real singlet. The
computation of the (N)2HDM EW corrections has shown that the corrections
can become very large for certain areas of the parameters space. There can be
several reasons for this. The corrections can be parametrically enhanced
due to involved couplings that can be large
\cite{Kanemura:2002vm,Kanemura:2004mg,Krause:2016xku,Krause:2017mal}. This
is in particular the case for the trilinear Higgs self-couplings that in
contrast to SUSY are not given in terms of the
gauge couplings of the theory and that are so far only weakly
constrained by the LHC Higgs data. The corrections can be large
due to light particles in the loop in combination with not too small
couplings, {\it e.g.}~light Higgs states of the extended Higgs
sector. Also an inapt choice of the renormalization
scheme can artificially enhance loop corrections. Thus we found for our investigated
processes that process-dependent renormalization schemes or
$\overline{\mbox{MS}}$ renormalization of the scalar
mixing angles can blow up the one-loop
corrections due to an insufficient cancellation of the large finite
contributions from wave function renormalization
constants \cite{Krause:2016oke,Krause:2016xku}. Moreover,
counterterms can blow up in certain parameter
regions because of small leading-order couplings, {\it e.g.}~in the 2HDM the coupling of
the heavy non-SM-like CP-even Higgs boson to gauge bosons, which in
the limit of a light SM-like CP-even Higgs boson is almost zero. The
same effects are observed for supersymmetric theories where a
badly chosen parameter set for the renormalization
can lead to very large counterterms and hence enhanced loop
corrections, {\it cf.}~Ref.~\cite{Belanger:2017rgu} for a recent analysis.
This discussion shows that the renormalization of the EW corrections to BSM Higgs
observables is a highly non-trivial task. In addition, there may be no unique
best renormalization scheme for the whole parameter space of a
specific model, and the user has to decide which scheme to choose to
obtain trustworthy predictions. With the publication of the new tool {\tt
2HDECAY} we aim to give an answer to this problematic task.
The program {\tt 2HDECAY} computes, for 17 different renormalization
schemes, the EW corrections to the Higgs decays of the 2HDM Higgs bosons
into all possible on-shell two-particle final states of the model that
are not loop-induced. It is
combined with the widely used Fortran code {\tt HDECAY} version 6.52
\cite{DJOUADI199856,Djouadi:2018xqq} which provides the loop-corrected
decay widths and branching ratios for the
SM, the MSSM and 2HDM incorporating the state-of-the-art higher-order QCD
corrections including also loop-induced and off-shell decays. Through the combination
of these corrections with the
2HDM EW corrections {\tt 2HDECAY} becomes a code for the prediction of
the 2HDM Higgs boson decay widths at the presently highest possible level of
precision. Additionally, the separate output of the leading order (LO)
and next-to-leading order (NLO) EW corrected decay widths allows to
perform studies on the importance of the relative EW corrections (as function
of the parameter choices), comparisons with the relative EW corrections
within the MSSM or investigations on the most suitable renormalization
scheme for specific parameter regions. The comparison
of the results for different renormalization schemes moreover
permits to estimate the remaining theoretical error due to missing
higher-order corrections. To that end, {\texttt{2HDECAY}} includes a parameter conversion routine which performs the automatic conversion of input parameters from one renormalization scheme to another for all 17 renormalization schemes that are implemented. With this tool we contribute to
the effort of improving the theory predictions for BSM Higgs physics
observables so that in combination with sophisticated experimental
techniques Higgs precision physics becomes possible and the gained
insights may advance us in our understanding of the mechanism of
electroweak symmetry breaking (EWSB) and the true underlying theory.
The program package was developed and tested under
{\texttt{Windows 10}}, {\texttt{openSUSE Leap 15.0}} and
{\texttt{macOS Sierra 10.12}}. It requires an up-to-date version of
{\texttt{Python 2}} or {\texttt{Python 3}} (tested with versions
{\texttt{2.7.14}} and {\texttt{3.5.0}}), the {\texttt{FORTRAN}}
compiler {\texttt{gfortran}} and the {\texttt{GNU C}} compilers
{\texttt{gcc}} (tested for compatibility with versions
{\texttt{6.4.0}} and {\texttt{7.3.1}}) and {\texttt{g++}}. The latest
version of the package can be downloaded from
\begin{center}
\href{https://github.com/marcel-krause/2HDECAY}{https://github.com/marcel-krause/2HDECAY} ~.
\end{center}
The paper is organized as follows. The subsequent
Sec.\,\ref{sec:EWQCD2HDMMain} forms the theoretical background for our
work. We briefly introduce the 2HDM, all
relevant parameters and particles and set our notation.
We give a summary of all counterterms that are needed for the computation
of the EW corrections and state them explicitly. The relevant formulae
for the computation of the partial decay widths at one-loop level are
presented and the combination of the electroweak corrections with the
QCD corrections already implemented in {\texttt{HDECAY}} is described. In
Sec.\,\ref{sec:programDescriptionMain}, we introduce
{\texttt{2HDECAY}} in detail, describe the structure of the program
package and the input and output file formats. Additionally, we
provide installation and usage manuals. We conclude with a short
summary of our work in Sec.\,\ref{sec:summary}. As reference for the
user, we list exemplary input and output files in Appendices
\ref{sec:AppendixInputFile} and \ref{sec:AppendixOutputFile},
respectively.
\section{One-Loop Electroweak and QCD Corrections in the 2HDM}
\label{sec:EWQCD2HDMMain}
In the following, we briefly set up our notation and introduce the
2HDM along with the input parameters used in our
parametrization. We give details on the EW one-loop
renormalization of the 2HDM. We discuss how the calculation of the
one-loop partial decay widths is performed. At the end of the section,
we explain how the EW corrections are combined with the existing
state-of-the-art QCD corrections already implemented in {\texttt{HDECAY}} and present the automatic parameter conversion routine that is implemented in {\texttt{2HDECAY}}.
\subsection{Introduction of the 2HDM}
\label{sec:setupOfModel}
For our work, we consider a general CP-conserving 2HDM
\cite{Lee:1973iz,Branco:2011iw} with a global discrete $\mathbb{Z}_2$
symmetry that is softly broken. The model consists of two complex
$SU(2)_L$ doublets $\Phi _1$ and $\Phi _2$, both with hypercharge
$Y=+1$. The electroweak part of the 2HDM can be described by the
Lagrangian
\begin{equation}
\mathcal{L} ^\text{EW}_\text{2HDM} = \mathcal{L} _\text{YM} + \mathcal{L} _\text{F} + \mathcal{L} _\text{S} + \mathcal{L} _\text{Yuk} + {\mathcal{L}} _\text{GF} + \mathcal{L} _\text{FP}
\label{eq:electroweakLagrangian}
\end{equation}
in terms of the Yang-Mills Lagrangian $\mathcal{L} _\text{YM}$ and the fermion
Lagrangian $\mathcal{L} _\text{F}$ containing the kinetic terms of the
gauge bosons and fermions and their interactions, the Higgs Lagrangian
$\mathcal{L} _\text{S}$, the Yukawa
Lagrangian $\mathcal{L}_{\text{yuk}}$ with the Higgs-fermion
interactions, the gauge-fixing and the Fadeev-Popov Lagrangian,
${\mathcal{L}} _\text{GF}$ and $\mathcal{L}_\text{FP}$,
respectively. Explicit forms of $\mathcal{L} _\text{YM}$ and
$\mathcal{L} _\text{F}$ can be found {\it
e.g.}~in~\cite{Peskin:1995ev, Denner:1991kt} and of the general 2HDM
Yukawa Lagrangian {\it e.g.}~in \cite{Aoki:2009ha, Branco:2011iw}. We
do not give them explicitly here.
For the renormalization of the 2HDM, we follow the approach of
Ref.~\cite{Santos:1996vt} and apply the gauge-fixing only \textit{after}
the renormalization of the theory, {\it i.e.}~$\mathcal{L} _\text{GF}$
contains only fields that are already renormalized. For the purpose of
our work we do not present ${\mathcal{L}} _\text{GF}$ nor $\mathcal{L}_\text{FP}$ since
their explicit forms are not needed in the following.
The scalar Lagrangian $\mathcal{L} _\text{S}$ introduces the kinetic
terms of the Higgs doublets and their scalar potential. With the
the covariant derivative
\begin{equation}
D_\mu = \partial _\mu + \frac{i}{2} g \sum _{a=1}^3 \sigma ^a W_\mu ^a + \frac{i}{2} g{'} B_\mu
\end{equation}
where $W_\mu ^a$ and $B_\mu$ are the gauge bosons of the $SU(2)_L$ and
$U(1)_Y$ respectively, $g$ and $g{'}$ are the corresponding coupling
constants of the gauge groups and $\sigma ^a$ are the Pauli matrices,
the scalar Lagrangian is given by
\begin{equation}
\mathcal{L} _S = \sum _{i=1}^2 (D_\mu \Phi _i)^\dagger (D^\mu \Phi _i) - V_\text{2HDM} ~.
\label{eq:scalarLagrangian}
\end{equation}
The scalar potential of the CP-conserving 2HDM reads
\cite{Branco:2011iw}
\begin{equation}
\begin{split}
V_\text{2HDM} =&~ m_{11}^2 \left| \Phi _1 \right| ^2 + m_{22}^2 \left|
\Phi _2 \right| ^2 - m_{12}^2 \left( \Phi _1 ^\dagger \Phi _2 +
\textit{h.c.} \right) + \frac{\lambda _1}{2} \left( \Phi _1^\dagger
\Phi _1 \right) ^2 + \frac{\lambda _2}{2} \left( \Phi _2^\dagger
\Phi _2 \right) ^2 \\
&+ \lambda _3 \left( \Phi _1^\dagger \Phi _1 \right) \left( \Phi
_2^\dagger \Phi _2 \right) + \lambda _4 \left( \Phi _1^\dagger \Phi
_2 \right) \left( \Phi _2^\dagger \Phi _1 \right) + \frac{\lambda
_5}{2} \left[ \left( \Phi _1^\dagger \Phi _2 \right) ^2 +
\textit{h.c.} \right] ~.
\end{split}
\label{eq:scalarPotential}
\end{equation}
Since we consider a CP-conserving model, the 2HDM potential can be
parametrized by three real-valued mass parameters $m_{11}$, $m_{22}$
and $m_{12}$ as well as five real-valued dimensionless coupling
constants $\lambda _i$ ($i=1,...,5$). For later convenience, we define
the frequently appearing combination of three of these coupling
constants as
\begin{equation}
\lambda _{345} \equiv \lambda _3 + \lambda _4 + \lambda _5 ~.
\end{equation}
For $m_{12}^2=0$, the potential $V_\text{2HDM}$ exhibits a discrete $\mathbb{Z}_2$
symmetry under the simultaneous field transformations $\Phi _1
\rightarrow - \Phi _1$ and $\Phi _2 \rightarrow \Phi
_2$. This
symmetry, implemented in the scalar potential in order to avoid
flavour-changing neutral currents (FCNC) at tree level, is softly
broken by a non-zero mass parameter $m_{12}$.
After EWSB the neutral components of
the Higgs doublets develop vacuum expectation values (VEVs) which are
real in the CP-conserving case. After expanding about the real VEVs $v_1$
and $v_2$, the Higgs doublets $\Phi_i$ ($i=1,2$) can be expressed in
terms of the charged complex field $\omega_i^+$ and the real neutral
CP-even and CP-odd fields $\rho_i$ and $\eta_i$, respectively as
\begin{equation}
\Phi _1 = \begin{pmatrix} \omega ^+ _1 \\ \frac{v_1 + \rho _1 + i \eta
_1 }{\sqrt{2}} \end{pmatrix} ~~~\text{and}~~~~ \Phi _2 = \begin{pmatrix}
\omega ^+ _2 \\ \frac{v_2 + \rho _2 + i \eta
_2}{\sqrt{2}} \end{pmatrix}
\label{eq:vevexpansion}
\end{equation}
where
\begin{equation}
v^2 = v_1^2 + v_2^2 \approx (246.22~\text{GeV})^2
\label{eq:vevRelations}
\end{equation}
is the SM VEV obtained from the Fermi constant
$G_F$ and we define the ratio of the VEVs through the mixing angle $\beta$
as
\beq
\tan \beta = \frac{v_2}{v_1}
\eeq
so that
\beq
v_1 = v c_\beta \quad \mbox{and} \quad v_2 = v s_\beta~.
\eeq
Insertion of Eq.~(\ref{eq:vevexpansion}) in the kinetic part of the scalar
Lagrangian in Eq.~(\ref{eq:scalarLagrangian}) yields after rotation to
the mass eigenstates the tree-level relations for the masses of the
electroweak gauge bosons
\begin{align}
m_W^2 &= \frac{g^2v^2}{4} \\
m_Z^2 &= \frac{(g^2 + g{'}^2)v^2}{4} \\
m_\gamma ^2 &= 0 ~.
\end{align}
The electromagnetic coupling constant $e$ is connected to the
fine-structure constant $\alpha _\text{em}$ and to the gauge boson
coupling constants through the tree-level relation
\begin{equation}
e = \sqrt{4\pi \alpha _\text{em} } = \frac{g g{'}}{\sqrt{g^2 + g{'} ^2}}
\label{eq:electromagneticCouplingDefinition}
\end{equation}
which allows to replace $g{'}$ in favor of $e$ or
$\alpha_\text{em}$. In our work, we use the fine-structure constant $\alpha
_\text{em}$ as an independent input. Alternatively, one could use the
tree-level relation to the Fermi constant
\begin{equation}
G_F \equiv \frac{\sqrt{2} g^2}{8m_W^2} = \frac{\alpha _\text{em} \pi
}{\sqrt{2} m_W^2 \left( 1 - \frac{m_W^2}{m_Z^2} \right)}
\label{eq:definitionFermiConstant}
\end{equation}
to replace one of the parameters of the electroweak sector in favor of
$G_F$. Since $G_F$ is used as an input value for {\texttt{HDECAY}}, we
present the formula here explicitly and explain the conversion between
the different parametrizations in Sec.\,\ref{sec:connectionHDECAY}.
Inserting Eq.~(\ref{eq:vevexpansion}) in the scalar potential in
\eqref{eq:scalarPotential} leads to
\begin{equation}
\begin{split}
V_\text{2HDM} =&~ \frac{1}{2} \left( \rho _1 ~~ \rho _2 \right) M_\rho ^2 \begin{pmatrix} \rho _1 \\ \rho _2 \end{pmatrix} + \frac{1}{2} \left( \eta _1 ~~ \eta _2 \right) M_\eta ^2 \begin{pmatrix} \eta _1 \\ \eta _2 \end{pmatrix} + \frac{1}{2} \left( \omega ^\pm _1 ~~ \omega ^\pm _2 \right) M_\omega ^2 \begin{pmatrix} \omega ^\pm _1 \\ \omega ^\pm _2 \end{pmatrix} \\
& + T_1 \rho _1 + T_2 \rho _2 + ~~ \cdots
\end{split}
\label{eq:scalarPotentialMultilinearFields}
\end{equation}
where the terms $T_1$ and $T_2$ and the matrices $M_\omega ^2$,
$M_\rho ^2$ and $M_\eta ^2$ are defined below. By requiring the
VEVs of \eqref{eq:vevexpansion} to
represent the minimum of the potential, the minimum conditions for the
potential can be expressed as
\begin{equation}
\frac{\partial V_\text{2HDM}}{\partial \Phi _i} \Bigg| _{\left\langle \Phi _j \right\rangle} = 0~.
\end{equation}
This is equivalent to the statement that the two terms linear in the
CP-even fields $\rho _1$ and $\rho _2$, the tadpole terms,
\begin{align}
\frac{T_1}{v_1} &\equiv m_{11}^2 - m_{12}^2 \frac{v_2}{v_1} + \frac{v_1^2 \lambda _1}{2} + \frac{v_2 ^2 \lambda _{345}}{2} \label{eq:tadpoleCondition1} \\
\frac{T_2}{v_2} &\equiv m_{22}^2 - m_{12}^2 \frac{v_1}{v_2} + \frac{v_2^2 \lambda _2}{2} + \frac{v_1 ^2 \lambda _{345}}{2} \label{eq:tadpoleCondition2}
\end{align}
have to vanish at tree level:
\begin{equation}
T_1 = T_2 = 0 ~~~ (\text{at tree level}) ~.
\label{eq:tadpoleVanishAtTreelevel}
\end{equation}
The tadpole equations can be solved for $m_{11}^2$ and $m_{22}^2$ in
order to replace these two parameters by the tadpole parameters $T_1$
and $T_2$.
The terms bilinear in the fields given in
\eqref{eq:scalarPotentialMultilinearFields} define the scalar mass
matrices
\begin{align}
M_\rho ^2 &\equiv \begin{pmatrix}
m_{12}^2 \frac{v_2}{v_1} + \lambda _1 v_1^2 & -m_{12}^2 + \lambda _{345} v_1 v_2 \\ -m_{12}^2 + \lambda _{345} v_1 v_2 & m_{12}^2 \frac{v_1}{v_2} + \lambda _2 v_2^2
\end{pmatrix} + \begin{pmatrix}
\frac{T_1}{v_1} & 0 \\ 0 & \frac{T_2}{v_2}
\end{pmatrix} \label{eq:massMatrices1} \\
M_\eta ^2 &\equiv \left( \frac{m_{12}^2}{v_1v_2} - \lambda _5 \right) \begin{pmatrix}
v_2^2 & -v_1 v_2 \\ -v_1 v_2 & v_1 ^2
\end{pmatrix} + \begin{pmatrix}
\frac{T_1}{v_1} & 0 \\ 0 & \frac{T_2}{v_2}
\end{pmatrix} \\
M_\omega ^2 &\equiv \left( \frac{m_{12}^2}{v_1v_2} - \frac{\lambda _4 + \lambda _5}{2} \right) \begin{pmatrix}
v_2^2 & -v_1 v_2 \\ -v_1 v_2 & v_1 ^2
\end{pmatrix} + \begin{pmatrix}
\frac{T_1}{v_1} & 0 \\ 0 & \frac{T_2}{v_2}
\end{pmatrix} \label{eq:massMatrices3}
\end{align}
where Eqs.\,(\ref{eq:tadpoleCondition1}) and
(\ref{eq:tadpoleCondition2}) have already been applied to replace the
parameters $m_{11}^2$ and $m_{22}^2$ in favor of $T_1$ and
$T_2$. Keeping the latter explicitly in the expressions of the
mass matrices is crucial for the correct renormalization of the scalar
sector, as explained in Sec.\,\ref{sec:renormalization2HDM}. By means
of two mixing angles $\alpha$ and $\beta$ which define the rotation
matrices\footnote{Here and in the following, we use the short-hand
notation $s_x \equiv \sin (x)$, $c_x \equiv \cos (x)$, $t_x \equiv
\tan (x)$ for convenience.}
\begin{equation}
R (x) \equiv \begin{pmatrix} c_x & - s_x \\ s_x & c_x \end{pmatrix}
\end{equation}
the fields $\omega ^+ _i$, $\rho _i$ and $\eta _i$ are rotated to the mass basis according to
\begin{align}
\begin{pmatrix} \rho _1 \\ \rho _2 \end{pmatrix} &= R(\alpha ) \begin{pmatrix} H \\ h \end{pmatrix} \label{eq:rotationCPEven} \\
\begin{pmatrix} \eta _1 \\ \eta _2 \end{pmatrix} &= R(\beta ) \begin{pmatrix} G^0 \\ A \end{pmatrix} \\
\begin{pmatrix} \omega ^\pm _1 \\ \omega ^\pm _2 \end{pmatrix} &= R(\beta ) \begin{pmatrix} G^\pm \\ H^\pm \label{eq:rotationCharged} \end{pmatrix}
\end{align}
with the two CP-even Higgs bosons $h$ and $H$, the CP-odd Higgs boson
$A$, the CP-odd Goldstone boson $G^0$ and the charged Higgs bosons
$H^\pm$ as well as the charged Goldstone bosons $G^\pm$. In the mass
basis, the diagonal mass matrices are given by
\begin{align}
D_\rho ^2 &\equiv \begin{pmatrix} m_H^2 & 0 \\ 0 & m_h^2 \end{pmatrix} \\
D_\eta ^2 &\equiv \begin{pmatrix} m_{G^0}^2 & 0 \\ 0 & m_A^2 \end{pmatrix} \\
D_\omega ^2 &\equiv \begin{pmatrix} m_{G^\pm}^2 & 0 \\ 0 & m_{H^\pm}^2 \end{pmatrix}
\end{align}
with the diagonal entries representing the squared masses of the
respective particles. The Goldstone bosons are massless,
\begin{equation}
m_{G^0}^2 = m_{G^\pm}^2 = 0~.
\end{equation}
The squared masses expressed in terms of the potential parameters and
the mixing angle $\alpha$ can be cast into the form~\cite{Kanemura:2004mg}
\begin{align}
m_H^2 &= c_{\alpha - \beta}^2 M_{11}^2 + s_{2(\alpha - \beta )} M_{12}^2 + s_{\alpha - \beta } ^2 M_{22}^2 \label{eq:parameterTransformationInteractionToMass1} \\
m_h^2 &= s_{\alpha - \beta}^2 M_{11}^2 - s_{2(\alpha - \beta )} M_{12}^2 + c_{\alpha - \beta } ^2 M_{22}^2 \\
m_A^2 &= \frac{m_{12}^2}{s_\beta c_\beta} - v^2 \lambda _5 \\
m_{H^\pm } ^2 &= \frac{m_{12}^2}{s_\beta c_\beta} - \frac{v^2}{2}
\left( \lambda _4 + \lambda _5 \right) \\
t_{2(\alpha - \beta ) } &= \frac{2M_{12}^2}{M_{11}^2 - M_{22}^2} \label{eq:parameterTransformationInteractionToMass5}
\end{align}
where we have introduced
\begin{align}
M_{11}^2 &\equiv v^2 \left( c_\beta ^4 \lambda _1 + s_\beta ^4 \lambda _2 + 2 s_\beta ^2 c_\beta ^2 \lambda _{345} \right) \\
M_{12}^2 &\equiv s_\beta c_\beta v^2 \left( - c_\beta ^2 \lambda _1 + s_\beta ^2 \lambda _2 + c_{2\beta } \lambda _{345} \right) \\
M_{22}^2 &\equiv \frac{m_{12}^2}{s_\beta c_\beta } + \frac{v^2}{8} \left( 1 - c_{4\beta } \right) \left( \lambda _1 + \lambda _2 - 2\lambda _{345} \right) ~.
\end{align}
Inverting these relations, the quartic couplings $\lambda _i$
($i=1,...,5$) can be expressed in terms of the mass
parameters $m_h^2$, $m_H^2$, $m_A^2$, $m_{H^\pm}^2$ and the CP-even
mixing angle $\alpha$ as \cite{Kanemura:2004mg}
\begin{align}
\lambda _1 &= \frac{1}{v^2 c_\beta ^2} \left( c_\alpha ^2 m_H^2 + s_\alpha ^2 m_h^2 - \frac{s_\beta}{c_\beta} m_{12}^2 \right) \label{eq:parameterTransformationMassToInteraction1} \\
\lambda _2 &= \frac{1}{v^2 s_\beta ^2} \left( s_\alpha ^2 m_H^2 + c_\alpha ^2 m_h^2 - \frac{c_\beta}{s_\beta} m_{12}^2 \right) \\
\lambda _3 &= \frac{2m_{H^\pm}^2}{v^2} + \frac{s_{2\alpha}}{s_{2\beta} v^2} \left( m_H^2 - m_h^2\right) - \frac{m_{12}^2}{s_\beta c_\beta v^2} \\
\lambda _4 &= \frac{1}{v^2} \left( m_A^2 - 2m_{H^\pm}^2 + \frac{m_{12}^2}{s_\beta c_\beta} \right) \\
\lambda _5 &= \frac{1}{v^2} \left( \frac{m_{12}^2}{s_\beta c_\beta} - m_A^2 \right) ~. \label{eq:parameterTransformationMassToInteraction5}
\end{align}
In order to avoid tree-level FCNC currents, as introduced by the most
general 2HDM Yukawa Lagrangian, one type of fermions is allowed to
couple only to one Higgs doublet by imposing a global $\mathbb{Z}_2$
symmetry under which $\Phi_{1,2} \to \mp \Phi_{1,2}$. Depending on the
$\mathbb{Z}_2$ charge assignments, there are four phenomenologically
different types of 2HDMs summarized in Tab.\,\ref{tab:yukawaDefinitions}.
\begin{table}[tb]
\centering
\begin{tabular}{ c c c c }
\hline
& $u$-type & $d$-type & leptons \\ \hline
I & $\Phi _2$ & $\Phi _2$ & $\Phi _2$ \\
II & $\Phi _2$ & $\Phi _1$ & $\Phi _1$ \\
lepton-specific & $\Phi _2$ & $\Phi _2$ & $\Phi _1$ \\
flipped & $\Phi _2$ & $\Phi _1$ & $\Phi _2$ \\
\hline
\end{tabular}
\caption{The four Yukawa types of the $\mathbb{Z}_2$-symmetric
2HDM defined by the Higgs doublet that couples to each kind of fermions.}
\label{tab:yukawaDefinitions}
\end{table}
For the four 2HDM types considered in this work, all Yukawa couplings can be
parametrized through six different Yukawa coupling parameters $Y_i$
($i=1,...,6$) whose values for the different types are presented
in Tab.\,\ref{tab:yukawaCouplings}. They are introduced here for later
convenience.
\begin{table}[tb]
\centering
\begin{tabular}{ c c c c c c c }
\hline
2HDM type & $Y_1$ & $Y_2$ & $Y_3$ & $Y_4$ & $Y_5$ & $Y_6$ \\ \hline
I & $\frac{c_\alpha }{s_\beta }$ & $\frac{s_\alpha }{s_\beta }$ & $-\frac{1}{t_\beta}$ & $\frac{c_\alpha }{s_\beta }$ & $\frac{s_\alpha }{s_\beta }$ & $-\frac{1}{t_\beta}$ \\
II & $-\frac{s_\alpha }{c_\beta }$ & $\frac{c_\alpha }{c_\beta }$ & $t_\beta $ & $-\frac{s_\alpha }{c_\beta }$ & $\frac{c_\alpha }{c_\beta }$ & $t_\beta $ \\
lepton-specific & $\frac{c_\alpha }{s_\beta }$ & $\frac{s_\alpha }{s_\beta }$ & $-\frac{1}{t_\beta}$ & $-\frac{s_\alpha }{c_\beta }$ & $\frac{c_\alpha }{c_\beta }$ & $t_\beta $ \\
flipped & $-\frac{s_\alpha }{c_\beta }$ & $\frac{c_\alpha }{c_\beta }$ & $t_\beta $ & $\frac{c_\alpha }{s_\beta }$ & $\frac{s_\alpha }{s_\beta }$ & $-\frac{1}{t_\beta}$ \\
\hline
\end{tabular}
\caption{Parametrization of the Yukawa coupling parameters in
terms of six parameters $Y_i$ ($i=1,...,6$)
for each 2HDM type.}
\label{tab:yukawaCouplings}
\end{table}
We conclude this section with an overview over the full set of
independent parameters that is used as input for the computations in
{\texttt{2HDECAY}}. Additionally to the parameters defined by
${\mathcal L}_{\text{2HDM}}^{\text{EW}}$, {\texttt{HDECAY}} requires
the electromagnetic coupling constant
$\alpha_{\text{em}}$ in the Thomson limit for the calculation of
the loop-induced decays into a photon pair and into $Z\gamma$, the
strong coupling constant
$\alpha _s$ for the loop-induced decay into gluons
and the QCD corrections as well as the total decay widths of
the $W$ and $Z$ bosons, $\Gamma _W$ and $\Gamma _Z$, for the
computation of the off-shell decays into
massive gauge boson final states. In the mass basis of the
scalar sector, the set of independent parameters is given by
\begin{equation}
\{ G_F, \alpha _s , \Gamma _W , \Gamma _Z , \alpha _\text{em} , m_W ,
m_Z , m_f, V_{ij} , t_\beta , m_{12}^2 , \alpha , m_h , m_H , m_A ,
m_{H^\pm} \} ~.
\label{eq:inputSetMassBase}
\end{equation}
Here $m_f$ denote the fermion masses of the strange, charm, bottom and
top quarks and of the $\mu$ and $\tau$ leptons
($f=s,c,b,t,\mu,\tau$). All other fermion
masses are assumed to be zero in {\texttt{HDECAY}} and will also be
assumed to be zero in our computation of the EW corrections to the
decay widths. The fermion and gauge boson masses are defined in
accordance with the recommendations of the LHC Higgs cross section
working group \cite{Denner:2047636}. The $V_{ij}$ denote the CKM
mixing matrix elements. All {\texttt{HDECAY}} decay widths are
computed in terms of the Fermi constant $G_F$
except for processes involving on-shell external photon vertices that are
expressed by $\alpha_\text{em}$ in the Thomson limit. In the
computation of the EW corrections, however, we require the on-shell
masses $m_W$ and $m_Z$ and the electromagnetic coupling at
the $Z$ boson mass scale,
$\alpha_{\text{em}}(m_Z^2)$ (not to be confused with the mixing angle
$\alpha$ in the Higgs sector), as input parameters for our
renormalization conditions. We will come back to this
point later.
Alternatively, the original parametrization of the scalar
potential in the interaction basis can be
used\footnote{{\texttt{HDECAY}} internally translates the parameters
from the interaction
to the mass basis, in terms of which the decay widths are
implemented.}. In this case, the set of
independent parameters is given by
\begin{equation}
\{ G_F, \alpha _s , \Gamma _W , \Gamma _Z , \alpha _\text{em} , m_W , m_Z ,
m_f, V_{ij} , t_\beta , m_{12}^2 , \lambda _1 , \lambda _2 , \lambda _3
, \lambda _4 , \lambda _5 \} ~.
\label{eq:inputSetInteractionBase}
\end{equation}
However, we want to emphasize that the automatic parameter conversion routine in {\texttt{2HDECAY}} is only performed when the parameters are given in the mass basis of \eqref{eq:inputSetMassBase}.
Actually, also the tadpole parameters $T_1$ and $T_2$ should be
included in the two sets as independent parameters of the Higgs
potential. However, as described in
Sec.\,\ref{sec:renormalization2HDM}, the treatment of the minimum of
the Higgs potential at higher orders requires special care, and in an
alternative treatment of the minimum conditions, the tadpole
parameters disappear as independent parameters. In any case, after the
renormalization procedure is completely performed, the tadpole
parameters vanish again and hence, do not count as input parameters
for {\texttt{2HDECAY}}.
\subsection{Renormalization}
\label{sec:renormalization2HDM}
We focus on the calculation of EW one-loop corrections to decay widths
of Higgs particles in the 2HDM. Since the higher-order (HO)
corrections of these decay widths are in general ultraviolet (UV)-divergent, a
proper regularization and renormalization of the UV divergences is
required. In the following, we briefly present the definition of the
counterterms (CTs) needed for the calculation of the EW one-loop
corrections. For a thorough derivation and presentation of the
gauge-independent renormalization of the 2HDM, we refer the reader to
\cite{Krause:2016gkg, Krause:2016oke, Krause:2016xku}.\footnote{See
also Refs.~\cite{Denner:2016etu,Altenkamp:2017ldc,Denner2018}
for a discussion of the renormalization of the 2HDM. For recent
works discussing gauge-independent renormalization within multi-Higgs
models, see \cite{Fox:2017hbw,Grimus:2018rte}.}
All input parameters that are renormalized for the calculation of the
EW corrections (apart from the mixing angles $\alpha $ and
$\beta$ and the soft-$\mathbb{Z}_2$-breaking scale $m_{12}$) are
renormalized in the on-shell (OS) scheme. For the physical fields,
we employ the conditions that any mixing of fields with the same
quantum numbers shall vanish on the mass shell of the respective
particles and that the fields are normalized by fixing the
residue of their corresponding propagators at their poles to unity. Mass
CTs are fixed through the condition that the masses are defined as the
real parts of the poles of the renormalized propagators. These OS
conditions suffice to renormalize most of the parameters of the 2HDM
necessary for our work. The renormalization of the mixing angles
$\alpha$ and $\beta$ follows an OS-motivated approach, as discussed in
Sec.\,\ref{sec:renormalizationMixingAngles}, while $m_{12}$ is
renormalized via an $\overline{\text{MS}}$ condition as discussed in
Sec.\,\ref{sec:renormalizationSoftm12Squared}.
\subsubsection{Renormalization of the Tadpoles}
\label{sec:renormalizationTadpoles}
As shown for the 2HDM for the first time in \cite{Krause:2016gkg,
Krause:2016oke}, the proper treatment of the
tadpole terms at one-loop order is crucial for the
gauge-independent definition of the CTs of the mixing angles $\alpha$
and $\beta$. This allows for the calculation of one-loop partial decay widths with
a manifestly gauge-independent relation between input variables and
the physical observable.
In the following, we briefly repeat the different renormalization
conditions for the tadpoles that can be employed in the 2HDM.
The \textit{standard tadpole scheme} is a commonly used
renormalization scheme for the tadpoles ({\it cf.}~{\it e.g.}~\cite{Denner:1991kt}
for the SM or \cite{Kanemura:2004mg, Kanemura:2015mxa} for the
2HDM). While the tadpole parameters vanish at tree level, as stated in
\eqref{eq:tadpoleVanishAtTreelevel}, they are in general non-vanishing
at higher orders in perturbation theory. Since the tadpole terms,
being the terms linear in the Higgs potential, define the minimum
of the potential, it is necessary to employ a renormalization of the
tadpoles in such a way that the ground state of the potential still
represents the minimum at higher orders. In the standard tadpole
scheme, this condition is imposed on the loop-corrected potential. By
replacing the tree-level tadpole terms at one-loop order with the
physical ({\it i.e.}~renormalized) tadpole terms and the tadpole CTs $\delta
T _i$,
\begin{equation}
T_i ~\rightarrow ~ T_i + \delta T_i ~~~ (i = 1,2)
\end{equation}
the correct minimum of the loop-corrected potential
is obtained by demanding the renormalized tadpole
terms $T_i$ to vanish. This directly connects the tadpole CTs $\delta
T_i$ with the corresponding one-loop tadpole diagrams,
\begin{equation}
i\delta T_{H/h} = \mathord{ \left(\rule{0cm}{30px}\right. \vcenter{
\hbox{ \includegraphics[height=57px , trim = 16.6mm 12mm 16.6mm
10mm, clip]{TadpoleDiagramHiggsBasis.pdf} } }
\left.\rule{0cm}{30px}\right) }
\label{eq:tadpoleCountertermDefinition}
\end{equation}
where we switched the tadpole terms from the interaction basis to the
mass basis by means of the rotation matrix $R(\alpha )$, as indicated
in \eqref{eq:rotationCPEven}. Since the tadpole terms explicitly
appear in the mass matrices in
Eqs.\,(\ref{eq:massMatrices1})-(\ref{eq:massMatrices3}), their CTs
explicitly appear in the mass matrices at one-loop
order. The rotation from the interaction to the mass basis yields
nine tadpole CTs in total which depend
on the two tadpole CTs $\delta T_{H/h}$ defined by the one-loop
tadpole diagrams in \eqref{eq:tadpoleCountertermDefinition}:
\begin{mdframed}[frametitle={Renormalization of the tadpoles (standard scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt]\begin{align}
\delta T_{HH} &= \frac{c _\alpha ^3 s _\beta + s _\alpha ^3 c _\beta }{vs _\beta c _\beta } \delta T_{H} - \frac{s_{2\alpha} s_{\beta - \alpha} }{vs_{2\beta} } \delta T_{h} \label{eq:RenormalizationRadpolesTadpoleCountertermDeltaTH0H0ExplicitForm} \\
\delta T_{Hh} &= -\frac{s _{2\alpha} s _{\beta - \alpha} }{vs_{2\beta}} \delta T_{H} + \frac{s _{2\alpha} c _{\beta - \alpha} }{vs_{2\beta}} \delta T_{h} \\
\delta T_{hh} &= \frac{s_{2\alpha} c_{\beta - \alpha} }{vs_{2\beta}} \delta T_{H} - \frac{s_\alpha ^3 s _\beta - c _\alpha ^3 c _\beta }{vs_\beta c_\beta } \delta T_{h} \\
\delta T_{G^0G^0} &= \frac{c_{\beta -\alpha} }{v} \delta T_{H} + \frac{s _{\beta - \alpha} }{v} \delta T_{h} \label{eq:RenormalizationRadpolesTadpoleCountertermDeltaTG0G0ExplicitForm} \\
\delta T_{G^0A} &= -\frac{s_{\beta - \alpha} }{v} \delta T_{H} + \frac{c _{\beta - \alpha} }{v} \delta T_{h} \\
\delta T_{AA} &= \frac{c_\alpha s_\beta ^3 + s _\alpha c_\beta ^3 }{vs_\beta c_\beta } \delta T_{H} - \frac{s_\alpha s_\beta ^3 - c_\alpha c_\beta ^3 }{vs_\beta c_\beta } \delta T_{h} \label{eq:RenormalizationRadpolesTadpoleCountertermDeltaTA0A0ExplicitForm} \\
\delta T_{G^\pm G^\pm } &= \frac{c_{\beta -\alpha} }{v}
\delta T_{H} + \frac{s _{\beta -
\alpha} }{v} \delta T_{h} \\
\delta T_{G^\pm H^\pm } &= -\frac{s_{\beta - \alpha} }{v}
\delta T_{H} + \frac{c _{\beta -
\alpha} }{v} \delta T_{h} \\
\delta T_{H^\pm H^\pm } &= \frac{c_\alpha s_\beta ^3
+ s _\alpha c_\beta ^3
}{vs_\beta c_\beta }
\delta T_{H} - \frac{s_\alpha
s_\beta ^3 - c_\alpha
c_\beta ^3 }{vs_\beta
c_\beta } \delta
T_{h}
\label{eq:RenormalizationRadpolesTadpoleCountertermDeltaTHpHpExplicitForm}
\end{align}\end{mdframed}
Since the minimum of the potential is defined through the
loop-corrected scalar potential, which in general is a gauge-dependent
quantity, the CTs defined through this minimum ({\it e.g.}~the CTs of
the scalar or gauge boson masses) become manifestly gauge-dependent
themselves. This is no problem as long as all gauge dependences
arising in a fixed-order calculation cancel against each other. In the
2HDM, however, an improper renormalization condition for the mixing
angle CTs within the standard tadpole scheme can lead to uncanceled
gauge dependences in the calculation of partial decay widths. This is
discussed in more detail in
Sec.\,\ref{sec:renormalizationMixingAngles}. Apart from the appearance
of the tadpole diagrams in
Eqs.\,(\ref{eq:RenormalizationRadpolesTadpoleCountertermDeltaTH0H0ExplicitForm})-(\ref{eq:RenormalizationRadpolesTadpoleCountertermDeltaTHpHpExplicitForm}),
and subsequently in the CTs and the wave function renormalization
constants (WFRCs) defined through these, the renormalization condition
in \eqref{eq:tadpoleCountertermDefinition} ensures that all other
appearances of tadpoles are canceled in the one-loop calculation,
{\it i.e.}~tadpole diagrams in the self-energies or vertex corrections
do not have to be taken into account.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth, trim=2.2cm 2.2cm 0cm 2.2cm, clip]{Propagators.pdf}
\caption{Generic definition of the self-energies $\Sigma$ and $\Sigma
^\text{tad}$ as function of the external momentum $p^2$
used in our CT definitions of the 2HDM. While $\Sigma$
is the textbook definition of the one-particle irreducible
self-energy, the self-energy $\Sigma ^\text{tad}$ additionally
contains tadpole diagrams, indicated by the gray
blob. For the actual calculation, the full particle content of the 2HDM has to be
inserted into the self-energy topologies depicted here.}
\label{fig:definitionOfSelfenergies}
\end{figure}
An alternative treatment of the tadpole renormalization was proposed
by J. Fleischer and F. Jegerlehner in the SM
\cite{PhysRevD.23.2001}. It was applied to the extended scalar sector
of the 2HDM for the first time in \cite{Krause:2016gkg,
Krause:2016oke} and is called \textit{alternative (FJ) tadpole
scheme} in the following. In this alternative approach, the VEVs
$v_{1,2}$ are considered as the fundamental quantities instead of the
tadpole terms. The \textit{proper} VEVs are the renormalized all-order
VEVs of the Higgs fields which represent the true ground state of the
theory and which are connected to the particle masses and the
couplings of the electroweak sector. Since the alternative approach
relies on the minimization of the gauge-independent tree-level scalar
potential, the mass CTs defined in this framework become manifestly
gauge-independent quantities by themselves. Moreover, the alternative
tadpole scheme connects the all-order renormalized VEVs directly to
the corresponding tree-level VEVs. Since the tadpoles are not the
fundamental quantities of the Higgs minimum in this framework, they do
not receive CTs. Instead, CTs for the VEVs are introduced by replacing
the VEVs with the renormalized VEVs and their CTs,
\begin{equation}
v_i ~ \rightarrow ~ v_i + \delta v_i
\end{equation}
and by fixing the latter in such a way that it is ensured that the
renormalized VEVs represent the proper tree-level minima to all
orders. At one-loop level, this leads to the following connection
between the VEV CTs in the interaction basis and the one-loop tadpole
diagrams in the mass basis,
\begin{equation}
\delta v_1 = \frac{-i c_\alpha }{m_{H}^2} \mathord{
\left(\rule{0cm}{30px}\right. \vcenter{
\hbox{ \includegraphics[height=57px , trim = 18mm 12mm 17.4mm
10mm, clip]{TadpoleDiagramHiggsBasisHH.pdf} } }
\left.\rule{0cm}{30px}\right) } - \frac{-i s_\alpha }{m_{h}^2}
\mathord{ \left(\rule{0cm}{30px}\right. \vcenter{
\hbox{ \includegraphics[height=57px , trim = 18mm 12mm 17.4mm
10mm, clip]{TadpoleDiagramHiggsBasish0.pdf} } }
\left.\rule{0cm}{30px}\right) } ~~~\text{and} ~~~ \delta v_2 = \frac{-i
s_\alpha }{m_{H}^2} \mathord{
\left(\rule{0cm}{30px}\right. \vcenter{
\hbox{ \includegraphics[height=57px , trim = 18mm 12mm 17.4mm
10mm, clip]{TadpoleDiagramHiggsBasisHH.pdf} } }
\left.\rule{0cm}{30px}\right) } + \frac{-i c_\alpha }{m_{h}^2}
\mathord{ \left(\rule{0cm}{30px}\right. \vcenter{
\hbox{ \includegraphics[height=57px , trim = 18mm 12mm 17.4mm
10mm, clip]{TadpoleDiagramHiggsBasish0.pdf} } }
\left.\rule{0cm}{30px}\right) } ~.
\label{eq:vevCountertermDefinition}
\end{equation}
The renormalization of the VEVs in the alternative tadpole scheme
effectively shifts the VEVs by tadpole contributions. As a
consequence, tadpole diagrams have to be considered wherever they can
appear in the 2HDM. For the self-energies, this means that the
fundamental self-energies used to define the CTs are the ones defined
as $\Sigma ^\text{tad}$ in \figref{fig:definitionOfSelfenergies}
instead of the usual one-particle irreducible self-energies
$\Sigma$. Additionally, tadpole diagrams have to be considered in the
calculation of the one-loop vertex corrections to the Higgs decays. In
summary, the renormalization of the tadpoles in the alternative scheme
leads to the following conditions:
\begin{mdframed}[frametitle={Renormalization of the tadpoles (alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt]\begin{align}
\delta T_{ij} &= 0 \\
\Sigma (p^2) ~ &\rightarrow ~\Sigma ^\text{tad} (p^2) \\
\text{Tadpole diagrams have to be }&\text{considered in the vertex
corrections.}
\nonumber
\end{align}\end{mdframed}
\subsubsection{Renormalization of the Gauge Sector}
\label{sec:renormalizationGaugeSector}
For the renormalization of the gauge sector, we introduce CTs and
WFRCs for all parameters and fields of the electroweak sector of the
2HDM by applying the shifts
\begin{align}
m_W^2 ~ &\rightarrow ~ m_W^2 + \delta m_W^2 \\
m_Z^2 ~ &\rightarrow ~ m_Z^2 + \delta m_Z^2 \\
\alpha _\text{em} ~ &\rightarrow ~ \alpha _\text{em} + \delta \alpha _\text{em} \equiv \alpha _\text{em} + 2 \alpha _\text{em} \delta Z_e \\
W^\pm _\mu ~ &\rightarrow ~ \left( 1 + \frac{\delta Z_{WW} }{2} \right) W^\pm _\mu \\
\begin{pmatrix} Z \\ \gamma \end{pmatrix} ~ &\rightarrow ~ \begin{pmatrix} 1 + \frac{\delta Z_{ZZ} }{2} & \frac{\delta Z_{Z \gamma }}{2} \\ \frac{\delta Z_{\gamma Z }}{2} & 1 + \frac{\delta Z_{\gamma \gamma }}{2} \end{pmatrix} \begin{pmatrix} Z \\ \gamma \end{pmatrix}
\;
\end{align}
where for convenience, we additionally introduced the shift
\begin{equation}
e ~ \rightarrow ~ e\,( 1 + \delta Z_e )
\end{equation}
for the electromagnetic coupling constant by using
\eqref{eq:electromagneticCouplingDefinition}. Applying OS
conditions to the gauge sector of the 2HDM leads to equivalent
expressions for the CTs as derived in Ref.~\cite{Denner:1991kt} for the
SM\footnote{In contrast to Ref.~\cite{Denner:1991kt}, however, we choose a
different sign for the $SU(2)_L$ term of the covariant derivative,
which subsequently leads to a different sign in front of the second
term of \eqref{eq:RenormalizationGaugeSectorExplicitFormDeltaZe}.},
for the standard and alternative tadpole scheme, respectively,
\begin{mdframed}[frametitle={Renormalization of the gauge sector (standard scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt]\begin{align}
\delta m_W^2 &= \textrm{Re} \left[ \Sigma _{WW} ^{T} \left( m_W ^2 \right) \right] \\
\delta m_Z^2 &= \textrm{Re} \left[ \Sigma _{ZZ} ^{T} \left( m_{Z} ^2 \right) \right]
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={Renormalization of the gauge sector (alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt]\begin{align}
\delta m_W^2 &= \textrm{Re} \left[ \Sigma _{WW} ^{\textrm{tad},T} \left( m_W ^2 \right) \right] \\
\delta m_Z^2 &= \textrm{Re} \left[ \Sigma _{ZZ} ^{\textrm{tad},T} \left( m_{Z} ^2 \right) \right]
\end{align}\end{mdframed}
The WFRCs are the same in both tadpole schemes,
\begin{mdframed}[frametitle={Renormalization of the gauge sector
(standard and alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta Z_e (m_Z^2) &= \frac{1}{2} \left. \frac{\partial \Sigma ^T _{\gamma \gamma } \left( p^2 \right) }{\partial p^2 } \right| _{p^2 = 0} + \frac{s_W }{c_W } \frac{\Sigma ^T _{\gamma Z} \left( 0\right) }{m_Z ^2 } - \frac{1}{2} \Delta \alpha (m_Z^2) \label{eq:RenormalizationGaugeSectorExplicitFormDeltaZe} \\
\delta Z_{WW} &= - \textrm{Re} \left[ \frac{\partial \Sigma ^T _{WW} \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = m_W^2 } \label{eq:RenormalizationGaugeSectorExplicitFormDeltaWW} \\
\begin{pmatrix} \delta Z_{ZZ} & \delta Z _{Z \gamma } \\ \delta Z _{\gamma Z } & \delta Z_{ \gamma \gamma } \end{pmatrix} &= \begin{pmatrix} - \textrm{Re} \left[ \frac{\partial \Sigma ^T _{ZZ} \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = m_Z^2 } & \frac{2}{m_Z ^2 } \Sigma ^{T} _{Z \gamma } \left( 0 \right) \\ -\frac{2}{m_Z ^2 } \textrm{Re} \left[ \Sigma ^{T} _{Z \gamma } \left( m_Z ^2 \right) \right] & - \textrm{Re} \left[ \frac{\partial \Sigma ^T _{\gamma \gamma } \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = 0} \end{pmatrix} \label{eq:RenormalizationGaugeSectorExplicitFormDeltaZZ}
\end{align}\end{mdframed}
The superscript $T$ indicates that only the transverse parts of the
self-energies are taken into account. The CT for the
electromagnetic coupling $\delta Z_e (m_Z^2)$ is defined at the scale
of the $Z$ boson mass instead of the Thomson limit. For this, the additional term
\begin{equation}
\Delta \alpha (m_Z^2) = \frac{\partial \Sigma _{\gamma \gamma} ^{\text{light},T} (p^2) }{\partial p^2} \Bigg| _{p^2 = 0} - \frac{\Sigma ^T _{\gamma \gamma } (m_Z^2) }{m_Z^2}
\label{eq:lightContributions}
\end{equation}
is required, where the transverse photon self-energy $\Sigma _{\gamma \gamma}
^{\text{light},T} (p^2)$ in \eqref{eq:lightContributions} contains
solely light fermion contributions ({\it i.e.}~contributions from all
fermions apart from the $t$ quark). This ensures that the results of
our EW one-loop computations are independent of large
logarithms due to light fermion contributions \cite{Denner:1991kt}.
For later convenience, we additionally introduce the shift of the weak coupling constant
\begin{equation}
g ~ \rightarrow ~ g + \delta g ~.
\end{equation}
Since $g$ is not an independent parameter in our approach, {\it
cf.}~\eqref{eq:electromagneticCouplingDefinition}, the CT $\delta g$
is not independent either and can be expressed through the other CTs
derived in this subsection as
\begin{equation}
\frac{\delta g}{g} = \delta Z_e (m_Z^2) + \frac{1}{2( m_Z^2 - m_W^2)} \left( \delta m_W^2 - \frac{m_W^2}{m_Z^2} \delta m_Z^2 \right) ~.
\end{equation}
\subsubsection{Renormalization of the Scalar Sector}
\label{sec:renormalizationScalarSector}
In the scalar sector of the 2HDM, the masses and fields of the scalar particles are shifted as
\begin{align}
m_{H}^2 ~ &\rightarrow ~ m_{H}^2 + \delta m_{H}^2 \\
m_{h}^2 ~ &\rightarrow ~ m_{h}^2 + \delta m_{h}^2 \\
m_{A}^2 ~ &\rightarrow ~ m_{A}^2 + \delta m_{A}^2 \\
m_{H^\pm }^2 ~ &\rightarrow ~ m_{H^\pm }^2 + \delta m_{H^\pm }^2 \\
\begin{pmatrix} H \\ h \end{pmatrix} ~ &\rightarrow ~ \begin{pmatrix} 1 + \frac{\delta Z _{{H} {H}}}{2} & \frac{\delta Z _{{H} {h}}}{2} \\ \frac{\delta Z _{{h} {H}}}{2} & 1 + \frac{\delta Z _{{h} {h}}}{2} \end{pmatrix} \renewcommand*{\arraystretch}{1.6} \begin{pmatrix} H \\ h \end{pmatrix} \label{RenormalizationOnShellLabelSectionScalarSectorFieldRenormalizationConstantsCPEvenHiggses} \\
\begin{pmatrix} G^0 \\ A \end{pmatrix} ~ &\rightarrow ~ \begin{pmatrix} 1 + \frac{\delta Z _{{G}^0 {G}^0}}{2} & \frac{\delta Z _{{G}^0 {A}}}{2} \\ \frac{\delta Z _{{A} {G}^0}}{2} & 1 + \frac{\delta Z _{{A} {A}}}{2} \renewcommand*{\arraystretch}{1.6} \end{pmatrix} \begin{pmatrix} G^0 \\ A \end{pmatrix} \label{RenormalizationOnShellLabelSectionScalarSectorFieldRenormalizationConstantsCPOddHiggses} \\
\begin{pmatrix} G^\pm \\ H^\pm \end{pmatrix} ~ &\rightarrow ~ \begin{pmatrix} 1 + \frac{\delta Z _{{G}^\pm {G}^\pm }}{2} & \frac{\delta Z _{{G}^\pm {H}^\pm }}{2} \\ \frac{\delta Z _{{H}^\pm {G}^\pm }}{2} & 1 + \frac{\delta Z _{{H}^\pm {H}^\pm }}{2} \renewcommand*{\arraystretch}{1.6} \end{pmatrix} \begin{pmatrix} G^\pm \\ H^\pm \end{pmatrix} \label{RenormalizationOnShellLabelSectionScalarSectorFieldRenormalizationConstantsChargedHiggses} ~.
\end{align}
Applying OS renormalization conditions leads to the following CT
definitions \cite{Krause:2016gkg},
\begin{mdframed}[frametitle={Renormalization of the scalar sector (standard scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta Z_{Hh} &= \frac{2}{m_{H}^2 - m_{h}^2} \textrm{Re} \Big[ \Sigma _{Hh} (m_{h}^2) - \delta T_{Hh} \Big] \label{RenormalizationScalarFieldsMassesExplicitFormWaveFunctionRenormalizationConstantH0h0} \\
\delta Z_{hH} &= -\frac{2}{m_{H}^2 - m_{h}^2} \textrm{Re} \Big[ \Sigma _{Hh} (m_{H}^2) - \delta T_{Hh} \Big] \label{RenormalizationScalarFieldsMassesExplicitFormWaveFunctionRenormalizationConstanth0H0} \\
\delta Z_{G^0A} &= -\frac{2}{m_{A}^2} \textrm{Re} \Big[ \Sigma _{G^0A} (m_{A}^2) - \delta T_{G^0A} \Big] \\
\delta Z_{AG^0} &= \frac{2}{m_{A}^2} \textrm{Re} \Big[ \Sigma _{G^0A} (0) - \delta T_{G^0A} \Big] \\
\delta Z_{G^\pm H^\pm} &= -\frac{2}{m_{H^\pm}^2} \textrm{Re} \Big[ \Sigma _{G^\pm H^\pm} (m_{H^\pm}^2) - \delta T_{G^\pm H^\pm} \Big] \\
\delta Z_{H^\pm G^\pm} &= \frac{2}{m_{H^\pm}^2} \textrm{Re} \Big[ \Sigma _{G^\pm H^\pm} (0) - \delta T_{G^\pm H^\pm} \Big] \\
\delta m_{H}^2 &= \textrm{Re} \Big[ \Sigma _{HH} (m_{H}^2) - \delta T_{HH} \Big] \\
\delta m_{h}^2 &= \textrm{Re} \Big[ \Sigma _{hh} (m_{h}^2) - \delta T_{hh} \Big] \\
\delta m_{A}^2 &= \textrm{Re} \Big[ \Sigma _{AA} (m_{A}^2) - \delta T_{AA} \Big] \\
\delta m_{H^\pm }^2 &= \textrm{Re} \Big[ \Sigma _{H^\pm H^\pm } (m_{H^\pm }^2) - \delta T_{H^\pm H^\pm } \Big]
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={Renormalization of the scalar sector (alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta Z_{Hh} &= \frac{2}{m_{H}^2 - m_{h}^2} \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{Hh} (m_{h}^2) \Big] \label{RenormalizationScalarFieldsMassesExplicitFormWaveFunctionRenormalizationConstantH0h0Alt} \\
\delta Z_{hH} &= -\frac{2}{m_{H}^2 - m_{h}^2} \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{Hh} (m_{H}^2) \Big] \label{RenormalizationScalarFieldsMassesExplicitFormWaveFunctionRenormalizationConstanth0H0Alt} \\
\delta Z_{G^0A} &= -\frac{2}{m_{A}^2} \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{G^0A} (m_{A}^2) \Big] \\
\delta Z_{AG^0} &= \frac{2}{m_{A}^2} \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{G^0A} (0) \Big] \\
\delta Z_{G^\pm H^\pm} &= -\frac{2}{m_{H^\pm}^2} \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{G^\pm H^\pm} (m_{H^\pm}^2) \Big] \\
\delta Z_{H^\pm G^\pm} &= \frac{2}{m_{H^\pm}^2} \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{G^\pm H^\pm} (0) \Big] \\
\delta m_{H}^2 &= \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{HH} (m_{H}^2) \Big] \label{RenormalizationScalarFieldsMassesExplicitFormMassCountertermH0} \\
\delta m_{h}^2 &= \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{hh} (m_{h}^2) \Big] \label{RenormalizationScalarFieldsMassesExplicitFormMassCountertermh0} \\
\delta m_{A}^2 &= \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{AA} (m_{A}^2) \Big] \\
\delta m_{H^\pm }^2 &= \textrm{Re} \Big[ \Sigma ^\textrm{tad} _{H^\pm H^\pm } (m_{H^\pm }^2) \Big] \label{RenormalizationScalarFieldsMassesExplicitFormMassCountertermHp}
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={Renormalization of the scalar sector
(standard and alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta Z_{HH} &= - \textrm{Re} \left[ \frac{\partial \Sigma _{H H } \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = m_{H} ^2} \label{RenormalizationScalarFieldsMassesExplicitFormWaveFunctionRenormalizationConstantH0H0} \\
\delta Z_{hh} &= - \textrm{Re} \left[ \frac{\partial \Sigma _{h h } \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = m_{h} ^2} \\
\delta Z_{G^0G^0} &= - \textrm{Re} \left[ \frac{\partial \Sigma _{G^0 G^0 } \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = 0} \\
\delta Z_{AA} &= - \textrm{Re} \left[ \frac{\partial \Sigma _{A A } \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = m_{A} ^2} \\
\delta Z_{G^\pm G^\pm } &= - \textrm{Re} \left[ \frac{\partial \Sigma _{G^\pm G^\pm } \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = 0} \\
\delta Z_{H^\pm H^\pm } &= - \textrm{Re} \left[ \frac{\partial \Sigma _{H^\pm H^\pm } \left( p^2 \right) }{\partial p^2 } \right] _{p^2 = m_{H^\pm} ^2} \label{RenormalizationScalarFieldsMassesExplicitFormWaveFunctionRenormalizationConstantHpHp}
\end{align}\end{mdframed}
with the tadpole CTs in the standard scheme defined in Eqs.\,(\ref{eq:RenormalizationRadpolesTadpoleCountertermDeltaTH0H0ExplicitForm})-(\ref{eq:RenormalizationRadpolesTadpoleCountertermDeltaTHpHpExplicitForm}).
\subsubsection{Renormalization of the Scalar Mixing Angles}
\label{sec:renormalizationMixingAngles}
In the following, we describe the renormalization of the scalar mixing
angles $\alpha$ and $\beta$ in the 2HDM. In our approach, we perform
the rotation from the interaction to the mass basis,
{\it cf.}~Eqs.\,(\ref{eq:rotationCPEven})-(\ref{eq:rotationCharged}), before
renormalization so that the mixing angles need to be renormalized.
At one-loop level, the bare mixing angles are replaced by their
renormalized values and counterterms as
\begin{align}
\alpha ~ &\rightarrow ~ \alpha + \delta \alpha \\
\beta ~ &\rightarrow ~ \beta + \delta \beta ~.
\end{align}
The renormalization of the mixing angles in the 2HDM is a non-trivial
task and several different schemes have been proposed in the
literature. In the following, we only briefly present the definition
of the mixing angle CTs in all different schemes that are implemented
in {\texttt{2HDECAY}} and refer to \cite{Krause:2016gkg,
Krause:2016oke} for details on the derivation of these
schemes. \\
\textbf{$\overline{\text{MS}}$ scheme.} It was shown in \cite{Lorenz:2015, Krause:2016gkg} that an
$\overline{\text{MS}}$ condition for $\delta \alpha$ and $\delta
\beta$ can lead to one-loop corrections that
are orders of magnitude larger than the LO result\footnote{In \cite{Denner:2016etu},
an $\overline{\text{MS}}$ condition for the scalar mixing angles
in certain processes led to corrections that are numerically
well-behaving due to a partial cancellation of large contributions
from tadpoles. In the decays considered in our work, an
$\overline{\text{MS}}$ condition of $\delta \alpha$ and $\delta
\beta$ in general leads to very large corrections, however.}. We implemented this scheme in {\texttt{2HDECAY}} for reference, as the $\overline{\text{MS}}$ CTs contain only the UV divergences of the CTs, but no finite parts $\left. \delta \alpha \right| _\text{fin}$ and $\left. \delta \beta \right| _\text{fin}$. After having checked for UV finiteness of the full decay width, the CTs of the mixing angles $\alpha$ and $\beta$ are effectively set to zero in {\texttt{2HDECAY}} in this scheme for the numerical evaluation of the partial decay widths.
\begin{mdframed}[frametitle={ Renormalization of $\delta \alpha$ and $\delta \beta$: $\overline{\text{MS}}$ scheme (both schemes) },frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\left. \delta \alpha \right| _\text{fin} &= 0 \\
\left. \delta \beta \right| _\text{fin} &= 0
\end{align}\end{mdframed}
The $\overline{\text{MS}}$ CTs of $\alpha$ and
$\beta$ depend on the renormalization scale $\mu_R$. The user has to
specifiy in the input file the scale at which $\alpha$ and $\beta$
are understood to be given when the $\overline{\text{MS}}$ renormalization scheme is
chosen. The one-loop corrected decay widths that contain these CTs,
then additionally depend on the renormalization scale of $\alpha$ and
$\beta$. The scale at which the decays are evaluated is also defined
by the user in the input file and should be chosen appropriately in
order to avoid the appearance of large logarithms in the EW one-loop
corrections. In case this scale differs from the scale of the
$\overline{\text{MS}}$ mixing angles
$\alpha$ and $\beta$, the automatic parameter conversion
routine converts $\alpha$ and $\beta$ to the scale of the
loop-corrected decay widths, as further described in
Sec.\,\ref{sec:ParameterConversion}. For the
conversion, the UV-divergent terms for the CTs $\delta \alpha$ and
$\delta \beta$ are needed, \textit{i.e.}\,the terms proportional to
$1/\varepsilon$. These UV-divergent terms are presented analytically
for both the standard and alternative tadpole scheme in
Ref.\,\cite{Altenkamp:2017ldc}. We cross-checked these terms
analytically in an independent calculation. \\
\textbf{KOSY scheme.} The KOSY scheme (denoted by the authors' initials)
was suggested in \cite{Kanemura:2004mg}. It combines the standard
tadpole scheme with the definition of the counterterms
through off-diagonal wave function renormalization constants.
As shown in \cite{Krause:2016gkg,Krause:2016oke}, the KOSY scheme not
only implies a gauge-dependent definition of the mixing angle CTs but also
leads to explicitly gauge-dependent decay amplitudes. The CTs are
derived by temporarily switching from the mass to the gauge
basis. Since $\beta$ diagonalizes both the charged and CP-odd
sector not all scalar fields can be defined OS at the same time,
unless a systematic modification of the $SU(2)$
relations is performed which we do not do here.
We implemented two different CT definitions where $\delta \beta$ is defined
through the CP-odd or the charged sectors, indicated by superscripts $o$
and $c$, respectively. The KOSY scheme is implemented in
{\texttt{2HDECAY}} both in the standard and in the alternative FJ
scheme as a benchmark scheme for comparison with other schemes, but
for actual computations, we do not recommend to use it due to the
explicit gauge dependence of the decay amplitudes. In the KOSY
scheme, the mixing angle CTs are defined as
\begin{mdframed}[frametitle={Renormalization of $\delta \alpha$ and $\delta \beta$: KOSY scheme (standard scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= \frac{1}{2(m_H^2 - m_h^2)} \text{Re} \left[ \Sigma _{Hh} (m_H^2) + \Sigma _{Hh} (m_h^2) - 2\delta T_{Hh} \right] \\
\delta \beta ^o &= -\frac{1}{2m_A^2} \text{Re} \left[ \Sigma _{G^0A} (m_A^2) + \Sigma _{G^0A} (0) - 2\delta T_{G^0A} \right] \\
\delta \beta ^c &= -\frac{1}{2m_{H^\pm}^2} \text{Re} \left[ \Sigma _{G^\pm H^\pm} (m_{H^\pm}^2) + \Sigma _{G^\pm H^\pm} (0) - 2\delta T_{G^\pm H^\pm} \right]
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={Renormalization of $\delta \alpha$ and $\delta \beta$: KOSY scheme (alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= \frac{1}{2(m_H^2 - m_h^2)} \text{Re} \left[ \Sigma ^\text{tad} _{Hh} (m_H^2) + \Sigma ^\text{tad} _{Hh} (m_h^2) \right] \\
\delta \beta ^o &= -\frac{1}{2m_A^2} \text{Re} \left[ \Sigma ^\text{tad} _{G^0A} (m_A^2) + \Sigma ^\text{tad} _{G^0A} (0) \right] \\
\delta \beta ^c &= -\frac{1}{2m_{H^\pm}^2} \text{Re} \left[ \Sigma ^\text{tad} _{G^\pm H^\pm} (m_{H^\pm}^2) + \Sigma ^\text{tad} _{G^\pm H^\pm} (0) \right]
\end{align}\end{mdframed}
\vspace*{0.14cm}
\textbf{$p_{*}$-pinched scheme.} One possibility to avoid
gauge-parameter-dependent mixing angle CTs was
suggested in \cite{Krause:2016gkg, Krause:2016oke}. The main idea is
to maintain the OS-based definition of $\delta \alpha$ and $\delta
\beta$ of the KOSY scheme, but instead of using the usual
gauge-dependent off-diagonal WFRCs, the WFRCs are defined through
pinched self-energies in the alternative FJ scheme by applying the pinch
technique (PT) \cite{Binosi:2004qe, Binosi:2009qm, Cornwall:1989gv,
Papavassiliou:1989zd, Degrassi:1992ue, Papavassiliou:1994pr,
Watson:1994tn, Papavassiliou:1995fq}. As worked out for the
2HDM for the first time in \cite{Krause:2016gkg, Krause:2016oke}, the
pinched scalar self-energies are equivalent to the usual scalar
self-energies in the alternative FJ scheme, evaluated in Feynman-'t
Hooft gauge ($\xi =1$), up to additional UV-finite self-energy contributions
$\Sigma ^\text{add} _{ij} (p^2)$. The mixing angle
CTs depend on the scale where the pinched self-energies are
evaluated. In the $p_{*}$-pinched scheme, we follow the approach of
\cite{Espinosa:2002cd} in the MSSM, where the self-energies $\Sigma
^\text{tad} _{ij} (p^2)$ are evaluated at the scale
\begin{equation}
p_{*}^2 \equiv \frac{m_i^2 + m_j^2}{2} ~.
\end{equation}
At this scale, the additional contributions $\Sigma ^\text{add} _{ij}
(p^2)$ vanish. Using the $p_{*}$-pinched scheme at
one-loop level yields explicitly gauge-parameter-independent partial
decay widths. The mixing angle CTs are defined as
\begin{mdframed}[frametitle={Renormalization of $\delta \alpha$ and $\delta \beta$: $p_{*}$-pinched scheme (alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= \frac{1}{m_H^2 - m_h^2} \text{Re} \left[ \Sigma ^\text{tad} _{Hh} \left( \frac{m_H^2 + m_h^2}{2} \right) \right] _{\xi = 1} \\
\delta \beta ^o &= -\frac{1}{m_A^2} \text{Re} \left[ \Sigma ^\text{tad} _{G^0A} \left( \frac{m_A^2 }{2} \right) \right] _{\xi = 1} \\
\delta \beta ^c &= -\frac{1}{m_{H^\pm}^2} \text{Re} \left[ \Sigma ^\text{tad} _{G^\pm H^\pm} \left( \frac{m_{H^\pm}^2 }{2} \right) \right] _{\xi = 1}
\end{align}\end{mdframed}
\textbf{OS-pinched scheme.} In order to allow for the analysis of the effects
of different scale choices of the mixing angle CTs, we implemented
another OS-motivated scale choice, which is called the OS-pinched
scheme. Here, the additional terms do not vanish and are given by
\cite{Krause:2016gkg}
\begin{align}
\Sigma ^\textrm{add} _{H h } (p^2) &= \frac{\alpha _\text{em} m_Z^2 s_{\beta - \alpha} c_{\beta - \alpha} }{8 \pi m_W^2 \left( 1 - \frac{m_W^2}{m_Z^2} \right) } \left( p^2 - \frac{ m_{H}^2 + m_{h}^2}{2} \right) \bigg\{ \left[ B_0( p^2; m_Z^2, m_{A}^2) - B_0( p^2; m_Z^2, m_{Z}^2) \right] \nonumber \\
&\hspace*{0.4cm} + 2\frac{m_W^2}{m_Z^2} \left[ B_0( p^2; m_W^2, m_{H^\pm }^2) - B_0( p^2; m_W^2, m_{W}^2) \right] \bigg\} \label{eq:RenormalizationScalarAnglesAdditionalTermCPEven} \\
\Sigma ^\textrm{add} _{G^0 A } (p^2) &= \frac{\alpha _\text{em} m_Z^2 s_{\beta - \alpha} c_{\beta - \alpha}}{8 \pi m_W^2 \left( 1 - \frac{m_W^2}{m_Z^2} \right) } \left( p^2 - \frac{m_{A}^2}{2} \right) \left[ B_0( p^2; m_Z^2, m_{H}^2) - B_0( p^2; m_Z^2, m_{h}^2) \right] \label{eq:RenormalizationScalarAnglesAdditionalTermCPOdd} \\
\Sigma ^\textrm{add} _{G^\pm H^\pm } (p^2) &= \frac{\alpha _\text{em} s_{\beta - \alpha} c_{\beta - \alpha}}{4 \pi \left( 1 - \frac{m_W^2}{m_Z^2} \right) } \left( p^2 - \frac{m_{H^\pm }^2}{2} \right) \left[ B_0( p^2; m_W^2, m_{H}^2) - B_0( p^2; m_W^2, m_{h}^2) \right] ~. \label{eq:RenormalizationScalarAnglesAdditionalTermCharged}
\end{align}
The mixing angle CTs in the OS-pinched scheme are then defined as
\begin{mdframed}[frametitle={Renormalization of $\delta \alpha$ and $\delta \beta$: OS-pinched scheme (alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= \frac{\textrm{Re} \Big[ \left[ \Sigma ^\textrm{tad} _{Hh} (m_{H}^2) + \Sigma ^\textrm{tad} _{Hh} (m_{h}^2) \right] _{\xi = 1} + \Sigma ^\textrm{add} _{Hh} (m_{H}^2) + \Sigma ^\textrm{add} _{Hh} (m_{h}^2) \Big]}{2\left( m_{H}^2 - m_{h}^2\right) } \label{eq:RenormalizationScalarAnglesDeltaAlphaOSPinchedResult} \\
\delta \beta ^o &= -\frac{\textrm{Re} \Big[ \left[ \Sigma ^\textrm{tad} _{G^0A} (m_{A}^2) + \Sigma ^\textrm{tad} _{G^0A} (0) \right] _{\xi = 1} + \Sigma ^\textrm{add} _{G^0A} (m_{A}^2) + \Sigma ^\textrm{add} _{G^0A} (0) \Big]}{2 m_{A}^2} \label{eq:RenormalizationScalarAnglesDeltaBeta1OSPinchedResult} \\
\delta \beta ^c &= -\frac{\textrm{Re} \Big[ \left[ \Sigma ^\textrm{tad} _{G^\pm H^\pm } (m_{H^\pm }^2) + \Sigma ^\textrm{tad} _{G^\pm H^\pm } (0) \right] _{\xi = 1} + \Sigma ^\textrm{add} _{G^\pm H^\pm } (m_{H^\pm }^2) + \Sigma ^\textrm{add} _{G^\pm H^\pm } (0) \Big]}{2 m_{H^\pm }^2} \label{eq:RenormalizationScalarAnglesDeltaBeta2OSPinchedResult}
\end{align}\end{mdframed}
\textbf{Process-dependent schemes.} The definition of the mixing angle
CTs through observables, like {\it e.g.}~partial decay widths of Higgs
bosons, was proposed for the MSSM in \cite{Coarasa:1996qa,
Freitas:2002um} and for the 2HDM in \cite{Santos:1996hs}. This
scheme leads to explicitly gauge-independent partial
decay widths per construction. Moreover, the connection of the mixing angle CTs with
physical observables allows for a more physical interpretation of the
unphysical mixing angles $\alpha $ and $\beta$. However, as
it was shown in \cite{Krause:2016gkg, Krause:2016oke},
process-dependent schemes can in general lead to very large one-loop
corrections. We implemented three different process-dependent schemes
for $\delta \alpha$ and $\delta \beta$ in
{\texttt{2HDECAY}}. The schemes differ in the processes that are used
for the definition of the CTs. In all cases we have chosen leptonic
Higgs boson decays. For these, the QED corrections can be
separated in a UV-finite way from the rest of the EW corrections and therefore be
excluded from the counterterm
definition. This is necessary to avoid
the appearance of infrared (IR) divergences in the CTs \cite{Freitas:2002um}. The
NLO corrections to the partial decay widths of the leptonic decay
of a Higgs particle $\phi _i$ into a pair of leptons $f_j$, $f_k$ can
then be cast into the form
\begin{equation}
\Gamma _{\phi _i f_j f_k}^{\text{NLO,weak}} = \Gamma _{\phi _i f_j f_k}^{\text{LO}} \left( 1 + 2\text{Re} \left[ \mathcal{F}_{\phi _i f_j f_k}^\text{VC} + \mathcal{F}_{\phi _i f_j f_k}^\text{CT} \right] \right)
\end{equation}
where $\mathcal{F}_{\phi _i f_j f_k}^\text{VC}$ and $\mathcal{F}_{\phi
_i f_j f_k}^\text{CT}$ are the form factors of the vertex
corrections and the CT, respectively, and the superscript weak
indicates that in the vertex corrections IR-divergent QED
contributions are excluded. The form factor $\mathcal{F}_{\phi _i f_j
f_k}^\text{CT}$ contains either $\delta \alpha$ or $\delta \beta$ or
both simultaneously as well as other CTs that are fixed as described
in the other subsections of
Sec.\,\ref{sec:renormalization2HDM}. Employing the renormalization
condition
\begin{equation}
\Gamma ^\text{LO} _{\phi _i f_j f_k} \equiv \Gamma ^\text{NLO,weak} _{\phi _i f_j f_k}
\end{equation}
for two different decays then allows for a process-dependent
definition of the mixing angle CTs. For more details on the
calculation of the CTs in process-dependent schemes in the 2HDM, we
refer to \cite{Krause:2016gkg, Krause:2016oke}. In {\texttt{2HDECAY}},
we have chosen the following three different combinations of processes
as definition for the CTs,
\begin{enumerate}
\item $\delta \beta$ is first defined by $A \rightarrow \tau ^+ \tau ^-$ and $\delta \alpha$ is subsequently defined by $H \rightarrow \tau ^+ \tau ^-$.
\item $\delta \beta$ is first defined by $A \rightarrow \tau ^+ \tau ^-$ and $\delta \alpha$ is subsequently defined by $h \rightarrow \tau ^+ \tau ^-$.
\item $\delta \beta$ and $\delta \alpha$ are simultaneously defined by $H \rightarrow \tau ^+ \tau ^-$ and $h \rightarrow \tau ^+ \tau ^-$.
\end{enumerate}
Employing these renormalization conditions yields the following
definitions of the mixing angle CTs\footnote{While the definition of
the CTs is generically the same for both tadpole schemes, their actual analytic
forms differ in both schemes since some of the CTs used in the
definition differ in the two schemes, as well. However, when choosing a process-dependent scheme for the mixing angle CTs, the full partial decay width is independent of the chosen tadpole scheme, which was checked explicitly by us. Therefore, in {\texttt{2HDECAY}} we implemented the process-dependent schemes in the alternative tadpole scheme, only. }:
\begin{mdframed}[frametitle={Renormalization of $\delta \alpha$ and $\delta \beta$: process-dependent scheme 1 (both schemes)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= \frac{- Y_5}{Y_4} \bigg[ \mathcal{F}^\textrm{VC}_{H \tau \tau } + \frac{\delta g}{g} + \frac{\delta m_\tau }{m_\tau } - \frac{\delta m_W^2}{2m_W^2} + Y_6 \delta \beta + \frac{\delta Z_{HH}}{2} + \frac{Y_4}{Y_5} \frac{\delta Z_{hH}}{2} + \frac{\delta Z^\textrm{L} _{\tau \tau}}{2} \\
&\hspace*{1.4cm} + \frac{\delta Z^\textrm{R} _{\tau \tau}}{2} \bigg] \nonumber \\
\delta \beta &= \frac{- Y_6}{1+Y_6^2} \bigg[ \mathcal{F}^\textrm{VC}_{A \tau \tau } + \frac{\delta g}{g} + \frac{\delta m_\tau }{m_\tau } - \frac{\delta m_W^2}{2m_W^2} + \frac{\delta Z_{AA}}{2} - \frac{1}{Y_6} \frac{\delta Z_{G^0A}}{2} + \frac{\delta Z^\textrm{L} _{\tau \tau}}{2} + \frac{\delta Z^\textrm{R} _{\tau \tau}}{2} \bigg]
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={Renormalization of $\delta \alpha$ and $\delta \beta$: process-dependent scheme 2 (both schemes)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= \frac{Y_4}{Y_5} \bigg[ \mathcal{F}^\textrm{VC}_{h \tau \tau } + \frac{\delta g}{g} + \frac{\delta m_\tau }{m_\tau } - \frac{\delta m_W^2}{2m_W^2} + Y_6 \delta \beta + \frac{\delta Z_{hh}}{2} + \frac{Y_5}{Y_4} \frac{\delta Z_{Hh}}{2} + \frac{\delta Z^\textrm{L} _{\tau \tau}}{2} \\
&\hspace*{1.1cm} + \frac{\delta Z^\textrm{R} _{\tau \tau}}{2} \bigg] \nonumber \\
\delta \beta &= \frac{- Y_6}{1+Y_6^2} \bigg[ \mathcal{F}^\textrm{VC}_{A \tau \tau } + \frac{\delta g}{g} + \frac{\delta m_\tau }{m_\tau } - \frac{\delta m_W^2}{2m_W^2} + \frac{\delta Z_{AA}}{2} - \frac{1}{Y_6} \frac{\delta Z_{G^0A}}{2} + \frac{\delta Z^\textrm{L} _{\tau \tau}}{2} + \frac{\delta Z^\textrm{R} _{\tau \tau}}{2} \bigg]
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={Renormalization of $\delta \alpha$ and $\delta \beta$: process-dependent scheme 3 (both schemes)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= \frac{Y_4 Y_5}{Y_4^2 + Y_5^2} \bigg[ \mathcal{F}^\textrm{VC}_{h \tau \tau } - \mathcal{F}^\textrm{VC}_{H \tau \tau } + \frac{\delta Z_{hh}}{2} - \frac{\delta Z_{HH}}{2} + \frac{Y_5}{Y_4} \frac{\delta Z_{Hh}}{2} - \frac{Y_4}{Y_5} \frac{\delta Z_{hH}}{2} \bigg] \\
\delta \beta &= \frac{- 1}{Y_6(Y_4^2+Y_5^2)} \bigg[ (Y_4^2 + Y_5^2) \left( \frac{\delta g}{g} + \frac{\delta m_\tau }{m_\tau } - \frac{\delta m_W^2}{2m_W^2} + \frac{\delta Z^\textrm{L} _{\tau \tau}}{2} + \frac{\delta Z^\textrm{R} _{\tau \tau}}{2} \right) \\
&\hspace*{0.4cm}+ Y_4Y_5 \left( \frac{\delta Z_{Hh}}{2} + \frac{\delta Z_{hH}}{2} \right) + Y_4^2 \left( \frac{\delta Z_{hh}}{2} + \mathcal{F}^\textrm{VC}_{h \tau \tau } \right) + Y_5^2 \left( \frac{\delta Z_{HH}}{2} + \mathcal{F}^\textrm{VC}_{H \tau \tau } \right) \bigg] \nonumber
\end{align}\end{mdframed}
Note that for the process-dependent schemes, decays have to be chosen
that are experimentally accessible. This may not be the case for
certain parameter configurations, in which case the user has to
choose, if possible, the decay combination that leads to large enough
decay widths to be measurable. \\
\textbf{Physical (on-shell) schemes.} In order to exploit the advantages of process-dependent schemes, \textit{i.e.}\,gauge independence of the mixing angle CTs that are defined within these schemes, while simultaneously avoiding possible drawbacks, \textit{e.g.}\,potentially large NLO corrections, the mixing angle CTs can be defined through certain observables or combinations of $S$ matrix elements in such a way that the CTs of all other parameters of the theory do not contribute to the mixing angle CTs. Such a scheme was proposed for the quark mixing within the SM in \cite{Denner:2004bm} and for the mixing angle CTs in the 2HDM in \cite{Denner2018}, where the derivation of the scheme is presented in detail. Here, we only recapitulate the key ideas and state the relevant formulae. For the sole purpose of renormalizing the mixing angles, two right-handed fermion singlets $\nu _{1\text{R}}$ and $\nu _{2\text{R}}$ are added to the 2HDM Lagrangian. An additional discrete $\mathbb{Z}_2$ symmetry is imposed under which the singlets transform as
\begin{align}
\nu _{1\text{R}} &\longrightarrow - \nu _{1\text{R}} \\
\nu _{2\text{R}} &\longrightarrow \nu _{2\text{R}}
\end{align}
which prevents lepton generation mixing. The two singlets are coupled via Yukawa couplings $y_{\nu _1}$ and $y_{\nu _2}$ to two arbitrary left-handed lepton doublets of the 2HDM, giving rise to two massive Dirac neutrinos $\nu _1$ and $\nu _2$. The CT of the mixing angle $\alpha$ can then be defined by demanding that the ratio of the decay amplitudes of the decays $H \rightarrow \nu _i \bar{\nu} _i$ and $h \rightarrow \nu _i \bar{\nu} _i$ (for either $i=1$ or $i=2$) is the same at tree level and at NLO. Taking the ratio of the decay amplitudes has the advantage that other CTs apart from some WFRCs and the mixing angle CTs cancel against each other. For the CT of the mixing angle $\beta$, analogous conditions are imposed, involving additionally the decay of the pseudoscalar Higgs boson $A$ into the pair of massive neutrinos in the ratios of the LO and NLO decay amplitudes. In all cases, the mixing angle CTs are then given as functions of the scalar WFRCs as well as the genuine one-loop vertex corrections to the decays of the scalar particles into the pair of massive neutrinos, namely $\delta _{H\nu _i \bar{\nu} _i}$, $\delta _{h\nu _i \bar{\nu} _i}$ and $\delta _{A\nu _i \bar{\nu} _i}$, as given in Ref.\,\cite{Denner2018}. In this reference, three combinations of ratios of decay amplitudes were chosen to define three different renormalization schemes for the mixing angle CTs in the physical (on-shell) scheme:
\begin{itemize}
\item ``OS1'' scheme: $\mathcal{A} _{H_1 \rightarrow \nu _1 \bar{\nu }_1} / \mathcal{A} _{H_2 \rightarrow \nu _1 \bar{\nu }_1} $ for $\delta \alpha$ and $\mathcal{A} _{A \rightarrow \nu _1 \bar{\nu }_1} / \mathcal{A} _{H_1 \rightarrow \nu _1 \bar{\nu }_1} $ for $\delta \beta$
\item ``OS2'' scheme: $\mathcal{A} _{H_1 \rightarrow \nu _2 \bar{\nu }_2} / \mathcal{A} _{H_2 \rightarrow \nu _2 \bar{\nu }_2} $ for $\delta \alpha$ and $\mathcal{A} _{A \rightarrow \nu _2 \bar{\nu }_2} / \mathcal{A} _{H_1 \rightarrow \nu _2 \bar{\nu }_2} $ for $\delta \beta$
\item ``OS12'' scheme: $\mathcal{A} _{H_1 \rightarrow \nu _2 \bar{\nu }_2} / \mathcal{A} _{H_2 \rightarrow \nu _2 \bar{\nu }_2} $ for $\delta \alpha$ and a specific combination of all possible decay amplitudes $\mathcal{A} _{H_i \rightarrow \nu _j \bar{\nu }_j}$ and $\mathcal{A} _{A \rightarrow \nu _j \bar{\nu }_j}$ ($i,j=1,2$) for $\delta \beta$ .
\end{itemize}
All three of these schemes were implemented in
{\texttt{2HDECAY}}\footnote{As for the process-dependent schemes
before, the generic form of the CTs is valid for both the standard
and alternative tadpole scheme, while the actual analytic
expressions differ between the schemes. Since the full partial decay
width is again independent of the tadpole scheme when using the
physical (on-shell) scheme, we implemented these schemes in the
alternative tadpole scheme,
only.}\textsuperscript{,}\footnote{Note
that the CTs of the physical on-shell schemes are defined in
\cite{Denner2018} in the framework of the complex mass scheme
\cite{Denner:2005fg, Denner:2006ic} while in {\texttt{2HDECAY}},
we take the real parts of the self-energies through which these
CTs are defined. These different definitions can lead to
different finite parts in the one-loop partial decay
widths. These differences are formally of
next-to-next-to leading order.} according to the following definitions of the mixing angle CTs:
\begin{mdframed}[frametitle={ Renormalization of $\delta \alpha$ and $\delta \beta$: physical (on-shell) scheme OS1 (both schemes) },frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= s_\alpha c_\alpha \left( \delta _{H \nu _1 \bar{\nu} _1} - \delta _{h \nu _1 \bar{\nu} _1} \right) + s_\alpha c_\alpha \frac{\delta Z _{HH} - \delta Z_{hh}}{2} + \frac{c_\alpha ^2 \delta Z_{Hh} - s_\alpha ^2 \delta Z_{hH}}{2} \\
\delta \beta &= t_\beta \Bigg[ c_\alpha ^2 \delta _{H\nu _1 \bar{\nu} _1} + s_\alpha ^2 \delta _{h\nu _1 \bar{\nu} _1} - \delta _{A \nu _1 \bar{\nu} _1} + \frac{ c_\alpha ^2 \delta Z_{HH} + s_\alpha ^2 \delta Z_{hh} -\delta Z_{AA}}{2} \\
&\hspace*{1.24cm} - s_\alpha c_\alpha \frac{\delta Z_{Hh} + \delta Z_{hH}}{2} \Bigg] + \frac{\delta Z_{G^0A}}{2} \nonumber
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={ Renormalization of $\delta \alpha$ and $\delta \beta$: physical (on-shell) scheme OS2 (both schemes) },frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= s_\alpha c_\alpha \left( \delta _{h \nu _2 \bar{\nu} _2} - \delta _{H \nu _2 \bar{\nu} _2} \right) + s_\alpha c_\alpha \frac{\delta Z _{hh} - \delta Z_{HH}}{2} + \frac{s_\alpha ^2 \delta Z_{Hh} - c_\alpha ^2 \delta Z_{hH}}{2} \\
\delta \beta &= \frac{1}{t_\beta } \Bigg[ \delta _{A \nu _2 \bar{\nu} _2} - s_\alpha ^2 \delta _{H\nu _2 \bar{\nu} _2} - c_\alpha ^2 \delta _{h\nu _2 \bar{\nu} _2} + \frac{ \delta Z_{AA} - s_\alpha ^2 \delta Z_{HH} - c_\alpha ^2 \delta Z_{hh} }{2} \\
&\hspace*{1.24cm} - s_\alpha c_\alpha \frac{\delta Z_{Hh} + \delta Z_{hH}}{2} \Bigg] + \frac{\delta Z_{G^0A}}{2} \nonumber
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={ Renormalization of $\delta \alpha$ and $\delta \beta$: physical (on-shell) scheme OS12 (both schemes) },frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= s_\alpha c_\alpha \left( \delta _{h \nu _2 \bar{\nu} _2} - \delta _{H \nu _2 \bar{\nu} _2} \right) + s_\alpha c_\alpha \frac{\delta Z _{hh} - \delta Z_{HH}}{2} + \frac{s_\alpha ^2 \delta Z_{Hh} - c_\alpha ^2 \delta Z_{hH}}{2} \\
\delta \beta &= s_\beta c_\beta \left[ c_{2\alpha} \frac{\delta Z_{HH} - \delta Z_{hh}}{2} - s_{2\alpha} \frac{\delta Z_{Hh} + \delta Z_{hH}}{2} \right] + \frac{\delta Z_{G^0A}}{2} \\
&\hspace*{0.4cm} + s_\beta c_\beta \left[ \delta _{A\nu _2 \bar{\nu } _2} - \delta _{A\nu _1 \bar{\nu } _1} + c_\alpha ^2 \delta _{H\nu _1 \bar{\nu } _1} - s_\alpha ^2 \delta _{H\nu _2 \bar{\nu } _2} + s_\alpha ^2 \delta _{h\nu _1 \bar{\nu } _1} - c_\alpha ^2 \delta _{h\nu _2 \bar{\nu } _2} \right] \nonumber
\end{align}\end{mdframed}
For $y _{\nu _i} \rightarrow 0$ ($i=1,2$), the two Dirac neutrinos become massless again, the right-handed neutrino singlets decouple and the original 2HDM Lagrangian is recovered. The vertex corrections $\delta _{H\nu _i \bar{\nu} _i}$, $\delta _{h\nu _i \bar{\nu} _i}$ and $\delta _{A\nu _i \bar{\nu} _i}$ are non-vanishing in this limit, however, so that the mixing angle CTs can still be defined through these processes. The mixing angle CTs defined in these physical (on-shell) schemes are manifestly gauge-independent. \\
\textbf{Rigid symmetry scheme.} The renormalization of mixing matrix elements, \textit{e.g.}\,of $\alpha$ and $\beta$ for the scalar sector of the 2HDM, can be connected to the renormalization of the WFRCs by using the rigid symmetry of the Lagrangian. More specifically, it is possible to renormalize the fields and dimensionless parameters of the unbroken gauge theory and to connect the renormalization of the mixing matrix elements of \textit{e.g.}\,the scalar sector through a field rotation from the symmetric to the broken phase of the theory. Such a scheme was applied for the renormalization of the SM in \cite{Bohm:1986rj}. In \cite{Denner2018}, the scheme was applied to the scalar mixing angles of the 2HDM within the framework of the background field method (BFM) \cite{Zuber:1975sa,Zuber:1975gh,Boulware:1981,Abbott:1981,Abbott:1982,Hart:1983,Denner:1994xt}, which allows to formulate the mixing angle CTs as functions of the WFRCs $\delta Z_{\hat{H} \hat{h}}$ and $\delta Z_{\hat{h} \hat{H}}
$ in the alternative tadpole scheme, where the hat denotes that the fields are given in the BFM framework. These WFRCs differ from the ones used in the non-BFM framework, \textit{i.e.}\,$\delta Z_{Hh}$ and $\delta Z_{hH}
$ as given by Eqs.\,(\ref{RenormalizationScalarFieldsMassesExplicitFormWaveFunctionRenormalizationConstantH0h0Alt}) and (\ref{RenormalizationScalarFieldsMassesExplicitFormWaveFunctionRenormalizationConstanth0H0Alt}), by some additional term as presented in App.\,B of \cite{Denner2018} which coincides with the additional term given in \eqref{eq:RenormalizationScalarAnglesAdditionalTermCPEven} derived by means of the PT. The scalar self-energies involved in defining the mixing angle CTs are evaluated in a specifically chosen gauge, \textit{e.g.}\, the Feynman-'t Hooft gauge. This leads to the following definition of the CTs according to \cite{Denner2018} which is implemented in {\texttt{2HDECAY}},
\begin{mdframed}[frametitle={ Renormalization of $\delta \alpha$ and $\delta \beta$: BFMS scheme (alternative FJ scheme) },frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta \alpha &= \frac{\textrm{Re} \Big[ \left[ \Sigma ^\textrm{tad} _{Hh} (m_{H}^2) + \Sigma ^\textrm{tad} _{Hh} (m_{h}^2) \right] _{\xi = 1} + \Sigma ^\textrm{add} _{Hh} (m_{H}^2) + \Sigma ^\textrm{add} _{Hh} (m_{h}^2) \Big]}{2\left( m_{H}^2 - m_{h}^2\right) } \\
\delta \beta &= \frac{s_{2\beta}}{s_{2\alpha}} \frac{\textrm{Re} \Big[ \left[ \Sigma ^\textrm{tad} _{Hh} (m_{h}^2) - \Sigma ^\textrm{tad} _{Hh} (m_{H}^2) \right] _{\xi = 1} + \Sigma ^\textrm{add} _{Hh} (m_{h}^2) - \Sigma ^\textrm{add} _{Hh} (m_{H}^2) \Big]}{2\left( m_{H}^2 - m_{h}^2\right) } \\
&\hspace*{0.4cm} + \frac{e}{2m_W \sqrt{ 1 - \frac{m_W^2}{m_Z^2} } } \left[ s_{\beta - \alpha} \frac{\delta T_H}{m_H^2} - c_{\beta - \alpha} \frac{\delta T_h}{m_h^2} \right] \nonumber
\end{align}\end{mdframed}
where we replaced the BFM WFRCs with the corresponding self-energies and additional terms. As mentioned in \cite{Denner2018}, we want to remark that the definition of $\delta \alpha$ in the BFMS scheme coincides with the definition in the OS-pinched scheme of \eqref{eq:RenormalizationScalarAnglesDeltaAlphaOSPinchedResult}, while the definition of $\delta \beta$ in the BFMS scheme is different from the one in the OS-pinched scheme.
\subsubsection{Renormalization of the Fermion Sector}
\label{sec:renormalizationFermionSector}
The masses $m_f$, where $f$ generically stands for any fermion of the
2HDM, the CKM matrix elements $V_{ij}$ ($i,j=1,2,3$), the Yukawa coupling parameters
$Y_k$ ($k=1,...,6)$ and the fields of the fermion sector are replaced by the
renormalized quantities and the respective CTs and WFRCs as
\begin{align}
m_f ~ &\rightarrow ~ m_f + \delta m_f \\
V_{ij} ~&\rightarrow ~ V_{ij} + \delta V_{ij} \\
Y_k ~& \rightarrow ~ Y_k + \delta Y_k \\
f_i^L ~& \rightarrow ~ \left(\delta _{ij} + \frac{\delta Z_{ij}^{f,L} }{2} \right) f_j^L \\
f_i^R ~& \rightarrow ~ \left(\delta _{ij} + \frac{\delta Z_{ij}^{f,R} }{2} \right) f_j^R
\end{align}
where we use Einstein's sum convention in the last two lines. The
superscripts $L$ and $R$ denote the left- and right-chiral component
of the fermion fields, respectively. The Yukawa coupling parameters
$Y_i$ are not independent input parameters, but functions of $\alpha$ and
$\beta$, {\it cf.}~Tab.~\ref{tab:yukawaCouplings}. Their one-loop
counterterms are therefore given in terms of $\delta \alpha$
and $\delta \beta$ defined in
Sec.\,\ref{sec:renormalizationMixingAngles} by the following formulae
which are independent of the 2HDM type,
\begin{align}
\delta Y_1 &= Y_1 \left( -\frac{Y_2}{Y_1} \delta \alpha + Y_3 \delta \beta \right) \\
\delta Y_2 &= Y_2 \left( \frac{Y_1}{Y_2} \delta \alpha + Y_3 \delta \beta \right) \\
\delta Y_3 &= \left( 1+ Y_3 ^2 \right) \delta \beta \\
\delta Y_4 &= Y_4 \left( -\frac{Y_5}{Y_4} \delta \alpha + Y_6 \delta \beta \right) \\
\delta Y_5 &= Y_5 \left( \frac{Y_4}{Y_5} \delta \alpha + Y_6 \delta \beta \right) \\
\delta Y_6 &= \left( 1+ Y_6 ^2 \right) \delta \beta ~.
\end{align}
Before presenting the renormalization conditions of the
mass CTs and WFRCs, we shortly discuss the
renormalization of the CKM matrix. In \cite{Denner:1991kt} the
renormalization of the CKM matrix is connected to the renormalization
of the fields, which in turn are renormalized in an OS approach,
leading to the definition ($i,j,k=1,2,3$)
\begin{equation}
\delta V_{ij} = \frac{1}{4} \left[ \left( \delta Z ^{u,L} _{ik} - \delta Z ^{u,L \dagger} _{ik} \right) V_{kj} - V_{ik} \left( \delta Z ^{d,L} _{kj} - \delta Z ^{d,L \dagger} _{kj} \right) \right]
\label{eq:CKMCTdefinition}
\end{equation}
where the superscripts $u$ and $d$ denote up-type and down-type
quarks, respectively. This definition of the CKM matrix CTs leads to
uncanceled explicit gauge dependences when used in the calculation of
EW one-loop corrections, however, \cite{Gambino:1998ec,
Barroso:2000is, Kniehl:2000rb, Pilaftsis:2002nc, Yamada:2001px,
Diener:2001qt}. Since the CKM matrix is approximately a unit matrix
\cite{1674-1137-38-9-090001}, the
numerical effect of this gauge dependence is typically very small, but
the definition nevertheless introduces uncanceled explicit gauge dependences
into the partial decay widths, which should be avoided. In our work,
we follow the approach of
Ref.~\cite{Yamada:2001px} and use pinched fermion self-energies for the
definition of the CKM matrix CT. An analytic analysis shows that this
is equivalent with defining the CTs in \eqref{eq:CKMCTdefinition} in
the Feynman-'t Hooft gauge.
Apart from the CKM matrix CT, all other CTs of the fermion sector are
defined through OS conditions. The resulting forms of the CTs are
analogous to the ones presented in \cite{Denner:1991kt} and given by
\begin{mdframed}[frametitle={Renormalization of the fermion sector (standard scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta m_{f, i} &= \frac{m_{f,i}}{2} \text{Re} \left( \Sigma _{ii}^{f,L} (m_{f,i}^2) + \Sigma _{ii}^{f,R} (m_{f,i}^2) + 2\Sigma _{ii}^{f,S} (m_{f,i}^2) \right) \\
\delta Z^{f,L}_{ij} &= \frac{2}{m_{f,i}^2 - m_{f,j}^2} \text{Re} \bigg[ m_{f,j}^2 \Sigma _{ij}^{f,L} (m_{f,j}^2) + m_{f,i} m_{f,j} \Sigma _{ij} ^{f,R} (m_{f,j}^2) \\
&\hspace*{3.2cm} + (m_{f,i}^2 + m_{f,j}^2) \Sigma _{ij}^{f,S} (m_{f,j}^2) \bigg] ~~~~~~ (i\neq j) \nonumber \\
\delta Z^{f,R}_{ij} &= \frac{2}{m_{f,i}^2 - m_{f,j}^2} \text{Re} \bigg[ m_{f,j}^2 \Sigma _{ij}^{f,R} (m_{f,j}^2) + m_{f,i} m_{f,j} \Sigma _{ij} ^{f,L} (m_{f,j}^2) \\
&\hspace*{3.2cm} + 2 m_{f,i}m_{f,j} \Sigma _{ij}^{f,S} (m_{f,j}^2) \bigg] ~~~~~~ (i\neq j) \nonumber
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={Renormalization of the fermion sector (alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta m_{f, i} &= \frac{m_{f,i}}{2} \text{Re} \left( \Sigma _{ii}^{f,L} (m_{f,i}^2) + \Sigma _{ii}^{f,R} (m_{f,i}^2) + 2\Sigma _{ii}^{\text{tad},f,S} (m_{f,i}^2) \right) \\
\delta Z^{f,L}_{ij} &= \frac{2}{m_{f,i}^2 - m_{f,j}^2} \text{Re} \bigg[ m_{f,j}^2 \Sigma _{ij}^{f,L} (m_{f,j}^2) + m_{f,i} m_{f,j} \Sigma _{ij} ^{f,R} (m_{f,j}^2) \\
&\hspace*{3.2cm} + (m_{f,i}^2 + m_{f,j}^2) \Sigma _{ij}^{\text{tad},f,S} (m_{f,j}^2) \bigg] ~~~~~~ (i\neq j) \nonumber \\
\delta Z^{f,R}_{ij} &= \frac{2}{m_{f,i}^2 - m_{f,j}^2} \text{Re} \bigg[ m_{f,j}^2 \Sigma _{ij}^{f,R} (m_{f,j}^2) + m_{f,i} m_{f,j} \Sigma _{ij} ^{f,L} (m_{f,j}^2) \\
&\hspace*{3.2cm} + 2 m_{f,i}m_{f,j} \Sigma _{ij}^{\text{tad},f,S} (m_{f,j}^2) \bigg] ~~~~~~ (i\neq j) \nonumber \label{RenormalizationFermionSectorExplicitFormMassCountertermTauAlternativeTadpoleScheme}
\end{align}\end{mdframed}
\begin{mdframed}[frametitle={Renormalization of the fermion sector
(standard and alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]\begin{align}
\delta V_{ij} &= \frac{1}{4} \left[ \left( \delta Z ^{u,L} _{ik} - \delta Z ^{u,L \dagger} _{ik} \right) V_{kj} - V_{ik} \left( \delta Z ^{d,L} _{kj} - \delta Z ^{d,L \dagger} _{kj} \right) \right] _{\xi = 1} \\
\delta Z^{f,L} _{ii} &= - \textrm{Re} \Big[ \Sigma ^{f,L} _{ii} (m_{f,i} ^2) \Big] - m_{f,i} ^2 \textrm{Re} \left[ \frac{\partial \Sigma ^{f,L} _{ii} (p ^2)}{\partial p^2} + \frac{\partial \Sigma ^{f,R} _{ii} (p ^2)}{\partial p^2} + 2\frac{\partial \Sigma ^{f,S} _{ii} (p ^2)}{\partial p^2} \right] _{p^2 = m_{f,i} ^2} \raisetag{2.2\baselineskip} \\
\delta Z^{f,R} _{ii} &= - \textrm{Re} \Big[ \Sigma ^{f,R} _{ii} (m_{f,i} ^2) \Big] - m_{f,i} ^2 \textrm{Re} \left[ \frac{\partial \Sigma ^{f,L} _{ii} (p ^2)}{\partial p^2} + \frac{\partial \Sigma ^{f,R} _{ii} (p ^2)}{\partial p^2} + 2\frac{\partial \Sigma ^{f,S} _{ii} (p ^2)}{\partial p^2} \right] _{p^2 = m_{f,i} ^2} \raisetag{2.2\baselineskip}
\end{align}\end{mdframed}
where as before, the superscripts $L$ and $R$ denote the left- and
right-chiral parts of the self-energies, while the superscript $S$
denotes the scalar part.
\subsubsection{Renormalization of the Soft-$\mathbb{Z}_2$-Breaking Parameter $m_{12}^2$}
\label{sec:renormalizationSoftm12Squared}
The last remaining parameter of the 2HDM that needs to be renormalized
is the soft-$\mathbb{Z}_2$-breaking parameter $m_{12}^2$. As before,
we replace the bare parameter by the renormalized one and its
corresponding CT,
\begin{equation}
m_{12}^2 ~\rightarrow ~ m_{12}^2 + \delta m_{12}^2 ~.
\end{equation}
In order to fix $\delta m_{12}^2$ in a physical way, one could use a
process-dependent scheme analogous to
Sec.\,\ref{sec:renormalizationMixingAngles} for the scalar mixing
angles. Since $m_{12}^2$ only appears in trilinear Higgs couplings, a
Higgs-to-Higgs decay width would have to be chosen as observable that
fixes the CT. However, as discussed in \cite{Krause:2016xku}, a
process-dependent definition of $\delta m_{12}^2$ can lead to very
large one-loop corrections in Higgs-to-Higgs decays. We therefore
employ an $\overline{\text{MS}}$ condition in {\texttt{2HDECAY}} to
fix the CT. This is done by calculating the
off-shell decay process $h \rightarrow hh$ at one-loop order and by
extracting all UV-divergent terms. This fixes the CT of $m_{12}^2$ to
\begin{mdframed}[frametitle={Renormalization of $m_{12}^2$ (standard
and alternative FJ scheme)},frametitlerule=true,frametitlebackgroundcolor=black!14,frametitlerulewidth=0.6pt,nobreak=true]
\begin{align}
\delta m_{12}^2 &= \frac{\alpha _\text{em} m_{12}^2 }{16\pi m_W^2 \left( 1 - \frac{m_W^2}{m_Z^2} \right)} \Big[ \frac{8m_{12}^2}{s_{2\beta }} - 2m_{H^\pm }^2 - m_A^2 + \frac{s_{2\alpha }}{s_{2\beta }} (m_H^2 - m_h^2) - 3(2m_W^2 + m_Z^2) \nonumber \\
&\hspace*{0.3cm} + \sum _u 3 m_u^2 \frac{1}{s_\beta ^2} - \sum _d 6 m_d^2 Y_3 \left( -Y_3 - \frac{1}{t_{2\beta}} \right) - \sum _l 2 m_l^2 Y_6 \left( -Y_6 - \frac{1}{t_{2\beta}} \right) \Big] \Delta \label{eq:renormalizationConditionm12Sq}
\end{align}\end{mdframed}
where the sum indices $u$, $d$ and $l$ indicate a summation over all
up-type and down-type quarks and charged leptons,
respectively, and
\begin{equation}
\Delta \equiv \frac{1}{\varepsilon } - \gamma _E + \ln (4\pi ) + \ln \left( \frac{\mu ^2}{\mu _R^2} \right) ~.
\end{equation}
Here, $\gamma _E$ is the Euler-Mascheroni
constant, $\varepsilon$ the dimensional shift when switching
from 4 physical to $D=4-2\varepsilon$ space-time dimensions in the framework of
dimensional regularization
\cite{Wilson1971,Wilson1972,Ashmore1972,Bollini:1972ui,THOOFT1972189} and $\mu$ is the
mass-dimensional 't Hooft scale which cancels in the calculation of
the decay amplitudes.
The result in \eqref{eq:renormalizationConditionm12Sq} is in
agreement with the formula presented in \cite{Kanemura:2015mxa}.
Since $m_{12}^2$ is $\overline{\text{MS}}$
renormalized, the user has to specify in the input file the scale at
which the parameter is understood to be
given.\footnote{All $\overline{\text{MS}}$ input
parameters are understood to be given at the same scale so that in
the input file there is only one entry for its specification.} Just as for the
$\overline{\text{MS}}$ renormalized mixing angles, the automatic
parameter conversion routine adapts $m_{12}^2$ to the scale at which
the EW one-loop corrected decay widths are evaluated in case the two
scales differ.
\subsection{Electroweak Decay Processes at LO and NLO}
\label{sec:decayProcessesAtLOandNLO}
Figure~\ref{fig:decayHiggsParticles} shows the topologies that
contribute to the tree-level and one-loop corrected decay of a scalar
particle $\phi$ with four-momentum $p_1$ into two other particles
$X _1$ and $X _2$ with four-momenta $p_2$ and $p_3$,
respectively. We emphasize that for the EW corrections, we restrict ourselves to
OS decays, {\it i.e.}~we demand
\begin{equation}
p _1^2 \ge (p_2 + p_3)^2
\end{equation}
with $p_i^2 = m_i^2$ ($i=1,2,3$) where $m_i$ denote the masses of the
three particles. Moreover, we do not calculate EW corrections to
loop-induced Higgs decays, which are of two-loop order. In particular,
we do not provide EW corrections to Higgs boson decays into two-gluon,
two-photon or $Z\gamma$ final states.
Note, however, that the decay widths implemented in
{\texttt{HDECAY}} include also loop-induced decay widths as well as
off-shell decays into heavy-quark,
massive gauge boson, neutral Higgs pair as well as Higgs and gauge boson final
states. We come back to this point in Sec.\,\ref{sec:connectionHDECAY}.
\begin{figure}[tb]
\centering
\includegraphics[width=12.5cm, trim=0cm 0cm 0cm 0.8cm, clip]{DecayAmplitudesLOandNLO.pdf}
\caption{Decay amplitudes at LO and NLO. The LO decay amplitude
$\mathcal{A}^\text{LO} _{\phi X _1 X _2}$ simply
consists of the trilinear coupling of the three particles $\phi
_1$, $X _1$ and $X _2$, while the one-loop amplitude is
given by the sum of the genuine vertex corrections
$\mathcal{A}^\text{VC} _{\phi X _1 X _2}$, indicated by
a grey blob, and the vertex counterterm $\mathcal{A}^\text{CT}
_{\phi X _1 X _2}$ which also includes all WFRCs
necessary to render the NLO amplitude UV-finite. We do not show
corrections on the external legs since in the decays we
consider, they vanish either due to OS renormalization
conditions or due to Slavnov-Taylor identities. In the case of
the alternative tadpole scheme, the vertex corrections
$\mathcal{A}^\text{VC} _{\phi X _1 X _2}$ also in
general contain tadpole diagrams.}
\label{fig:decayHiggsParticles}
\end{figure}
The LO and NLO partial decay widths were calculated by first
generating all Feynman diagrams and the corresponding amplitudes for
all decay modes that exist for the 2HDM, shown topologically in
\figref{fig:decayHiggsParticles}, with help of the tool
{\texttt{FeynArts 3.9}} \cite{Hahn:2000kx}. To that end, we used the
2HDM model file that is implemented in {\texttt{FeynArts}}, but
modified the Yukawa couplings to implement all four 2HDM
types. Diagrams that account for NLO corrections on the external legs
were not calculated since for all decay modes that we considered, they
either vanish due to OS renormalization conditions or due to
Slavnov-Taylor identities. All amplitudes were then calculated
analytically with {\texttt{FeynCalc 8.2.0}} \cite{MERTIG1991345,
Shtabovenko:2016sxi}, together with all self-energy amplitudes
needed for the CTs. For the numerical evaluation of all loop integrals involved in the analytic expression of the one-loop amplitudes, {\texttt{2HDECAY}} links {\texttt{LoopTools 2.14}} \cite{HAHN1999153}.
The LO partial decay width is obtained from the LO amplitude
$\mathcal{A}^\text{LO} _{\phi X _1 X _2}$, while the NLO
amplitude is given by the sum of all amplitudes stemming from
the vertex correction and the necessary CTs as defined in
Sec.\,\ref{sec:renormalization2HDM},
\begin{equation}
\mathcal{A}^\text{1loop} _{\phi X _1 X _2} \equiv \mathcal{A}^\text{VC} _{\phi X _1 X _2} + \mathcal{A}^\text{CT} _{\phi X _1 X _2} ~.
\end{equation}
By introducing the K\"{a}ll\'en phase space function
\begin{equation}
\lambda (x,y,z) \equiv \sqrt{x^2 + y^2 + z^2 - 2xy - 2xz - 2yz}
\end{equation}
the LO and NLO partial decay widths can be cast into the form
\begin{align}
\Gamma ^\text{LO} _{\phi X _1 X _2} &= S \frac{\lambda (m_1^2 , m_2^2 , m_3^2 )}{16\pi m_1^3} \sum _\text{d.o.f.} \left| \mathcal{A} _{\phi X _1 X _2}^\text{LO} \right| ^2 \label{eq:decayWidthLO} \\
\Gamma ^\text{NLO} _{\phi X _1 X _2} &= \Gamma ^\text{LO} _{\phi X _1 X _2} + S \frac{\lambda (m_1^2 , m_2^2 , m_3^2 )}{8\pi m_1^3} \sum _\text{d.o.f.} \text{Re} \left[ \left( \mathcal{A} _{\phi X _1 X _2}^\text{LO} \right) ^{*} \mathcal{A} _{\phi X _1 X _2}^\text{1loop} \right] + \Gamma _{\phi X _1 X _2 + \gamma} \label{eq:decayWidthNLO}
\end{align}
where the symmetry factor $S$ accounts for identical particles in the
final state and the sum extends over all degrees of freedom of the
final-state particles, {\it i.e.}~over spins or polarizations. The partial
decay width $\Gamma _{\phi X _1 X _2 + \gamma}$ accounts for
real corrections that are necessary for removing IR divergences in all
decays that involve charged particles in the initial or final
state. For this, we implemented the results given in
\cite{Goodsell:2017pdq} for generic one-loop two-body partial decay
widths. Since the involved integrals are analytically
solvable for two-body decays \cite{Denner:1991kt},
the IR corrections that are implemented in {\texttt{2HDECAY}} are
given in analytic form as well and do not require numerical
integration. Additionally, since the implemented integrals account for
the full phase-space of the radiated photon, {\it i.e.} both the ``hard''
and ``soft'' parts, our results do not depend on arbitrary cuts in the
photon phase-space.
In the following, we present all decay channels for which the
EW corrections were calculated at one-loop order:
\begin{itemize}
\item $h/H/A \to f\bar{f}$ ~ ($f=u,d,c,s,t,b,e, \mu ,\tau $)
\item $h/H \to VV$ ~ ($V=W^\pm ,Z$)
\item $h/H \to VS$ ~ ($V=Z, W^\pm$, $S=A, H^\pm$)
\item $h/H \to SS$ ~ ($S = A, H^\pm$)
\item $H \to hh$
\item $H^\pm \to VS$ ~ ($V=W^\pm$, $S=h,H,A$)
\item $H^+ \to f\bar{f}$ ~ ($f=u,c,t, \nu _e , \nu _\mu ,
\nu _\tau $ , $\bar{f} = \bar{d}, \bar{s}, \bar{b}, e^+ ,
\mu ^+ , \tau ^+ $)
\item $A \to VS$ ~ ($V=Z,W^\pm$, $S=h,H,H^\pm$)
\end{itemize}
All analytic results of these decay processes are stored in
subdirectories of {\texttt{2HDECAY}}. For a consistent connection with
{\texttt{HDECAY}}, {\it cf.}\,also Sec.\,\ref{sec:connectionHDECAY}, not all
of these decay processes are used for the calculation of the decay
widths and branching ratios, however. Decays containing pairs of
first-generation fermions are neglected, {\it i.e.}\,in {\texttt{2HDECAY}},
the EW corrections of the following processes are not used
for the calculation of the partial decay widths and branching ratios:
$h/H/A \to f\bar{f}$ ($f=u,d,e $) and $H^+ \to f\bar{f}$
($f\bar{f}=u\bar{d}, \nu _e e^+ $). The reason is
that they are overwhelmed by the Dalitz decays $\Phi \to f\bar{f}^{(')}
\gamma$ ($\Phi=h,H,A,H^\pm$) that are induced {\it
e.g.}~by off-shell $\gamma^* \to f\bar{f}$ splitting.
\subsection{Link to HDECAY, Calculated Higher-Order Corrections and Caveats}
\label{sec:connectionHDECAY}
The EW one-loop corrections to the Higgs decays in the 2HDM derived in
this work are combined with
{\texttt{HDECAY}} version 6.52 \cite{DJOUADI199856,
Djouadi:2018xqq}\footnote{The program code for
{\texttt{HDECAY}} can be downloaded from the URL
\href{http://tiger.web.psi.ch/hdecay/}{http://tiger.web.psi.ch/hdecay/}.}
in form of the new tool {\texttt{2HDECAY}}. The Fortran code {\texttt{HDECAY}}
provides the LO and QCD corrected decay widths.
As outlined in
Sec.\,\ref{sec:renormalizationGaugeSector} the EW corrections use
$\alpha _\text{em}$ at the $Z$ boson mass scale as input parameter
instead of $G_F$ as used in {\tt HDECAY}. For a
consistent combination of the EW corrected decay widths with the {\tt HDECAY}
implementation in the $G_F$ scheme we would have to convert between
the $\{\alpha_{\text{em}}, m_W, m_Z\}$ and the $\{ G_F, m_W, m_Z\}$
scheme including 2HDM higher-order corrections in the conversion
formulae. Since these conversion
formulae are not implemented yet, we chose a
pragmatic approximate solution:
In the configuration of {\texttt{2HDECAY}} with {\texttt{OMIT ELW2=0}}
being set ({\it cf.}~the input file format described in
Sec.\,\ref{sec:InputFileFormat}), the EW corrections to the decay
widths are calculated automatically. This setting
also overwrites the value that the user chooses for the input
{\texttt{2HDM}}. If {\it e.g.}~the user does not choose the 2HDM by
setting {\texttt{2HDM=0}} but at the same time chooses {\texttt{OMIT
ELW2=0}} in order
to calculate the EW corrections, then a warning is printed and
{\texttt{2HDM=1}} is automatically set internally. In
this configuration, the value of
$G_F$ given in the input file of {\texttt{2HDECAY}} is ignored by the
part of the program that calculates the EW
corrections. Instead, $\alpha _\text{em} (m_Z^2)$, given in line 26 of
the input file, is taken as independent input. This $\alpha _\text{em}
(m_Z^2)$ is used for the calculation of all electroweak
corrections. Subsequently, for the consistent combination with the
decay widths of {\texttt{HDECAY}} computed in terms of the Fermi
constant $G_F$, the latter decay widths are adapted to the input
scheme of the EW corrections by rescaling the {\texttt{HDECAY}} decay
widths with $G_F^\text{calc}/G_F$, where $G_F^\text{calc}$
is calculated by means of the tree-level relation
\eqref{eq:definitionFermiConstant} as a function of $\alpha_\text{em}
(m_Z^2)$. We expect the differences between the observables within
these two schemes to be small.
On the other hand, if {\texttt{OMIT ELW2=1}} is set, no
EW corrections are computed and {\texttt{2HDECAY}}
reduces to the original program code {\texttt{HDECAY}}, including
(where applicable)
the QCD corrections in the decay widths, the off-shell decays and the
loop-induced decays. In this case, the value of
$G_F$ given in line 27 of the input file is used as input parameter
instead of being calculated through the input value of $\alpha
_\text{em} (m_Z^2)$, and no rescaling with $G_F^\text{calc}$ is
performed. We note in particular that therefore the QCD corrected
decay widths, printed out separately by {\texttt{2HDECAY}}, will be
different in the two input options {\texttt{OMIT ELW2=0}} and {\texttt{OMIT ELW2=1}}.
Another comment is at order in view of the fact that we
implemented EW corrections to OS decays only, while
{\texttt{HDECAY}} also features the computation of
off-shell decays.
More specifically, {\texttt{HDECAY}} includes off-shell decays into
final states with an off-shell top-quark $t^*$, {\it i.e.}~$\phi \to t^*
\bar{t}$ ($\phi=h,H,A$), $H^+ \to t^* + \bar{d},\bar{s},\bar{b}$, into gauge and Higgs
boson final states with an off-shell gauge boson, $h/H \to Z^* A, A
\to Z^* h/H, \phi \to H^- W^{+*}, H^+ \to \phi W^{+*}$, and into
neutral Higgs pairs with one off-shell Higgs boson that is assumed to
predominantly decay into the $b\bar{b}$ final state, $h/H \to AA^*$,
$H \to hh^*$. The top quark total width within the 2HDM, required for the off-shell
decays with top final states, is calculated internally in {\tt HDECAY}.
In {\texttt{2HDECAY}}, we combine the EW
and QCD corrections in such a way that {\texttt{HDECAY}} still
computes the decay widths of off-shell decays, while the electroweak corrections
are added only to OS decay channels. It
is important to keep this restriction in mind when performing the
calculation for large varieties of input data. If {\it e.g.}~the lighter
Higgs boson $h$ is chosen to be the SM-like Higgs boson, then the OS
decay $h \rightarrow
W^+ W^-$ would be kinematically forbidden while the heavier Higgs boson
decay $H \rightarrow W^+ W^-$ might be OS. In such
cases, {\texttt{2HDECAY}} calculates the EW NLO corrections only for the latter decay
channel, while the LO (and QCD decay widths where applicable) are calculated
for both. The same is true for any other decay channel for which we
implemented EW corrections but which are off-shell in certain
input scenarios. Note, that the NLO EW corrections for the off-shell decays
into the massive gauge boson final states have been provided for the
2HDM in \cite{Altenkamp:2017ldc,Denner2018,Altenkamp:2017kxk}. For
the SM, the combination of {\texttt{HDECAY}} and {\texttt{Prophecy4f}}
\cite{Bredenstein:2006rh,Bredenstein:2006nk,Bredenstein:2006ha}
provides the decay widths including EW corrected off-shell decays into
these final states. In a similar way, a combination of {\texttt{2HDECAY}} and
{\texttt{Prophecy4f}} with the 2HDM decays may be envisaged in future.
For the combination of the QCD and EW corrections finally, we assume
that these corrections factorize. We denote by $\delta^{\text{QCD}}$
and $\delta^{\text{EW}}$ the relative QCD and EW corrections,
respectively. Here $\delta^{\text{QCD}}$ is normalized to the LO width
$\Gamma^{\text{HD,LO}}$,
calculated internally by {\tt HDECAY}. This means for example in the
case of quark pair final states that the LO width includes the running
quark mass in order to improve the perturbative
behaviour.
The relative EW corrections $\delta^{\text{EW}}$ on the other hand are obtained by
normalization to the LO width with on-shell particle masses. With these
definitions the QCD and EW corrected decay width into a specific final
state, $\Gamma^{\text{QCD\&EW}}$, is given by
\beq
\Gamma^{\text{QCD\&EW}} = \frac{G_F^{\text{calc}}}{G_F} \Gamma^{\text{HD,LO}}
[1+\delta^{\text{QCD}}] [1+ + \delta^{\text{EW}}]
\equiv \frac{G_F^\text{calc}}{G_F}
\Gamma^{\text{HD,QCD}}
[1 + \delta^{\text{EW}}] \;.
\eeq
We have included the rescaling factor $G_F^\text{calc}/G_F$ which is
necessary for the consistent connection of our EW corrections with the
decay widths obtained from {\tt HDECAY}, as outline above.
\underline{QCD\&EW-corrected branching ratios:}
The program code will provide the branching ratios calculated
originally by {\tt HDECAY}, which, however, for {\tt OMIT ELW2=0} are
rescaled by $G_F^\text{calc}/G_F$. They include all loop decays,
off-shell decays and QCD corrections where applicable. We summarize
these branching ratios under the name 'QCD-corrected' branching ratios
and call their associated decay widths $\Gamma^{\text{HD,QCD}}$,
keeping in mind that the QCD corrections are included only where
applicable.
Furthermore, the EW and QCD corrected branching ratios will be given
out. Here, we add the EW corrections to the decay widths calculated
internally by {\tt HDECAY} where possible, {\it i.e.}~for non-loop
induced and OS decay widths. We summarize these branching
ratios under the name 'QCD\&EW-corrected' branching ratios and call
their associated decay widths $\Gamma^{\text{QCD\&EW}}$. In
Table~\ref{tab:brs} we summarize all details and caveats on their
calculation that we described here above. All these branching ratios
are written to the output file carrying the suffix '\_BR' with its
filename, see also end of section~\ref{sec:InputFileFormat} for details.
\begin{table}[tb]
\centering
\begin{tabular}{ c c c }
\hline
{\tt IELW2=0} & QCD-corrected & QCD\&EW-corrected \\ \hline
on-shell and & $\Gamma^{\text{HD,QCD}} \frac{G_F^\text{calc}}{G_F}$ &
$\Gamma^{\text{HD,QCD}} [1+\delta^{\text{EW}}] \frac{G_F^\text{calc}}{G_F}$ \\
non-loop induced & & \\ \hline
off-shell or & $\Gamma^{\text{HD,QCD}}
\frac{G_F^\text{calc}}{G_F}$ & $\Gamma^{\text{HD,QCD}} \frac{G_F^\text{calc}}{G_F}$ \\
loop-induced & & \\ \hline
\end{tabular}
\caption{The QCD-corrected and the QCD\&EW-corrected decay widths
as calculated by {\tt 2HDECAY} for {\tt IELW2=0}. The label QCD
is in the sense that the QCD corrections are included where applicable.}
\label{tab:brs}
\end{table}
\underline{NLO EW-corrected decay widths:} For ${\tt IELW2=0}$,
we additionally give out the LO and the EW-corrected NLO decay widths
as calculated by the new addition to {\tt HDECAY}. Here the LO widths
do not include any running of the quark masses in the case of quark
final states, but are obtained for OS masses. They can hence
differ quite substantially from the LO widths as calculated in the
original {\tt HDECAY} version. These LO and EW-corrected NLO widths are
computed in the $\{\alpha_{\text{em}}, m_W, m_Z\}$ scheme and therefore
obviously do not need the inclusion of the rescaling factor
$G_F^\text{calc}/G_F$. The
decay widths are written to the output file carrying the suffix
'\_EW' with its filename. While the widths given out here are not meant
to be applied in Higgs observables as they do not include the
important QCD corrections, the study of the NLO EW-corrected decay
widths for various renormalization schemes, as provided by {\tt 2HDECAY}, allows to
analyze the importance of the EW corrections and estimate the
remaining theoretical error due to missing higher-order EW
corrections. The decay widths can also be used for phenomenological studies like
{\it e.g.}~the comparison with the EW-corrected decay widths in the
MSSM in the limit of large supersymmetric particle masses, or the
investigation of specific 2HDM parameter regions at LO and NLO as {\it e.g.}~the
alignment limit, the non-decoupling limit or the wrong-sign limit.
\underline{Caveats:} We would like to point out to
the user that it can happen that the EW-corrected decay widths become
negative because of too large negative EW corrections compared to
the LO width. There can be several reasons for this: $(i)$ The LO width
may be very small in parts of the parameter space due to
suppressed couplings. For example the decay of the heavy Higgs boson
$H$ into massive vector bosons is very small in the region where the lighter $h$
becomes SM-like and takes over almost the whole coupling to massive
gauge bosons. If the NLO EW width is not suppressed by the same
power of the relevant coupling or if at NLO there are cancellations
between the various terms that remove the
suppression, the NLO width can largely exceed the LO width. $(ii)$
The EW corrections are artificially enhanced due to a badly chosen renormalization
scheme, {\it
cf.}~Refs.~\cite{Krause:2016oke,Krause:2016xku,Krause:2017mal} for
investigations on this
subject. The choice of a different renormalization scheme may cure
this problem, but of course raises also the question for the remaining
theoretical error due to missing higher-order corrections.
$(iii)$ The EW corrections are parametrically enhanced due to
involved couplings that are large, because of small coupling
parameters in the denominator or due to light particles in the loop, see also
Refs.~\cite{Krause:2016oke,Krause:2016xku,Krause:2017mal} for discussions. This
would call for the resummation of EW corrections
beyond NLO to improve the behaviour. It is obvious that the EW
corrections should not be trusted in case of extremely large positive
or negative corrections and rather be discarded, in particular in the
comparison with experimental observables, unless some of the suggested
measures are taken to improve the behaviour.
\subsection{Parameter Conversion}
\label{sec:ParameterConversion}
Through the higher-order corrections the decay widths depend on the
renormalization scale. In {\texttt{2HDECAY}} the user can choose this
scale, called $\mu_{\text{out}}$ in the following, in the input file. It can either
chosen to be a fixed scale or the mass of the decaying Higgs
boson. Input parameters in the $\overline{\mbox{MS}}$ scheme depend
explicitly on the renormalization scale $\mu_R$. This scale also has to be
given by the user in the input scale, and is called $\mu_R$ in the
following. The value of the scale becomes particularly important when the values of
$\mu_R$ and $\mu_{\text{out}}$ differ. In this case the $\overline{\mbox{MS}}$
parameters have to be evolved from the scale $\mu_R$ to the scale $\mu_{\text{out}}$. This
applies for $m_{12}^2$ which is always understood to be an
$\overline{\mbox{MS}}$ parameter, and for $\alpha$ and $\beta$ in case
they are chosen to be $\overline{\mbox{MS}}$ renormalized.
{\texttt{2HDECAY}} internally converts the $\overline{\mbox{MS}}$
parameters from $\mu_R$ to $\mu_{\text{out}}$ by means of a linear
approximation, applying the formula
\begin{equation}
\varphi \left( \{ \mu _\text{out} \} \right) \approx \varphi
\left( \{ \mu _R \} \right) + \ln \left( \frac{\mu _\text{out}
^2}{\mu _R ^2} \right) \delta \varphi^\text{div} \left( \{
\varphi \} \right) \label{eq:scalechange}
\end{equation}
where $\varphi$ and $\delta \varphi$ denote the $\overline{\mbox{MS}}$
parameters ($\alpha$ and $\beta$, if chosen as such, $m_{12}^2$) and
their respective counterterms. The index 'div' means that only the
divergent part of the counterterm, {\it i.e.}~the terms proportional
to $1/\varepsilon$ (or equivalently $\Delta$), is taken. \newline \vspace*{-3.5mm}
In addition, a parameter conversion has to be performed, when the
chosen renormalization scheme of the input parameter differs from the
renormalization scheme at which the EW corrected decay widths are
chosen to be evaluated. {\texttt{2HDECAY}} performs this conversion
automatically which is necessary for a consistent interpretation of
the results. The renormalization schemes implemented in {\texttt{2HDECAY}} differ solely in their definition of the scalar mixing angle CTs, while the defnition of all
other CTs is fixed. Therefore, the values of $\alpha$ and $\beta$ must
be converted when switching from one renormalization scheme to
another. For this conversion, we follow the linearized approach described in Ref.\,\cite{Altenkamp:2017ldc}. Since the bare mixing angles are independent of the
renormalization scheme, their values $\varphi_i$ in a different renormalization scheme
are given by the values $\varphi_{\text{ref}}$ in the input scheme (called reference scheme in the following) and the corresponding counterterms $\delta \varphi_{\text{ref}}$ and $\delta \varphi_i$ in the reference and the other renormalization scheme, respectively, as
\begin{equation}
\varphi _i \left( \{ \mu _\text{out} \} \right) \approx \varphi _\text{ref} \left( \{ \mu _R \} \right) + \delta \varphi _\text{ref} \left( \{ \varphi _\text{ref} , \mu _R \} \right) - \delta \varphi _i \left( \{ \varphi _\text{ref} , \mu _\text{out} \} \right) ~. \label{eq:convertedParameterValues}
\end{equation}
Note, that Eq.~(\ref{eq:convertedParameterValues}) also contains the
dependence on the scales $\mu_R$ and $\mu_{\text{out}}$ introduced
above. They are relevant in case $\alpha$ and $\beta$ are
understood as $\overline{\mbox{MS}}$ parameters and additionally
depend on the renormalization scale, at which they are defined.
The relation Eq.~(\ref{eq:convertedParameterValues}) holds
approximately up to higher-order terms, as the CTs
involved in this equation are all evaluated with the mixing angles
given in the reference scheme.
\section{Program Description}
\label{sec:programDescriptionMain}
In the following, we describe the system requirements needed for
compiling and running {\texttt{2HDECAY}}, the installation procedure
and the usage of the program. Additionally, we describe the input and
output file formats in detail.
\subsection{System Requirements}
The \texttt{Python/FORTRAN} program code {\texttt{2HDECAY}} was
developed under {\texttt{Windows 10}} and {\texttt{openSUSE Leap
15.0}}. The supported operating systems are:
\begin{itemize}
\item {\texttt{Windows 7}} and {\texttt{Windows 10}} (tested
with {\texttt{Cygwin 2.10.0}})
\item {\texttt{Linux}} (tested with {\texttt{openSUSE Leap 15.0}})
\item {\texttt{macOS}} (tested with {\texttt{macOS Sierra 10.12}})
\end{itemize}
In order to compile and run {\texttt{2HDECAY}} on {\texttt{Windows}}, you need to install
{\texttt{Cygwin}} first (together with the packages {\texttt{cURL}},
{\texttt{find}}, {\texttt{gcc}}, {\texttt{g++}} and
{\texttt{gfortran}}, which also are required to be installed on {\texttt{Linux}}
and {\texttt{{macOS}}). For the compilation,
the {\texttt{GNU C}} compilers {\texttt{gcc}} (tested with versions
{\texttt{6.4.0}} and {\texttt{7.3.1}}), {\texttt{g++}} and the {\texttt{FORTRAN}}
compiler {\texttt{gfortran}} are required. Additionally, an up-to-date
version of {\texttt{Python 2}} or {\texttt{Python 3}} is required
(tested with versions {\texttt{2.7.14}} and {\texttt{3.5.0}}). For an
optimal performance of {\texttt{2HDECAY}}, we recommend that the
program is installed on a solid state drive (SSD) with high reading
and writing speeds.
\subsection{License}
{\texttt{2HDECAY}} is released under the GNU General Public License
(GPL) ({\texttt{GNU GPL-3.0-or-later}}). {\texttt{2HDECAY}} is free
software, which means that anyone can redistribute it and/or modify it
under the terms of the GNU GPL as published by the Free Software
Foundation, either version 3 of the License, or any later
version. {\texttt{2HDECAY}} is distributed without any warranty. A
copy of the GNU GPL is included in the {\texttt{LICENSE.md}} file in
the root directory of {\texttt{2HDECAY}}.
\subsection{Download}
\label{sec:Download}
The latest version of the program as well as a short quick-start
documentation is given at
\href{https://github.com/marcel-krause/2HDECAY}{https://github.com/marcel-krause/2HDECAY}. To
obtain the code either the repository is cloned or the zip archive is
downloaded and unzipped to a directory of the user's choice, which
here and in the following will be referred to as
{\texttt{\$2HDECAY}}. The main folder of {\texttt{2HDECAY}} consists
of several subfolders:
\begin{description}
\item[{\texttt{BuildingBlocks}}] Contains the analytic electroweak
one-loop corrections for all decays considered, as well as the
real corrections and CTs needed to render the decay widths UV- and
IR-finite.
\item[{\texttt{Documentation}}] Contains this documentation.
\item[{\texttt{HDECAY}}] This subfolder contains a modified version
of
{\texttt{HDECAY}} 6.52~\cite{DJOUADI199856,Djouadi:2018xqq},
needed for the computation of the LO and (where applicable) QCD
corrected decay widths.
{\tt HDECAY} also provides off-shell
decay widths and the loop-induced decay widths into gluon and
photon pair final states and into $Z\gamma$. {\tt HDECAY} is
furthermore used for the computation of the branching ratios.
\item[{\texttt{Input}}] In this subfolder, at least one
or more input files can be stored that shall be used for the computation. The
format of the input file is explained in
Sec.\,\ref{sec:InputFileFormat}. In the Github repository, the
{\texttt{Input}} folder contains an exemplary input file which is
printed in App.\,\ref{sec:AppendixInputFile}.
\item[{\texttt{Results}}] All results of a successful run of
{\texttt{2HDECAY}} are stored as output files in this subfolder
under the same name as the corresponding input files in the
{\texttt{Input}} folder, but with the file extension {\texttt{.in}}
replaced by {\texttt{.out}} and a suffix ``\_BR'' and ``\_EW'' for
the branching ratios and electroweak partial decay widths,
respectively. In the Github repository, the
{\texttt{Results}} folder contains two exemplary output files which
are given in App.\,\ref{sec:AppendixOutputFile}.
\end{description}
The main folder {\texttt{\$2HDECAY}} itself also contains several
files:
\begin{description}
\item[{\texttt{2HDECAY.py}}] Main program file of
{\texttt{2HDECAY}}. It serves as a wrapper file for calling
{\texttt{HDECAY}} in order to convert the charm and bottom quark
masses from the $\overline{\text{MS}}$ input values to the
corresponding OS values and to calculate the LO widths, QCD
corrections, off-shell and loop-induced decays, the branching ratios as well as
{\texttt{electroweakCorrections}} for the calculation of
the EW one-loop corrections.
\item[{\texttt{Changelog.md}}] Documentation of all changes made in
the program since version \newline {\texttt{2HDECAY\,1.0.0}}.
\item[{\texttt{CommonFunctions.py}}] Function library of
{\texttt{2HDECAY}}, providing functions frequently used in the
different files of the program.
\item[{\texttt{Config.py}}] Main configuration file. If
{\texttt{LoopTools}} is not installed automatically by the installer
of {\texttt{2HDECAY}}, the paths to the {\texttt{LoopTools}}
executables and libraries have to be set manually in this file.
\item[{\texttt{constants.F90}}] Library for all constants
used in {\texttt{2HDECAY}}.
\item[{\texttt{counterterms.F90}}] Definition of all fundamental CTs
necessary for the EW one-loop renormalization of the Higgs boson
decays. The CTs defined in this file require the analytic results
saved in the {\texttt{BuildingBlocks}} subfolder.
\item[{\texttt{electroweakCorrections.F90}}] Main file for the
calculation of the EW one-loop corrections to the Higgs boson
decays. It combines the EW one-loop corrections to the decay
widths with the necessary CTs and IR corrections and calculates
the EW contributions to the tree-level decay widths that
are then combined with the QCD corrections in {\texttt{HDECAY}}.
\item[{\texttt{getParameters.F90}}] Routine to
read in the input values
given by the user in the input files that are needed by
{\texttt{2HDECAY}}.
\item[{\texttt{LICENSE.md}}] Contains the full GNU General Public
License ({\texttt{GNU GPL-3.0-or-later}}) agreement under which
{\texttt{2HDECAY}} is published.
\item[{\texttt{README.md}}] Provides an overview over basic
information about the program as well as a quick-start guide.
\item[{\texttt{setup.py}}] Main setup and installation file of
{\texttt{2HDECAY}}. For a guided installation, this file should be
called after downloading the program.
\end{description}
\subsection{Installation}
\label{sec:Installation}
We highly recommend to use the automatic installation script
{\texttt{setup.py}} that is part of the {\texttt{2HDECAY}}
download. The script guides the user through the installation and asks
what components should be installed. For an installation under
{\texttt{Windows}}, the user should open the configuration file
{\texttt{\$2HDECAY/Config.py}} and check that the path to the
{\texttt{Cygwin}} executable in line 36 is set correctly before
starting the installation. In order to initiate the installation, the
user navigates to the {\texttt{\$2HDECAY}} folder and executes the
following in the command-line shell:
\begin{lstlisting}[numbers=none,language=bash,frame=single,backgroundcolor=\color{mygray}]
python setup.py
\end{lstlisting}
The script first asks the user if {\texttt{LoopTools}} should be
downloaded and installed. By entering {\texttt{y}}, the installer
downloads the {\texttt{LoopTools}} version that is specified in the
{\texttt{\$2HDECAY/Config.py}} file in line 37 and starts the
installation automatically. {\texttt{LoopTools}} is then installed in
a subdirectory of {\texttt{2HDECAY}}. Further information about the
installation of the program can be found in \cite{HAHN1999153}.
If the user already has a working version of {\texttt{LoopTools}} on the system,
this step of the installation can be skipped. In this case, the user
has to open the file {\texttt{\$2HDECAY/Config.py}} in an editor and
change the lines 33-35 to the absolute path of the
{\texttt{LoopTools}} root directory and to the {\texttt{LoopTools}}
executables and libraries on the system. Additionally, line 32 has
to be changed to
\begin{lstlisting}[language=Python,frame=single,backgroundcolor=\color{mygray},numbers=none]
useRelativeLoopToolsPath = False
\end{lstlisting}
This step is important if {\texttt{LoopTools}} is not installed
automatically with the install script, since otherwise,
{\texttt{2HDECAY}} will not be able to find the necessary executables
and libraries for the calculation of the EW one-loop
corrections.
As soon as {\texttt{LoopTools}} is installed (or alternatively, as
soon as paths to the {\texttt{LoopTools}} libraries and executables on
the user's system are being set manually in {\texttt{\$2HDECAY/Config.py}}),
the installation script asks whether it should automatically create
the makefile and the main EW corrections file
{\texttt{electroweakCorrections.F90}} and whether the program shall be
compiled. For an automatic installation, the user should type
{\texttt{y}} for all these requests to compile the main program as
well as to compile the modified version of {\texttt{HDECAY}} that
is included in {\texttt{2HDECAY}. The compilation may take several
minutes to finish. At the end of the installation the
user has the choice to 'make clean' the installation. This is optional.
In order to test if the installation was successful, the user can type
\begin{lstlisting}[numbers=none,language=bash,frame=single,backgroundcolor=\color{mygray}]
python 2HDECAY.py
\end{lstlisting}
in the command-line shell, which runs the main program. The exemplary
input file provided by the default {\texttt{2HDECAY}} version is used for the
calculation. In the command window, the output of several steps of the
computation should be printed, but no errors. If the installation was
successful, {\texttt{2HDECAY}} terminates with no errors and the
existing output files in {\texttt{\$2HDECAY/Results}} are overwritten by
the newly created ones, which, however, are equivalent to the exemplary
output files that are provided with the program.
\subsection{Input File Format}
\label{sec:InputFileFormat}
\begin{table}[tb]
\centering
\begin{tabular}{ c c c c }
\hline
Line & Input name & Allowed values and meaning \\ \hline
\makecell[tc]{6} & \makecell[tc]{{\texttt{OMIT ELW2}}} &
\makecell[tl]{0: electroweak corrections (2HDM) are calculated \\ 1: electroweak
corrections (2HDM) are neglected} \\
\makecell[tc]{9} & \makecell[tc]{{\texttt{2HDM}}} &
\makecell[tl]{0: considered model is not the 2HDM \\ 1: considered
model is the 2HDM } \\
\makecell[tc]{56} & \makecell[tc]{{\texttt{PARAM}}} & \makecell[tl]{1: 2HDM Higgs masses and $\alpha$ (lines 66-70) are given as input \\ 2: 2HDM potential parameters (lines 72-76) are given as input} \\
\makecell[tc]{57} & \makecell[tc]{{\texttt{TYPE}}} &
\makecell[tl]{1: 2HDM type I \\ 2: 2HDM type II \\
3: 2HDM lepton-specific \\ 4: 2HDM flipped} \\
\makecell[tc]{58} & \makecell[tc]{{\texttt{RENSCHEM}}} &
\makecell[tl]{0: all renormalization schemes are calculated \\ 1-17: only the chosen scheme ({\it cf.}~Tab.\,\ref{tab:2HDECAYImplementedSchemes}) is calculated} \\
\makecell[tc]{59} & \makecell[tc]{ {\texttt{REFSCHEM}} } &
\makecell[tl]{1-17: the input values of
$\alpha$, $\beta$ and $m_{12}^2$
(\textit{cf.}\,Tab.\,\ref{tab:2HDECAYInputValues}) are given in
the \\ \hspace*{0.35cm} chosen reference
scheme and at the scale $\mu _R$ given by {\texttt{INSCALE}} in
\\ \hspace*{0.35cm} case of $\overline{\mbox{MS}}$
parameters; the values of $\alpha$, $\beta$ and $m_{12}^2$ in all other
\\ \hspace*{0.35cm} schemes and
at the scale $\mu_{\text{out}}$ at which the decays are calculated,
\\ \hspace*{0.35cm} are evaluated using
Eqs.~(\ref{eq:scalechange}) and (\ref{eq:convertedParameterValues})} \\
\hline
\end{tabular}
\caption{Input parameters for the basic control of
{\texttt{2HDECAY}}. The line number corresponds to the line of
the input file where the input value can be found. In order to
calculate the EW corrections for the 2HDM, the input parameter
{\texttt{OMIT ELW2}} has to be set to 0. In this case, the given
input value of {\texttt{2HDM}} is ignored and {\texttt{2HDM=1}}
is set automatically, independent of the chosen input value. All
input values presented in this table have to be entered as integer values.}
\label{tab:2HDECAYControlInputs}
\end{table}
The format of the input file is adopted from {\texttt{HDECAY}}
\cite{DJOUADI199856, Djouadi:2018xqq}, with minor modifications to
account for the EW corrections that are implemented. The
file has to be stored as a text-only file in UTF-8
format. Since {\texttt{2HDECAY}} is a program designed for the calculation of
higher-order corrections solely for the 2HDM, only a subset of input
parameters in comparison to the original {\texttt{HDECAY}} input file
is actually used ({\it e.g.}~SUSY-related input parameters are not needed
for {\texttt{2HDECAY}}). The input file nevertheless contains the full
set of input parameters from {\texttt{HDECAY}} to make
{\texttt{2HDECAY}} fully backwards-compatible,
{\it i.e.}\,{\texttt{HDECAY\,6.52}} is fully contained in
{\texttt{2HDECAY}}. The input file contains two classes of
input parameters. The first class are input values that control the
main flow of the program ({\it e.g.\,}whether corrections for the SM or the
2HDM are calculated). The control parameters relevant for
{\texttt{2HDECAY}} are shown in Tab.\,\ref{tab:2HDECAYControlInputs},
together with their line numbers in the input file, their allowed
values and the meaning of the input values. In order to choose the
2HDM as the model that is considered, the input value {\texttt{2HDM =
1}} has to be chosen. By setting {\texttt{OMIT ELW2 = 0}}, the
EW and QCD corrections are calculated for the 2HDM, whereas
for {\texttt{OMIT ELW2 = 1}}, only the QCD corrections are
calculated. The latter choice corresponds to the corrections for the
2HDM that are already implemented in
{\texttt{HDECAY\,6.52}}. If the user sets {\texttt{OMIT ELW2 = 0}} in
the input file, then {\texttt{2HDM = 1}} is automatically set
internally, independent of the input value of {\texttt{2HDM}} that the
user provides. The input value {\texttt{PARAM}}
determines which parametrization of the Higgs sector shall be
used. For {\texttt{PARAM = 1}}, the Higgs boson masses and mixing angle
$\alpha$ are chosen as input, while for {\texttt{PARAM =
2}}, the Higgs potential parameters $\lambda _i$ are used as
input. As described at the end of Sec.\,\ref{sec:setupOfModel},
however, it should be noted that the EW corrections in
{\texttt{2HDECAY}} are in both cases parametrized through the Higgs
masses and mixing angle. Hence, if {\texttt{PARAM = 2}} is chosen, the
masses and mixing angle are calculated as functions of $\lambda _i$ by
means of
Eqs.\,(\ref{eq:parameterTransformationInteractionToMass1})-(\ref{eq:parameterTransformationInteractionToMass5}). The
input value {\texttt{TYPE}} sets the type of the 2HDM, as described in
Sec.\,\ref{sec:setupOfModel}, and {\texttt{RENSCHEM}} determines the
renormalization schemes that are used for the calculation. By setting
{\texttt{RENSCHEM = 0}}, the EW corrections to the Higgs boson
decays are calculated for all 17 implemented renormalization
schemes. This allows for analyses of the renormalization scheme dependence
and for an estimate of the effects of missing higher-order EW
corrections, but this setting has the caveat of increasing
the computation time and output file size rather significantly. A
specific integer value of {\texttt{RENSCHEME}} between 1 and 17 sets
the renormalization scheme to the chosen one. An overview of all
implemented schemes and their identifier values between 1 and 17 is
presented in Tab.\,\ref{tab:2HDECAYImplementedSchemes}.
As discussed in
Sec.\,\ref{sec:ParameterConversion}, the consistent comparison of
partial decay widths calculated in different renormalization schemes
requires the conversion of the input parameters between these
schemes. By setting {\texttt{REFSCHEM}} to a value between 1 and 17,
the input parameters for $\alpha$ and $\beta$ ({\it cf.}
Tab.\,\ref{tab:2HDECAYInputValues}) are understood as input parameters in
the chosen reference scheme and the automatic parameter conversion
is activated. The input value of the $\overline{\mbox{MS}}$
parameter $m_{12}^2$ is given at the input scale $\mu _R$. The same
applies for the input values of $\alpha$ and $\beta$ when they are
chosen to be $\overline{\text{MS}}$ renormalized. The values of
$\alpha$, $\beta$ and $m_{12}^2$ in all other renormalization
schemes and at all other scales $\mu _\text{out}$ are then
calculated using Eqs.~(\ref{eq:scalechange}) and
(\ref{eq:convertedParameterValues}). The automatic parameter
conversion requires the input parameters to be given in the mass
basis of \eqref{eq:inputSetMassBase}, \textit{i.e.}\,for the
automatic parameter conversion to be active, it is necessary to set
{\texttt{PARAM = 1}}. If instead {\texttt{PARAM = 2}} is set, then
{\texttt{REFSCHEM = 0}} is set automatically internally so that the
automatic parameter conversion is deactivated. In this case, a
warning is printed in the console. All input
values of the first class must be entered as integers.
\begin{table}[tb]
\centering
\begin{tabular}{ c c c c }
\hline
Line & Input name & Name in Sec.\,\ref{sec:EWQCD2HDMMain} & Allowed values and meaning \\ \hline
\makecell[tc]{18} & \makecell[tc]{{\texttt{ALS(MZ)}}} & $\alpha_s (m_Z)$ & \makecell[tl]{strong coupling constant (at $m_Z$)} \\
\makecell[tc]{19} & \makecell[tc]{{\texttt{MSBAR(2)}}} & $m_s (2\,\text{GeV})$ & \makecell[tl]{$s$-quark $\overline{\text{MS}}$ mass at 2 GeV in GeV} \\
\makecell[tc]{20} & \makecell[tc]{{\texttt{MCBAR(3)}}} & $m_c (3\,\text{GeV})$ & \makecell[tl]{$c$-quark $\overline{\text{MS}}$ mass at 3 GeV in GeV} \\
\makecell[tc]{21} & \makecell[tc]{{\texttt{MBBAR(MB)}}} & $m_b (m_b)$ & \makecell[tl]{$b$-quark $\overline{\text{MS}}$ mass at $m_b$ in GeV} \\
\makecell[tc]{22} & \makecell[tc]{{\texttt{MT}}} & $m_t$ & \makecell[tl]{$t$-quark pole mass in GeV} \\
\makecell[tc]{23} & \makecell[tc]{{\texttt{MTAU}}} & $m_\tau $ & \makecell[tl]{$\tau$-lepton pole mass in GeV} \\
\makecell[tc]{24} & \makecell[tc]{{\texttt{MMUON}}} & $m_\mu $ & \makecell[tl]{$\mu $-lepton pole mass in GeV} \\
\makecell[tc]{25} & \makecell[tc]{{\texttt{1/ALPHA}}} & $\alpha _\text{em} ^{-1} (0)$ & \makecell[tl]{inverse fine-structure constant (Thomson limit)} \\
\makecell[tc]{26} & \makecell[tc]{{\texttt{ALPHAMZ}}} & $\alpha _\text{em} (m_Z)$ & \makecell[tl]{fine-structure constant (at $m_Z$)} \\
\makecell[tc]{29} & \makecell[tc]{{\texttt{GAMW}}} & $\Gamma _W$ & \makecell[tl]{partial decay width of the $W$ boson } \\
\makecell[tc]{30} & \makecell[tc]{{\texttt{GAMZ}}} & $\Gamma _Z$ & \makecell[tl]{partial decay width of the $Z$ boson } \\
\makecell[tc]{31} & \makecell[tc]{{\texttt{MZ}}} & $m_Z$ &
\makecell[tl]{$Z$ boson on-shell mass in GeV} \\
\makecell[tc]{32} & \makecell[tc]{{\texttt{MW}}} & $m_W$ &
\makecell[tl]{$W$ boson on-shell mass in GeV} \\
\makecell[tc]{33-41} & \makecell[tc]{{\texttt{Vij}}} & $V_{ij}$ & \makecell[tl]{CKM matrix elements ($i\in \{ u,c,t\}$ , $j\in \{ d,s,b\} $) } \\
\makecell[tc]{61} & \makecell[tc]{{\texttt{TGBET2HDM}}} & $t_\beta $ & \makecell[tl]{ratio of the VEVs in the 2HDM} \\
\makecell[tc]{62} & \makecell[tc]{{\texttt{M\_12\textasciicircum
2}}} & $m_{12}^2 $ & \makecell[tl]{squared soft-$\mathbb{Z}_2$-breaking scale in GeV$^2$} \\
\makecell[tc]{63} & \makecell[tc]{{\texttt{INSCALE}}} & $\mu_R $
& \makecell[tl]{renormalization scale for $\overline{\text{MS}}$ inputs in GeV} \\
\makecell[tc]{64} & \makecell[tc]{ {\texttt{OUTSCALE}} } & $\mu_\text{out} $
& \makecell[tl]{ renormalization scale for the evaluation of the } \\
& & & \makecell[tl]{ partial decay widths in GeV or in terms of {\texttt{MIN}} } \\
\makecell[tc]{66} & \makecell[tc]{{\texttt{ALPHA\_H}}} & $\alpha $ & \makecell[tl]{CP-even Higgs mixing angle in radians} \\
\makecell[tc]{67} & \makecell[tc]{{\texttt{MHL}}} & $m_h $ &
\makecell[tl]{light CP-even Higgs boson mass in GeV} \\
\makecell[tc]{68} & \makecell[tc]{{\texttt{MHH}}} & $m_H $ &
\makecell[tl]{heavy CP-even Higgs boson mass in GeV} \\
\makecell[tc]{69} & \makecell[tc]{{\texttt{MHA}}} & $m_A $ &
\makecell[tl]{CP-odd Higgs boson mass in GeV} \\
\makecell[tc]{70} & \makecell[tc]{{\texttt{MH+-}}} & $m_{H^\pm } $
& \makecell[tl]{charged Higgs boson mass in GeV} \\
\makecell[tc]{72-76} & \makecell[tc]{{\texttt{LAMBDAi}}} &
$\lambda_i $ & \makecell[tl]{Higgs potential parameters
[see Eq.~(\ref{eq:scalarPotential})]} \\
\hline
\end{tabular}
\caption{Shown are all relevant physical input parameters of
{\texttt{2HDECAY}} that are necessary for the calculation of the
QCD and EW corrections. The
line number corresponds to the line of the input file where the input value
can be found. Depending on the chosen value of {\texttt{PARAM}}
({\it cf.}~Tab.\,\ref{tab:2HDECAYControlInputs}), either the Higgs
masses and mixing angle $\alpha$ (lines 66-70) or the 2HDM
potential parameters (lines 72-76) are chosen as input, but
never both simultaneously. The value {\texttt{OUTSCALE}} is
entered either as a double-precision number or as
{\texttt{MIN}}, representing the mass scale of the decaying
Higgs boson. All other input values presented in this table are
entered as double-precision numbers.}
\label{tab:2HDECAYInputValues}
\end{table}
The second class of input values in the input file are the physical
input parameters shown in Tab.\,\ref{tab:2HDECAYInputValues}, together
with their line numbers in the input file, their allowed input values
and the meaning of the input values. This is the full set of input
parameters needed for the calculation of the electroweak and QCD
corrections. All other input parameters present in the input file that
are not shown in Tab.\,\ref{tab:2HDECAYInputValues} are neglected for
the calculation of the QCD and EW corrections in the
2HDM. We want to emphasize again that depending on the
choice of {{\texttt{PARAM}} ({\it cf.}~Tab.\,\ref{tab:2HDECAYControlInputs}),
either the Higgs masses and mixing angle $\alpha$ or the Higgs
potential parameters $\lambda _i$ are chosen as independent input,
but never both simultaneously, {\it i.e.}~if {{\texttt{PARAM = 1}} is
chosen, then the input values for $\lambda _i$ are ignored, while
for {\texttt{PARAM = 2}}, the input values of the Higgs masses
and $\alpha$ are ignored and instead calculated by means of
Eqs.\,(\ref{eq:parameterTransformationInteractionToMass1})-(\ref{eq:parameterTransformationInteractionToMass5}).
All input values of the second class are entered in
{\texttt{FORTRAN}} double-precision format, {\it i.e.}~valid input
formats are {\it e.g.}~{\texttt{MT = 1.732e+02}} or {\texttt{MHH =
258.401D0}}. Since $m_{12}^2$ and, in case of a chosen $\overline{\text{MS}}$ scheme, $\alpha$ and $\beta$ depend on the renormalization scale $\mu _R$ at which these parameters are given, the calculation of the partial decay widths depends on this scale. Moreover, since the partial decay widths are evaluated at the (potentially different) renormalization scale $\mu _\text{out}$, the decay widths and branching ratios depend on this scale as well. In order to avoid artificially large corrections, both scales should be chosen appropriately. The input value
{\texttt{INSCALE}} of $\mu _R$, \textit{i.e.}\,the scale at which all $\overline{\text{MS}}$ parameters are defined, is entered as a double-precision number. The input value
{\texttt{OUTSCALE}} of $\mu _\text{out}$, \textit{i.e.}\,the renormalization scale at which the partial decay widths are evaluated, can be entered either as a
double-precision number or it can be expressed in terms of the mass
scale {\texttt{MIN}} of the decaying Higgs boson, {\it i.e.}\,setting {\texttt{OUTSCALE=MIN}} sets $\mu _R = m_1$ for each decay
channel, where $m_1$ is the mass of the decaying Higgs boson in the
respective channel. Note finally, that the input
masses for the $W$ and $Z$ gauge bosons must be the on-shell
values for consistency with the renormalization conditions applied
in the EW corrections.
\begin{table}[tb]
\centering
\begin{tabular}{ c c c c c }
\hline
Input ID & Tadpole scheme & $\delta \alpha$ & $\delta \beta$ & Gauge-par.-indep. $\Gamma$ \\ \hline
\makecell[tc]{1} & \makecell[tc]{standard} & \makecell[tc]{KOSY} & \makecell[tc]{KOSY (odd)} & \makecell[tc]{\xmark } \\
\makecell[tc]{2} & \makecell[tc]{standard} & \makecell[tc]{KOSY} & \makecell[tc]{KOSY (charged)} & \makecell[tc]{\xmark } \\
\makecell[tc]{3} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{KOSY} & \makecell[tc]{KOSY (odd)} & \makecell[tc]{\xmark } \\
\makecell[tc]{4} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{KOSY} & \makecell[tc]{KOSY (charged)} & \makecell[tc]{\xmark } \\
\makecell[tc]{5} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{$p_{*}$-pinched} & \makecell[tc]{$p_{*}$-pinched (odd)} & \makecell[tc]{\cmark } \\
\makecell[tc]{6} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{$p_{*}$-pinched} & \makecell[tc]{$p_{*}$-pinched (charged)} & \makecell[tc]{\cmark } \\
\makecell[tc]{7} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{OS-pinched} & \makecell[tc]{OS-pinched (odd)} & \makecell[tc]{\cmark } \\
\makecell[tc]{8} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{OS-pinched} & \makecell[tc]{OS-pinched (charged)} & \makecell[tc]{\cmark } \\
\makecell[tc]{9} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{proc.-dep. 1} & \makecell[tc]{proc.-dep. 1} & \makecell[tc]{\cmark } \\
\makecell[tc]{10} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{proc.-dep. 2} & \makecell[tc]{proc.-dep. 2} & \makecell[tc]{\cmark } \\
\makecell[tc]{11} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{proc.-dep. 3} & \makecell[tc]{proc.-dep. 3} & \makecell[tc]{\cmark } \\
\makecell[tc]{12} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{OS1} & \makecell[tc]{OS1} & \makecell[tc]{\cmark } \\
\makecell[tc]{13} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{OS2} & \makecell[tc]{OS2} & \makecell[tc]{\cmark } \\
\makecell[tc]{14} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{OS12} & \makecell[tc]{OS12} & \makecell[tc]{\cmark } \\
\makecell[tc]{15} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{BFMS} & \makecell[tc]{BFMS} & \makecell[tc]{\cmark } \\
\makecell[tc]{16} & \makecell[tc]{standard} & \makecell[tc]{ $\overline{\text{MS}}$ } & \makecell[tc]{ $\overline{\text{MS}}$ } & \makecell[tc]{\xmark } \\
\makecell[tc]{17} & \makecell[tc]{alternative (FJ)} & \makecell[tc]{ $\overline{\text{MS}}$ } & \makecell[tc]{ $\overline{\text{MS}}$ } & \makecell[tc]{\cmark } \\
\hline
\end{tabular}
\caption{Overview over all renormalization schemes for the mixing
angles $\alpha$ and $\beta$ that are implemented in
{\texttt{2HDECAY}}. By setting {\texttt{RENSCHEM}} in the input
file, {\it cf.}~Tab.\,\ref{tab:2HDECAYControlInputs}, equal to the
Input ID the renormalization scheme is chosen. In case of 0 the
results for all renormalization schemes are given out. The
definition of the CTs $\delta \alpha$ and $\delta \beta$ in each
scheme is explained in
Sec.\,\ref{sec:renormalizationMixingAngles}. The crosses and
check marks in the column for gauge independence indicate
whether the chosen scheme in general yields explicitly gauge-independent
partial decay widths or not.}
\label{tab:2HDECAYImplementedSchemes}
\end{table}
The amount of input files that can be stored in the input folder is
not limited. The input files can have arbitrary non-empty names and
filename extensions\footnote{On some systems, certain filename
extensions should be avoided when naming the input files, as they
are reserved for certain types of files ({\it e.g.}~under {\texttt{Windows}},
the {\texttt{.exe}} file extension is automatically connected to
executables by the operating system, which can under certain
circumstances lead to runtime problems when trying to read the
file). Choosing text file extensions like {\texttt{.in}},
{\texttt{.out}}, {\texttt{.dat}} or {\texttt{.txt}} should in
general be unproblematic.}. The output files are saved in the
{\texttt{\$2HDECAY/Results}} subfolder under the same name as the
corresponding input files, but with their filename extension replaced
by {\texttt{.out}}.
For each input file, two output files are generated. The output file
containing the branching ratios is indicated by the filename suffix
'\_BR', while the output file containing the electroweak
partial decay widths is indicated by the filename suffix '\_EW'.
\subsection{Structure of the Program}
As briefly mentioned in Sec.\,\ref{sec:Download}, the main program
{\texttt{2HDECAY}} combines the already existing QCD corrections from
{\texttt{HDECAY}} with the full EW one-loop
corrections. Depicted in \figref{fig:flowchart2HDECAY} is the
flowchart of {\texttt{2HDECAY}} which shows how the QCD and
EW corrections are combined by the main wrapper file
{\texttt{2HDECAY.py}}.
\begin{figure}[tb]
\centering
\includegraphics[width=13.8cm]{flowchart.pdf}
\caption{Flowchart of {\texttt{2HDECAY}}. The main wrapper file
{\texttt{2HDECAY.py}} generates a list of input files, provided by the
user in the subfolder {\texttt{\$2HDECAY/Input}}, and iterates over
the list. For each selected input file in the list, the wrapper
calls {\tt HDECAY} and the subprogram {\tt
electroweakCorrections}. The computed branching ratios including
the EW and QCD corrections as described in the text are written
to the output file with suffix '\_BR', the calculated LO and NLO EW-corrected
partial decay widths are given out in the output file with suffix
'\_EW'. For further details, we refer to the text.}
\label{fig:flowchart2HDECAY}
\end{figure}
First, the wrapper file generates a list of all input files that the
user provides in {\texttt{\$2HDECAY/Input}}. The user can provide an
arbitrary non-zero amount of input files with arbitrary filenames, as
described in Sec.\,\ref{sec:InputFileFormat}. For any input file in
the list, the wrapper file first calls {\texttt{HDECAY}} in a
so-called minimal run, technically by calling {\texttt{HDECAY}} in the
subfolder {\texttt{\$2HDECAY/HDECAY}} with an additional flag ``1'':
\begin{lstlisting}[numbers=none,language=bash,frame=single,backgroundcolor=\color{mygray}]
run 1
\end{lstlisting}
With this flag, HDECAY reads the selected input file from the input
file list and uses the input values only to convert the
$\overline{\text{MS}}$ values of the $c$- and $b$-quark masses, as
given in the input file, to the corresponding pole masses, but no other computations are performed at
this step.
The wrapper file then calls the subprogram
{\texttt{electroweakCorrections}}, which reads the selected input file
as well as the OS values of the quark masses. With these input values,
the full EW one-loop corrections are calculated for all decays that
are kinematically allowed, as described in
Sec.\,\ref{sec:decayProcessesAtLOandNLO}, and the value of $G_F^{\text{calc}}$
at the $Z$ mass is calculated, as described in
Sec.\,\ref{sec:connectionHDECAY}. Subsequently, a temporary new input
file is created, which consists of a copy of the selected input file
with the calculated OS quark masses, the calculated value of
$G_F^{\text{calc}}$ and all EW corrections being appended.
Lastly, the wrapper file calls {\texttt{HDECAY}} without the minimal
flag. In this configuration, {\texttt{HDECAY}} reads the temporary
input file and calculates the LO widths and QCD corrections to the
decays. Moreover, the program calculates off-shell decay widths
as well as the loop-induced decays to final-state pairs
of gluons or photons and $Z \gamma$. Furthermore, the branching ratios
are calculated by {\texttt{HDECAY}}. The results of these computations
are consistently combined with the electroweak corrections, as
described in Sec.\,\ref{sec:connectionHDECAY}. The results are saved in an
output file in the {\texttt{\$2HDECAY/Results}} subfolder.
The wrapper file repeats these steps for each file in the input file
list until the end of the list is reached.
\subsection{Usage}
Before running the program, the user should check that all input files
for which the computation shall be performed are stored in the
subfolder {\texttt{\$2HDECAY/Input}}. The input files have to be
formatted exactly as described in Sec.\,\ref{sec:InputFileFormat} or
otherwise the input values are not read in correctly and the program
might crash with a segmentation error. The exemplary input file
printed in App.\,\ref{sec:AppendixInputFile} that is part of the
{\texttt{2HDECAY}} repository can be used as a template for generating
other input files in order to avoid formatting problems.
The user should check the output subfolder
{\texttt{\$2HDECAY/Results}} for any output files of previous runs of
{\texttt{2HDECAY}}. These previously created output files are
overwritten if in a new run input files with the same names as the
already stored output files are used. Hence, the user is advised to
create backups of the output files before starting a new run of
{\texttt{2HDECAY}}.
In order to run the program, open a terminal, navigate to the
{\texttt{\$2HDECAY}} folder and execute the following command:
\begin{lstlisting}[numbers=none,language=bash,frame=single,backgroundcolor=\color{mygray}]
python 2HDECAY.py
\end{lstlisting}
If {\texttt{2HDECAY}} was installed correctly according to
Sec.\,\ref{sec:Installation} and if the input files have the correct
format, the program should now compute the EW and/or QCD
corrections according to the flowchart shown in
\figref{fig:flowchart2HDECAY}. Several intermediate results and
information about the computation are printed in the terminal. As
soon as the computation for all input files is done,
{\texttt{2HDECAY}} is terminated and the resulting output files can be
found in the {\texttt{\$2HDECAY/Results}} subfolder.
\subsection{Output File Format}
\label{sec:OutputFileFormat}
For each input file, two output files with the suffixes '\_QCD'
and '\_EW' for the branching ratios and electroweak partial decay
widths, respectively, are generated in an SLHA format, as described in
Sec.~\ref{sec:connectionHDECAY}. The SLHA output
format \cite{Skands:2003cj,Allanach:2008qq,Mahmoudi:2010iz} in its strict and original
sense has only been designed for supersymmetric models. We have
modified the format to account for the EW corrections that are implemented in
{\texttt{2HDECAY}} in the 2HDM. As a reference for the following description,
exemplary output files are given in
App.\,\ref{sec:AppendixOutputFile}. These modified SLHA output files are
only generated if {\texttt{OMIT ELW2=0}} is set in the input file,
{\it i.e.}\,only if the electroweak corrections to the 2HDM decays are
taken into account. In the following we describe the changes that we
have applied.
The first block {\texttt{BLOCK DCINFO}} contains basic information
about the program itself, while the subsequent three blocks
{\texttt{SMINPUTS}}, {\texttt{2HDMINPUTS}} and {\texttt{VCKMIN}}
contain the input parameters used for the calculation that were
already described in Sec.\,\ref{sec:InputFileFormat}. As explained in
Sec.\,\ref{sec:connectionHDECAY}, the value of $G_F$ printed in the
output file is not necessarily the same as the one given in the input
file if {\texttt{OMIT ELW2=0}} is set, since in this case, $G_F$ is
calculated from the input value $\alpha _\text{em} (m_Z^2)$
instead, and this value is then given out. These four blocks are
given out in both output files.
In the output file containing the branching ratios, indicated by the
suffix '\_BR', subsequently two blocks follow for
each Higgs boson ($h,H,A$ and
$H^\pm$). They are called {\texttt{DECAY QCD}} and {\texttt{DECAY
QCD\&EW}}.
The block {\texttt{DECAY QCD}} contains the total decay
width, the mixing angles $\alpha$, $\beta$,
the $\overline{\mbox{MS}}$ parameter
$m_{12}^2$\footnote{Note that they differ from
the input values if $\mu_R \ne \mu_{\text{out}}$ or if the
reference/input scheme is different from the renormalization scheme in
which the decays are evaluated.}, and the
branching ratios of the decays of the respective Higgs boson, as
implemented in {\tt HDECAY}. These are in
particular the LO (loop-induced for the $gg$, $\gamma\gamma$ and
$Z\gamma$ final states) decay widths including the relevant and state-of-the
art QCD corrections where applicable ({\it
cf.}~\cite{DJOUADI199856,Djouadi:2018xqq} for further details).
For decays into heavy quarks, massive vector bosons, neutral Higgs
pairs as well as gauge and Higgs boson final states
also off-shell decays are computed if necessary. We want to emphasize
again that the partial and total decay widths differ from the ones of the original {\tt
HDECAY} version if {\texttt{OMIT ELW2=0}} is set, as for
consistency with the computed EW corrections in this case the {\tt
HDECAY} decay widths are rescaled by $G_F^\text{calc}/G_F$, as
explained in Sec.~\ref{sec:connectionHDECAY}. If {\texttt{OMIT
ELW2=1}} is set, no EW corrections are computed and the {\tt HDECAY}
decay widths are computed with $G_F$ as in the original {\tt HDECAY} version.
The block {\texttt{DECAY
QCD\&EW}} contains the total decay width, the
mixing angles $\alpha$, $\beta$, the $\overline{\mbox{MS}}$
parameter $m_{12}^2$, and the
branching ratios of the respective Higgs boson including both the QCD
corrections (provided by {\tt
HDECAY}) and the EW corrections (computed by {\tt 2HDECAY}) }to the
LO decay widths. Note that the
LO decay widths are also computed by {\tt 2HDECAY}. As an additional
cross-check, we internally compare the respective {\tt HDECAY} LO decay
width (rescaled by $G_F^\text{calc}/G_F$ and calculated with OS masses
for this comparison) with the one computed by
{\tt 2HDECAY}. If they differ (which they should not), a warning is
printed on the screen. As described in
Sec.\,\ref{sec:connectionHDECAY}, we emphasize again that the EW corrections are
calculated and included only for OS decay channels that are
kinematically allowed and for non-loop-induced decays. Therefore, some
of the branching ratios given out may be QCD-, but not
EW-corrected. The total decay width given out in this block is
the sum of all accordingly computed partial decay widths.
The last block at the end of the file with the branching ratios
contains the QCD-corrected branching ratios of the top-quark calculated in the 2HDM.
It is required for the computation of the Higgs decays into final states
with an off-shell top.
In the output file with the EW corrected NLO decay widths, indicated
by the suffix '\_EW', the first four blocks
described above are instead followed by the two blocks {\texttt{LO DECAY WIDTH}} and
{\texttt{NLO DECAY WIDTH}} for each Higgs boson ($h,H,A$ and
$H^\pm$). In these blocks, the partial decay widths at
LO and including the one-loop EW corrections are given out,
respectively, together with the
mixing angles $\alpha$, $\beta$ and the $\overline{\mbox{MS}}$
parameter $m_{12}^2$. These values of the widths
are particularly useful for studies of
the relative size of the EW corrections and for studying the
renormalization scheme dependence of the EW corrections. This allows
for a rough estimate of the remaining theoretical error due to missing higher-order
EW corrections. Since the EW corrections are calculated only for OS
decays and additionally only for decays that are not loop-induced,
these two blocks do not contain all final states written out in the
blocks {\texttt{DECAY QCD}} and {\texttt{DECAY
QCD\&EW}}. Hence, depending on the input values that are
chosen, it can happen that the two blocks {\texttt{DECAY QCD}} and {\texttt{DECAY
QCD\&EW}} contain decays that are not printed out in the blocks
{\texttt{LO DECAY WIDTH}} and {\texttt{NLO DECAY WIDTH}}, since for the calculation of
the branching ratios, off-shell and loop-induced decays are considered
by {\texttt{HDECAY}} as well.
\section{Summary}
\label{sec:summary}
We have presented the program package {\texttt{2HDECAY}} for the
calculation of the Higgs boson decays in the 2HDM. The tool computes the
NLO EW corrections to all 2HDM Higgs boson decays into OS final states
that are not loop-induced. The user can choose among 17 different
renormalization schemes that have been specified in the manual.
They are based on different renormalization schemes for the mixing
angles $\alpha $ and $\beta$, an $\overline{\text{MS}}$ condition for
the soft-$\mathbb{Z}_2$-breaking scale $m_{12}^2$ and an OS scheme for
all other counterterms and wave function renormalization constants of
the 2HDM necessary for calculating the EW corrections.
The EW corrections are combined with the state-of-the-art QCD
corrections obtained from {\texttt{HDECAY}}. The EW\&QCD-corrected
total decay widths and branching ratios are given out in an
SLHA-inspired output file format. Moreover, the tool provides
separately an SLHA-inspired output for the LO and
EW NLO partial decay widths to all OS and non-loop-induced
decays. This separate output enables {\it e.g.}~an
efficient analysis of the size of the EW corrections in the 2HDM or
the comparison with the relative
EW corrections in the MSSM as a SUSY benchmark model. The
implementation of several different renormalization schemes
additionally allows for the investigation of the numerical effects of
the different schemes and an estimate of the residual theoretical
uncertainty due to missing higher-order EW
corrections. For a consistent estimate of this error,
an automatic parameter conversion routine is implemented,
performing the automatic conversion of the input
values of $\alpha$, $\beta$ and $m_{12}^2$ from a reference scheme to
all other renormalization schemes that are implemented, as well as
from the $\overline{\text{MS}}$ input renormalization scale $\mu _R$
to the renormalization scale $\mu _\text{out}$ at which the partial
decay widths are evaluated. Being fast, our new
tool enables efficient phenomenological studies of the 2HDM Higgs
sector at high precision. The latter is necessary to reveal indirect
new physics effects in the Higgs sector and to identify the true
underlying model in case of the discovery of additional Higgs bosons. This
brings us closer to our goal of understanding electroweak symmetry
breaking and deciphering the physics puzzle in fundamental particle
physics.
\subsection*{Acknowledgments}
The authors thank David Lopez-Val and Jonas M\"{u}ller for
independently cross-checking some of the analytic results derived for
this work. The authors express gratitude to David Lopez-Val for his
endeavors on debugging the early alpha versions of {\texttt{2HDECAY}}
and to Stefan Liebler and Florian Staub for helpful discussions concerning the real corrections to the decays. The authors thank Ansgar Denner, Stefan Dittmaier and Jean-Nicolas Lang for helpful discussions and for providing the analytic results of their mixing angle counterterms to us for the implementation in {\texttt{2HDECAY}}. MK
and MM acknowledge financial support from the DFG project "Precision
Calculations in the Higgs Sector - Paving the Way to the New Physics
Landscape" (ID: MU 3138/1-1).
\begin{appendix}
\section{Exemplary Input File}
\label{sec:AppendixInputFile}
In the following, we present an exemplary input file
{\texttt{2hdecay.in}} as it is included in the subfolder
{\texttt{\$2HDECAY/Input}} in the {\texttt{2HDECAY}} repository. The
first integer in each line represents the line number and is not part
of the actual input file, but printed here for convenience. The
meaning of the input parameters is specified in
Sec.\,\ref{sec:InputFileFormat}. In comparison to the input file
format of the unmodified {\texttt{HDECAY}}
program\cite{DJOUADI199856,Djouadi:2018xqq}, the lines 6, 26, 28, 58, 59, 63 and 64 are new, but the rest of the input file format is unchanged. We
want to emphasize again that the value {\texttt{GFCALC}} in the input
file is overwritten by the program and thus not an input value that
is provided by the user, but it is calculated by {\texttt{2HDECAY}}
internally. The sample 2HDM parameter point has been checked
against all relevant theoretical and experimental constraints. In
particular it features a SM-like Higgs boson with a mass of
125.09~GeV which is given by the lightest CP-even neutral Higgs
boson $h$. For details on the applied constraints, we refer to
Refs.~\cite{Basler:2016obg,Muhlleitner:2017dkd}.
\lstinputlisting{2hdecay.in}
\section{Exemplary Output Files}
\label{sec:AppendixOutputFile}
In the following, we present exemplary output files
{\texttt{2hdecay\_BR.out}} and {\texttt{2hdecay\_EW.out}} as they
are generated from the sample input file
{\texttt{2hdecay.in}} and included in the subfolder \\
{\texttt{\$2HDECAY/Results}} in the
{\texttt{2HDECAY}} repository. The suffixes ``\_BR'' and ``\_EW''
stand for the branching ratios and electroweak partial decay widths,
respectively. The first integer in each line represents the line
number and is not part of the actual output file, but printed here for
convenience. The output file format is explained in detail in
Sec.\,\ref{sec:OutputFileFormat}. The exemplary output file was
generated for a specific choice of the renormalization scheme, {\it i.e.}~we
have set {\texttt{RENSCHEM = 7}} in line 58 of the input file,
{\it cf.}~App.\,\ref{sec:AppendixInputFile}. For {\texttt{RENSCHEM = 0}},
the output file becomes considerably longer, since the electroweak
corrections are calculated for all 17 implemented renormalization
schemes. We chose {\texttt{REFSCHEM = 5}} and
{\texttt{INSCALE = 125.09D0}}. This means that the input values for
$\alpha$ and $\beta$ are understood to be given in the renormalization
scheme 5 and the scale at which $\alpha$, $\beta$ and the
$\overline{\mbox{MS}}$ parameter $m_{12}^2$ are defined is equal to 125.09~GeV.
\subsection{Exemplary Output File for the Branching Ratios}
The exemplary output file {\texttt{2hdecay\_BR.out}} contains the
branching ratios without and with the electroweak corrections. The
content of the file is presented in the following.
\lstinputlisting{2hdecay_BR.out}
In the following, we make some comments on the output
files that partly pick up hints and caveats made in the main text of
the manual.
As can be inferred from the output, we give for the decays of each
Higgs boson the values of $\alpha$, $\beta$ and $m_{12}^2$. These
values change from the input values and for each Higgs boson as we
have to perform the parameter conversion from the input reference
scheme 5 to the renormalization scheme 7 and because we use for the
loop corrected widths the renormalization scale given by the mass of
the decaying Higgs boson, since we set
{\texttt{OUTSCALE = MIN}} while the input values for these parameters
are understood to be given at the mass of the SM-like Higgs boson.
Furthermore notice that indeed the branching ratios of
the lightest CP-even Higgs boson $h$ are SM-like. All branching ratios
presented in the blocks {\texttt{DECAY QCD}} can be compared to the
ones generated by the program code {\texttt{HDECAY}} version 6.52. The
user will notice that the partial widths related to the branching
ratios generated by {\texttt{2HDECAY}} and {\texttt{HDECAY}},
respectively, differ due to
the rescaling factor $G_F^\text{calc}/G_F =
1.026327$, which is applied in {\texttt{2HDECAY}}
for the consistent
combination of the EW-corrected decay widths with the decay widths generated by
{\texttt{HDECAY}}. Be aware that the rescaling factor
appears in the loop induced decay into $Z\gamma$ and in the off-shell decays non-linearly. This is why also the branching
ratios given here differ from the ones generated by
{\texttt{HDECAY}}6.52.
The comparison furthermore shows an additional difference
between the decay widths for the heavy CP-even Higgs boson $H$ into
massive vector bosons, $\Gamma (H \to VV)$ ($V=W,Z$), of around
2-3\%. The reason is that {\texttt{HDECAY}}
throughout computes these
decay widths using the double off-shell formula while
{\texttt{2HDECAY}} uses the on-shell formula for Higgs boson masses
above the threshold. Let us also note some phenomenological features of the chosen
parameter point. The $H$ boson with a mass of 382~GeV is heavy enough
to decay on-shell into $WW$ and $ZZ$,
and also into the 2-Higgs boson final state $hh$. It decays off-shell into
$AA$ and the gauge plus Higgs boson final state $ZA$ with branching
ratios of ${\cal O}(10^{-10})$ and ${\cal O}(10^{-4})$,
respectively. The pseudoscalar with a mass of 351~GeV decays on-shell
into the gauge plus Higgs boson final state $Zh$ with a branching ratio
at the per cent level. The charged Higgs boson has a mass of 414~GeV
allowing it to decay on-shell in the gauge plus Higgs boson final
state $W^+ h$ with a branching ratio at the per cent level. It decays
off-shell into the final states $W^+ H$ and $W^+ A$ with branching
ratios of ${\cal O}(10^{-5})$ and ${\cal O}(10^{-3})$,
respectively.
\subsection{Exemplary Output File for the Electroweak Partial Decay Widths}
The exemplary output file {\texttt{2hdecay\_EW.out}} contains the LO
and electroweak NLO partial decay widths. The content of the file is
presented in the following.
\lstinputlisting{2hdecay_EW.out}
The inspection of the output file shows that the EW
corrections reduce the $h$ decay widths, and the
relative NLO EW corrections,
$\Delta^{\text{EW}} = (\Gamma^{\text{EW}} -
\Gamma^{\text{LO}})/\Gamma^{\text{LO}}$, range between -6.3 and
-2.2\% for the decays $\Gamma(h \to \mu^+\mu^-)$ and $\Gamma(h\to
s\bar{s})$, respectively. Regarding $H$, the corrections can both
enhance and reduce the decay widths. The relative corrections range
between -11.5 and 27.7\% for the decays $\Gamma(H\to \mu^+ \mu^-)$
and $\Gamma(H \to hh)$, respectively. The relative corrections to
the $A$ decay widths vary between -31.2 and 0.3\% for the decays $\Gamma(A\to Zh)$
and $\Gamma(A \to t\bar{t})$, respectively. And those for the $H^\pm$
decays between -20.6 and 11.1\% for the decays $\Gamma(H^+ \to u\bar{b})$
and $\Gamma(H^+ \to W^+h)$, respectively. The EW corrections (for the
renormalization scheme number 7) of the
chosen parameter point can hence be sizeable. Finally, note also
that LO and NLO EW-corrected decay widths are given out for
on-shell and non-loop induced decays only.
\end{appendix}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,375 |
Dear Abby: Man who flirts online brushes aside…
Dear Abby: Man who flirts online brushes aside his girlfriend's concerns
By Jeanne Phillips |
DEAR ABBY: My boyfriend and I have been a couple for three years. We live together and have an incredible relationship and an amazing sex life. A while ago, he was approached by a strange woman on social media. Through Hangouts he told her she was beautiful and that he was looking for the right woman to be with. Their communication lasted about a week. It has now happened again. He handed out his phone number, and this one has sent him videos of her dancing wearing next to nothing. He tells her she has an amazing body and made comments to the effect that she must be wild in bed and he thinks only of her. When he talks to these other women, he tells them he lives alone. When I tell him this bothers me, he doesn't get upset. He swears he has feelings for only me and no one else, and that he's just having a little fun. I want to believe him, but I feel hurt and disrespected when I read what he's saying to these women. My heart is heavy because he used to talk to me like that and no longer does. Should I be worried? — SHARING HIM IN OHIO
DEAR SHARING HIM: You should not only be worried, you should be out of there. You may have invested three years in this person, but the sooner you divest yourself of him the better it will be for you. His actions show that his word cannot be trusted. He's not only lying to these women, he is also lying to you. Men who love and respect women do not treat them the way he is treating you.
DEAR ABBY: I'm a 13-year-old girl, and I'm bisexual. Some of my closest friends know, but that's it. Mom doesn't know, and neither do my gramma or papa. I'm afraid if I tell them they'll be disappointed in their little girl. Also, I'm growing up without a father, so that may have something to do with it. I wonder if not having a male role model is why I'm driven to like girls. It took me a while to figure out that I was bisexual. It was at the beginning of seventh grade, when people were talking about being bi. So I guess I need to find out who I am as a person. When I told my friend I was bi and I liked her, she was shocked and surprised. I think she took it the wrong way and thought I was asking her out. That afternoon she came up to me and said, "I like you, but only as a friend. I hope this doesn't damage our friendship." For me it did, and I haven't gotten the courage to go talk to her about it again. I was only saying that to tell her how I FEEL, not to ask her out. — INSECURE AND CONFUSED
DEAR INSECURE AND CONFUSED: You are right that you need to find out who you are as a person. You are very young and still discovering. People do NOT become gay or bisexual because of conversations they hear in the seventh grade or because their fathers are absent. Sexual orientation is simply a part of who we are.
You were clumsy about the way you "outed" yourself to your friend. Put aside your fears, talk to her again and explain that you weren't asking her out, and the feelings you were describing were not directed at her. If she's truly a friend, everything will be all right.
Contact Dear Abby at DearAbby.com or P.O. Box 69440, Los Angeles, CA 90069.
More in Advice
Bonnie Blodgett: Winter dreams: What I'm going to plant in 2023 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,882 |
Take children on an adventure without leaving the classroom with the addition of the Flagship Carpets Traveling the Solar System Photo Fun Classroom Rug. This detailed photographic rug features images of the planets on a starry background that is out of this world. Each planet is labeled so students can quickly recognize each heavenly body. The treated rug is constructed of dense nylon fiber that resists stains, mildew, odors and mold. The edges are tightly bound and double-stitched for maximum strength, and the skid-resistant Action Bac woven backing system prevents wrinkles and bunching. The Flagship Carpets Traveling the Solar System Photo Fun Classroom Rug is made in the USA and comes backed by a lifetime abrasive wear warranty. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,751 |
Koillissanomat är en finländsk dagstidning som utkommer i Kuusamo.
Tidningen, som grundades 1950, är en femdagarstidning, vars upplaga 2009 uppgick till omkring 7 358 exemplar.
Källor
Finländska dagstidningar
Finskspråkiga dagstidningar
Kuusamo
Dagstidningar startade 1950 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,684 |
Mamelodi Sundowns playmaker, Teko Modise has explained why his overseas move never happened and says it "kills" him that he couldn't leave Orlando Pirates at the time despite lots of interest.
The 32-year-old says he had offers from the likes of Panathinaikos, West Bromwich and Stoke City, among others, but ended up joining Masandawana because he didn't have a release clause at the time.
"A buy-out clause in my contract didn't exist at the time. There was no concrete number and there was just speculation. You know what these European teams are like – they lose interest if you mess them around," he told Soccer Laduma's David Minchella in an exclusive interview.
"They'll only chase you for a certain time. We tried everything we could! Playing overseas was one of my biggest ambitions.
"The fact that I could have made it there is what kills me.
"Even at Pirates some big European teams came through with offers. So many teams came through with offers – Panathinaikos, West Brom and Stoke City. There were others from smaller countries like Denmark as well, " Modise said.
'Donadoni' further explained how hard his failed move abroad hit him and how he struggled to remain focussed on playing for Pirates.
"I just wanted an opportunity, even if I'd failed! None of the teams that came through said I must come for trials. They just wanted me to come, do a medical and sign," he added.
Make sure to read the exclusive full length interview with Teko Modise, where he also reveals why Pitso Mosimane didn't speak to him for years and that he probably would have died without football. It's a must-read interview with Soccer Laduma's David Minchella in edition 918. It's in stores and on sale now. | {
"redpajama_set_name": "RedPajamaC4"
} | 713 |
Category: MLB
Double X, Hitting Home Runs and Some Guy You Never Heard of
By Blaidd Drwg
Part one of a two parter! In the immortal words of Mel Allen, "How about that?"
In the history of baseball, there have been 43 instances of a player hitting 50 or more home runs in a season. For reference, here is the breakdown of those seasons:
Number of HRs Number of Times Accomplished
Here is the same list with one additional column – Number of Times Lead League. That column represents the number of times that the corresponding number of home runs was the highest total in the AL or NL in its respective season.
Number of HRs Number of Times Accomplished Number of Times Lead League
58 4 3*
The asterisk is an unusual case – in 1997, thanks to a mid-season trade, Mark McGwire was in the unusual position of leading MLB in home runs with 58 without leading one of the individual leagues – he would hit 34 for the A's, which ranked 9th in the AL in 1997, and 24 for the Cardinals, which ranked outside of the top 10. For the sake of this article we won't consider that one.
This leaves 9 times that a player topped 50 HR's without leading the league, which leads to my favorite list, the players who did not lead the league when hitting 50 HRs:
Year Player HR Rank Leader (HR)
1998 Sammy Sosa 66 2nd (NL) Mark McGwire (70)
2001 Sammy Sosa 64 2nd (NL) Barry Bonds (73)
2001 Luis Gonzalez 57 3rd (NL) Mark McGwire (70)
1961 Mickey Mantle 54 2nd (NL) Roger Maris (61)
2002 Jim Thome 52 2nd (AL) Alex Rodriguez (57)
1996 Brady Anderson 50 2nd (AL) Mark McGwire (52)
1938 Jimmie Foxx 50 2nd (AL) Hank Greenberg (58)
1998 Greg Vaughn 50 3rd (NL) Mark McGwire (70)
I feel bad for Luis Gonzalez and Greg Vaughn. They both had the misfortune of hitting a ton of HRs in years where two other player hit a ton more home runs than they did. Of course, then you have Sammy Sosa. Sosa lead the league twice in home runs – in 2000 and 2002, with totals of 50 and 49 and managed to lead the NL in HRs exactly 0 times in years that he hit 60+ home runs. Talk about bad timing.
Double X, also known as Beast – and a beast he was.
The player that intrigued me the most is Jimmie Foxx and his 1938 campaign. In that season, Foxx would reach 50 home runs for the second time, to go along with 139 runs, 197 hits, 175 RBI, 119 BB, a .349 BA, 1.166 OPS, 182 OPS+ and 7.6 WAR to win his 3rd MVP award, missing out on the triple crown in only the home run category. It would be the second time that Foxx missed out in one category – he finished 2nd in the AL to Dale Alexander in 1932 in BA by .003 points. You shouldn't feel too bad for Foxx though, he did win the 1932 MVP award (posting an insane 10.5 WAR, a number that has only been reached 36 times in baseball history) and he did manage to secure the Triple Crown the following season as well as also winning an MVP. The 1930's were good for offense, what can I say, but Foxx is still arguably one of the greatest hitters in baseball history. If that isn't enough, Foxx was also good enough to appear as a pitcher in 11 games in his career, including 9 in 1945, including 2 starts, when he was 37 years old. He posted a 1.59 ERA (albeit with not so stellar peripheral stats). It is also worth noting that when Foxx retired, he was #2 on the all-time career HR list, 170ish HR's behind some guy named Ruth. For nearly 20 years, the 500 HR club was Foxx, Ruth and Mel Ott.
So why is this part 1 of a 2 parter? Well, it goes back to the guy who had a slightly higher BA in 1932 than Jimmie Foxx. Part 2 is going to answer the question I had when I was writing this piece – "Who the hell is Dale Alexander and why have I never heard of him?"
AJ Athlete in Retrospect, MLB, Sports 1 Comment February 4, 2016
Memories of Monte Irvin
Yesterday I as saddened to hear of the passing of Monte Irvin on the 11th of January at age 96. If you don't know who Irvin was, he was a former NY Giant great and the first baseball Hall of Famer I ever met, back in the early 1980's at a baseball card show at St. Peter's College in New Jersey (I was probably 10 or 11 at the time). He appeared with another former Giant teammate Dusty Rhodes. I had the pleasure of sitting down with the both of them for about 20 minutes to talk baseball since there was no one there getting autographs. I remember Rhodes talking about how he became a tugboat captain after he retired from baseball and Irvin talked about his experiences in the Negro Leagues, which I knew very little about at the time. Thanks to Irvin, I became interested in Negro League history, which at the time, was not easy to find any information about and it is directly responsible for me being a long time supporter of the Negro League Museum in KC, a place that I sadly have not yet been to. Rhodes passed away in 2009 (not before I had a series of correspondences with him about his post-baseball career and I still have the letters that he sent – yep, we corresponded old school and I do have an unhealthy obsession with tugboats) and with Irvin passing on the 11th, we lost yet another link to the Negro Leagues.
I am sure that Irvin had no idea that a 20 minute conversation he had with a kid 30 years ago would have such an impact, and, frankly, I had not realized it myself until I reflected on my interaction with Irvin. A friend of mine sent me a link to an article on nj.com about Irvin's passing, written by someone who actually knew him. I would suggest reading the article, but I wanted to include a snippet of it just to make a point about how much the Negro Leagues meant to Irvin:
There was a moment about a decade ago when I researched all the Hall of Famers who played in Newark for the New York Yankees farm team called the Bears, the Eagles and the turn-of-the-century Newark Indians. Joe DiVincenzo, the Essex County Executive had hung plaques for a ring of honor just above façade behind home plate at the Bears and Eagles baseball stadium in Newark.
Monte was one of them. He was in a wheelchair, and afterward I walked over to him and hugged him. Monte, being Monte, the conversation went like this:
"Listen," he said, "I've got to tell you something. You can't die."
"Ever?" I said.
"Never," he said. "You are the last writer to ever see us play in the Negro Leagues. You die and that leaves nobody to tell our story. The kids won't even believe we had a league. Don't die."
So in honor of Irvin's passing, go read a book on tugboats, or the a book I highly recommend on the Negro Leagues, "When Only the Ball Was White." Better yet, go make a donation to the Negro League Museum – a place that would not existed if it were not for the efforts of guys like Buck O'Neil and Monte Irvin and lets make sure a very important piece of American history does not get forgotten once those who were part of it are all gone.
AJ MLB, Sports Leave a comment January 13, 2016
TBT: September 20th, 2009
I was looking through some old scorecards that I have saved and I came across one from 2009, September 20th to be exact, for a game between the Mariners and Yankees. That season, the Yankees would win 103 games and the World Series and the Mariners would win 85 through shear dumb luck (they had a -52 run differential but happened to go 35-20 in one run games.) It was a stellar pitching matchup between Ian Snell, who seemed to managed to stay in the M's rotation despite being horrible and Joba Chamberlain, who was a mega-prospect at the time and the Yankees were in the process of trying to turn him into a starter.
The Yankees lineup was an impressive one:
Jeter, Damon, Teixieria, Arod, Matsui, Posada, Cano, Melky Cabrera, Gardner
The Mariners, not so much:
Ichiro, Gutierrez, Lopez, Griffey, Beltre, Hall, Carp, Moore, Jack Wilson
So predictably, the Mariners jumped all over Chamberlain, scoring 7 runs in 2 innings and chasing him out of the game. Somehow Snell managed to limit the Yankees to 1 run in 5 1/3 innings despite 4 hits and 4 walks allowed. The M's bullpen shut down the Yankees the rest of the way and the final score was 7-1.
Why is this game of note, well, because the pitching line for the Yankees bullpen, specifically Sergio Mitre (who was the only Yankees pitcher to appear after Chamberlain was chased) caught my eye:
IP H R ER BB SO Pitches-Strikes
Mitre 5 1 0 0 1 5 65-43
There was an error on the scorecard which I had Mitre throwing 5 no-hit innings, which is why I was even interested. If you don't remember Mitre, he was a Cubs and Marlins prospect who never quite put it together in the bigs. Coming into the game on the 20th, Mitre had been the Yankees 5th starter and had a 7.63 ERA and had given up 18 runs in his last 2 starts.
Pitching down 6 runs in this game, Mitre got Ichiro to bounce out, gave up a single to Guti, struck out Lopez, walked Griffey and then ended the inning on a Beltre fielder's choice in his first inning of work. Nothing spectacular and then he proceeded to do what you expect that Mariners lineup to do the rest of the game – nothing. He put down the next 12 hitters in order. It is how he ended the game that I thought was interesting – the last 4 hitters he faced were strikeouts. So Mitre recorded 13 consecutive outs – the first 9 were on balls in play and the last 4 were strikeouts. Nothing earthshaking or anything that is ever going to appear in a record book, but just one of those little weird things about baseball that I love. I reminded me of this game that I wrote about a few years back.
AJ Mariners, MLB, Sports Leave a comment August 13, 2015
The Mariners and Their Playoff Chances
There are Mariners fans out there who are still entertaining hopes of getting into the playoffs despite a 50 – 59 (as of August 5th) record and being 7 games back from the 2nd wildcard spot. Why not. The AL is very mediocre this year and the 2 teams that currently possess the wild card spots are sporting .533 and .523 winning percentages, so one good win streak puts they M's into contention.
The problem here is that the Mariners need to pass 7 teams just to reach the 2nd playoff spot and that is no easy feat with 53 games remaining, although it has been done before. Here is how the M's remaining schedule breaks down:
15 games vs. teams with a record worse than the M's – 9 vs. Oakland, 3 vs. Boston and 3 vs. Colorado. They really need to come out of those 15 games with a 10-5 record.
13 games vs. Texas – the Rangers are currently 5 games ahead of the M's in the standings (and 6 ahead in the loss column) and significantly upgraded their starting pitching, so making the assumption that both teams play at the same level for the rest of the season (let's just say .500 for the sake of argument), the M's need to go 9-4 just to pass them in the standings.
7 games vs. Chicago – it is surprising that the White Sox are still in this, just 4.5 games back of the 2nd wild card despite being horrible. This is the team that the M's first need to pass, but once again, despite being just 2.5 games back of the Pale Hose, the M's are 4 back in the loss column (how have the M's managed to play that many more games than the other teams in the league?) It would take a 5-2 record in those 7 games to catch the Sox assuming they both play .500 ball the rest of the way.
3 games vs Baltimore – the O's are 6 games up on the M's and just 1 game back of the 2nd wild card. They do have a brutal schedule the rest of the way though – 28 of their remaining 55 games are against teams with better records and that doesn't count the 14 games against teams that are within 2 games of them in the wildcard race. The M's probably need a sweep here or at least winning the series and then hope that the O's split most of the games against the teams that are ahead of the M's in the wild card race. If the O's go into freefall or go on a tear, it will pretty much end the Mariners season.
6 games vs Houston – the Astros lead the M's by 10 games in the division and are 2 up on the first wild card spot. The 'Stros have been tough to figure out so these games don't help the M's other than potentially swapping the Angels and Astros as division leader/1st wild card team. A couple of important series, but the M's would do far more damage to themselves if they come out of those 6 with a losing record than the Astros would.
6 games vs. LA – They are up 8 on the M's for the first wild card spot. Same as the Astros – the 2 series against the Angels could do more harm than good to the M's chances, depending on how they play out.
3 games vs. KC – Same as the LA and Houston series, but just 3 games so it wouldn't be the end of the world for the M's as long as they don't get swept. KC is really good, so there is no guarantee there.
The M's have no games remaining against the following teams ahead of them in the wild card standing: Toronto (currently in the 2nd wild card spot, but a brutal schedule to end the season), Minnesota, Tampa Bay and Detroit. All of those teams are at least 3 games up on the M's in the standings and at least 4 up in the loss column.
It is a long road for the M's to even think about making the playoffs and they are going to need a ton of help.
My prediction: the M's make a brief run in the last 2 weeks of August and then fall apart, finishing somewhere around 78-84 for the season. McClendon will be looking for a new job but Zdrenzick will be safe for another year.
AJ Mariners, MLB, Sports Leave a comment August 8, 2015
Bye, Bye Bloomquist
The Mariners *finally* DFA'ed Lloyd McClendon's favorite waste of at bats utility player and called up Chris Taylor, who frankly can't be any worse than Willie and should, in theory, be at least slightly better. Bloomquist posted a 159/194/171 slash line with his usual lack of power, producing a whopping 8 OPS+ and -0.3 WAR is 72 plate appearances. While I appreciate the versatility that a guy like Bloomquist has, he is the worst kind of guy to have on a roster – an all-glove, no-hit veteran that plays multiple positions. For some reasons managers love to find ways to get these guys in the game far more often than they should, probably because most managers were the same way as players.
Now if the M's would just do something with Mike Zunino…
AJ Mariners, MLB, Sports Leave a comment July 2, 2015
Time to Hit the Panic Button
I had mostly written a post about how it was time for the Mariners to hit the panic button and then the Mariners went and did it in the worst possible way – they went out and traded for Mark Trumbo, which is exactly what the Mariners did not need to do. Granted, the Mariners traded a bunch of spare parts to get Trumbo (contrary what people may say, Gabby Guerrero, Vlad's Nephew, is probably not going to be anything beyond a 4th OFer in the majors.
David Schoenfield wraps up the Mariners issues really well in this article. Basically, the team doesn't get on base. The team has spent most of the Zdrienick era near or at the bottom of the AL in OBP which is why this team is consistently near or at the bottom of the AL in runs scored and near or at the bottom of the standings. This is a fundamental flaw in organizational philosophy. How many hitters have the Mariners developed into all-star caliber players under Z? One – Kyle Seager, despite having multiple top 10 picks. Ackley is looking like a 4th OFer, Zunino can't hit a curve ball and looks like he will struggle to get to .200 every season, Brad Miller is really a league average SS as a hitter, Chris Taylor looks like he is going to be the same, Nick Franklin, who was traded to TB last season is looking like a 4A player and there isn't much hope on the horizon.
Those who know me know that I wasn't high on the Mariners coming into 2015 – they didn't fix their on-base issues in the offseason and they were historically good last season at preventing runs, which meant that regression in that capacity is likely. Any offense that they added was going to be offset by a bigger increase in runs allowed, which is exactly what has happened so far this season.
Back to Trumbo. The guy has power, which puts him in 30 HR territory. That is great, except that the Mariners already hit a ton of home runs. Trumbo has a career .299 OBP which is horrible. He is a terrible defensive OF (although he isn't bad at 1B, but it seems that the Mariners are convinced that Logan Morrison is the answer there). He probably ends up as a full time DH and occasional 1B, which is probably about right for him. The other thing that Trumbo has going for him is that he hits right handed. The M's have an incredibly left handed heavy lineup and this will help balance it out. It doesn't solve the issue of getting on base.
While we are at it, it is past the time where Fernando Rodney is the Mariners closer. Rodney has converted 14 of 17 save opportunities, which makes him look better than he has been. He has managed to compile an ERA of near 7 and in 7 out of his last 9 save opportunities, he has given up at least 1 run. The scary thing is that he has entered every one of those games at the start of the inning with no runners on. I can't imagine that the Mariners even remotely trust him to come in with runners on at this point, yet McClendon still says Rodney is his closer.
I figured the Mariners were an 80-83 win team this season. I am willing to bet if they finish anywhere below 81 wins, that you will see a new manager and GM for the 2016 season.
AJ Mariners, MLB, Sports Leave a comment June 6, 2015
It's Time to Change Baseball
If MLB is going to insist on keeping its current configuration of 162 games and 15 teams in each league, there are a few things that they should change, and don't worry, what they should change won't affect the "integrity" of the game.
Change 1 – eliminate the divisions. When you have divisions, all you are doing is potentially recognizing mediocrity. Even with 2 wild card teams, the potential still exists to have a division winner having a worse record than a team that does not make the playoffs, you also have the potential of a division winner having a worse record than both of the wild card teams. That makes a huge difference since the wild card winners have to play in the "coin-toss" game (AKA the wild card game) and then get to not have home field advantage against a team with an inferior record. Granted, home field doesn't mean much in baseball (home teams win about 54% of the time) but I think most teams would rather have it. I am much more in favor of taking the 5 best teams in each league for the playoffs. Heck, I would even be in favor of adding another team and having the top 2 teams get a "bye" and the next 4 play a best of 3 series to advance. It would make the positioning for the playoffs much more important, forcing teams to play more meaningful games later in the year.
Change 2 – balance the schedule. If you eliminate the divisions, it becomes a no brainer to balance the schedules in the leagues. Even with the divisions, the balanced schedule makes more sense. You don't think that it gives a team in the NL East a competitive advantage this season to get to beat up on Philadelphia and Miami 18 times each than say the NL Central which only has Milwaukee as a whipping boy? You can never balance the interleague stuff, but it wouldn't hurt to have all of the AL teams play each other the same amount of times each season and eliminate the imbalance of the division strengths.
Change 3 – have both leagues use the DH. This one will upset people who complain that it will ruin the "sanctity of the game" or causes less strategy in the game. To that I say bullcrap. Here is how MLB pitcher have faired over the last 4 seasons (2015 stats through May 31).
Year BA OPB SLG OPS+
2015 .121 .144 .150 -16
2013 .132 .164 .169 -6
Honestly, let's say you were faced with a situation in a playoff game where you are down 1-0, a runner on 3rd, two out in the 8th inning. You have Felix Hernandez pitching and he is dealing – he gave up 1 hit (a HR in the first on a bad pitch) and he has retired every batter since and hasn't had a hard hit ball against him since the HR. Let's say you are facing elimination. Your chances of winning in the 9th aren't good because you are facing a Marino Rivera-like closer in the 9th. What would you do? Before you answer, here is what pinch hitters have done over the last 4 seasons:
2015 .219 .292 .335 78
With a pitcher hitting, you have pretty much no chance that you will be tying up the game. Granted, a pinch hitter is no real guarantee, but wouldn't you rather have someone who hits for a living up at the plate? So how does it not cause less strategy? Look at it from the other way – if you have the DH in play and don't have to worry about the pitcher coming up at bat, you have an opportunity to deploy your pitchers more effectively in critical situations. What made Mariano Rivera so great in the post season was that the Yankees were able to utilize him in multiple innings without having to worry about his spot coming up in the batting order. Look at the 2014 WS – the games in KC allowed both teams to deploy the 2 best bullpens in the league over a longer stretch and hit more strategic matchups than you ever would with allowing the pitcher to bat.
In addition to this, if you don't like the 2-1 games that dominate baseball these days, putting in a DH vs a pitcher is worth about half a run a game. Wouldn't you rather see a guy with 250/20/75 slash lines up at the plate rather than a guy who has a 100/0/0 slash line? A pitcher in the lineup generally means that you end up with effectively an 8 man lineup with an automatic out thrown in. That isn't all that much fun to watch.
AJ MLB, Sports Leave a comment June 3, 2015
Handicapping the MLB Awards
We are in MLB awards season. Yesterday we had the announcement of the AL and NL ROY, won by Jose Abreu and Jacob DeGrom. Abreu was a no-brainer and DeGrom, despite only pitching 144 innings, got 26 of the 30 first place votes over Billy Hamilton (remember him from this) and Kolten Wong, both of whom posted sub-300 OPB and sub-700 OPS. It was a weak year in the NL for rookies.
We have the Manager of the Year, MVP and Cy Young coming up this week and it should be interesting. Keep in mind that the voting is completed for all 3 of the awards on the last day of the regular season, so the playoffs have no impact. Here are my thoughts:
Predicted Winners: Mike Scioscia (AL), Bruce Bochy (NL)
I took a quick look at the ESPN 2014 predictions and virtually no one had either the Angels or the Giants winning their divisions or the wild card, so they both get the nod from me. It will be interesting to see who wins because you can make the argument than any of the 6 finalists deserve to win the award.
Predicted Winners: Felix Hernandez (AL), Clayton Kershaw (NL)
Let me just put this out there – Kershaw was far and away the best pitcher in baseball this year and he will be a unanimous winner. The AL is different. I actually would not have voted for Felix if I had the vote simply because he imploded in his next to last start and essentially cost the Mariners a legitimate shot at the playoffs. Neither Sale nor Kluber pitched for a contender, but here are the stats for all 3 of them:
W-L ERA SO WHIP IP ERA+ WAR K/9 K/BB
Felix 15-6 2.14 248 0.915 236 170 6.8 9.5 5.39
Kluber 18-9 2.44 269 1.095 235.2 152 7.4 10.3 5.27
Sale 12-4 2.17 208 0.966 174 178 6.6 10.8 5.33
There is a small piece of information that I wanted to share. A day after Felix had his disastrous start in Toronto, the official scorer changed a play that was (correctly) ruled a hit to (incorrectly) be an error. That resulted in 4 ER being removed from Felix's stats and gave him the ERA and WHIP titles. Without the scoring change, Felix essentially has the same ERA as Kluber. It is really a toss-up between Kluber and Felix though – they had virtually identical stat lines with Kluber striking out a few more hitters and Felix giving up a few less hits, but I think this will be a pretty close vote, with Felix coming out just ahead. Sale, while probably better than either Kluber of Felix this year, missed a month of the season and that will lead to him finishing third.
Predicted Winners: Mike Trout (AL), Clayton Kershaw (NL)
The NL is interesting since you have to compare 2 hitters to a pitcher. McCutchen and Stanton both had good, not great, seasons, pulling in a 6.4 and 6.5 WAR respectively. Kershaw, despite an early season injury and just under 200 IP, pulled in a 7.5 WAR. The selling point for me – the Pirates may or may not make the playoffs without McCutchen, the Marlins just suck worse without Stanton but the Dodgers don't win their division without Kershaw.
All 3 of the AL finalists had fine seasons, but the reality is Mike Trout was just a ton better than either Michael Brantley or Victor Martinez, neither of whom are very likely to be that good next year. Besides, I had to actually look up Brantley's stats to see how good he actually was, although I was tempted to give him points for being Mickey Brantley's kid. The scary thing about Trout is that he is just 22 and this was a "down year" for him, posting a career low 7.9 WAR, which was still good enough to lead all of baseball.
AJ MLB, Sports Leave a comment November 11, 2014
Don't Look Now, But The Mariners Aren't Terrible
Gene Brabender. Just because.
Every conversation I had about the Mariners this summer seemed to go the same way:
Me: "How about those M's.. they're doing ok!"
Them: "The M's suck."
Me: "No, they're really about a .500 team. They're average."
Them: "Eh. I'm pretty sure they suck, or they're going to suck soon."
Me: "No, the franchise has finally begun to recover from the damage that Bill Bavasi did."
Them: *Shrug*
(You'll notice it was always me initiating the topic. That shouldn't be a surprise, since I'm one of the handful of people that pays attention at all to the M's results.)
In all seriousness though, they're not bad. They have some good starting pitching, headlined by Felix. They have three position players that could be the base of a solid team in Cano, Seager, and Mike Zunino. The bullpen was excellent overall this year, though I always view the status of the bullpen as "transitory".
I'll be interested to see the Vegas line for wins next year. I'd guess it's going to be somewhere between 78 and 83 wins. 81 wins would be .500.
That sounds about right to me.
AJ Mariners, MLB, Sports 3 Comments September 30, 2014
How to Turn a Walk Into 2 Outs
With your every day 2-1-6-1-5 double play. The Giants managed to pick off 2 runners on the same play, which happened to be a walk. This is why you need to pay attention the game when you are running the bases kids.
AJ MLB, Sports 2 Comments August 6, 2014 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,374 |
Keyword Analysis & Research: gurgaon india population
gurgaon india population 1.06 0.5 6294 24 24
gurgaon 1.29 0.1 4747 21 7
india 0.81 0.4 2125 38 5
population 0.04 0.1 9479 43 10
Keyword Research: People who searched gurgaon india population also searched
gurgaon india population 0.88 0.2 8839 53
population of gurgaon india 0.71 0.6 5524 10
Search Results related to gurgaon india population on Search Engine
Gurgaon - Wikipedia
https://en.wikipedia.org/wiki/Gurgaon
Gurgaon, officially named Gurugram , is a city located in the northern Indian state of Haryana. It is situated near the Delhi–Haryana border, about 30 kilometres (19 mi) southwest of the national capital New Delhi and 268 km (167 mi) south of Chandigarh, the state capital. It is one of the major
Gurgaon Population 2021 - Current Population of Gurgaon
indiaonlinepages.com
https://www.indiaonlinepages.com/population/gurgaon-population.html
Current Population of Gurgaon in 2021: 2,982,928 (2.9 million) Population of Gurgaon in 2020: 2,770,450 (2.7 million) Population of Gurgaon in 2019: 2,563,906 (2.5 million) Population of Gurgaon in 2018: 2,360,761 (2.3 million) Sex Ratio in Gurgaon: 870 females per 1,000 males: Literacy Rate: 84.4% Population of Gurgaon in 2018: 2,360,761 (2.3 million) Population of Gurgaon in 2020: 2,770,450 (2.7 million) Population of Gurgaon in 2019: 2,563,906 (2.5 million) Sex Ratio in Gurgaon: 870 females per 1,000 males
Population of Gurgaon in 2018: 2,360,761 (2.3 million)
Sex Ratio in Gurgaon: 870 females per 1,000 males
Population of Gurgaon 2021 | Population of Gurugram District
findeasy.in
https://www.findeasy.in/population-of-gurgaon/
Jun 16, 2020 · Gurgaon, officially Gurugram, is a city in the Indian state of Haryana. Its main satellite city of Delhi, & situated just 30km from New Delhi and is part of the National Capital Region of India. With a population of close to 1.2 million, it's the second most populated city after Faridabad in Haryana. As per the provisional data of the 2011 census, Gurgaon had a …
Gurgaon City Population Census 2011-2022 | Haryana
census2011.co.in
https://www.census2011.co.in/census/city/46-gurgaon.html
The Gurgaon city is located in Haryana state of India. As per provisional reports of Census India, population of Gurgaon in 2011 is 876,969; of which male and female are 475,032 and 401,937 respectively. Although Gurgaon city has population of 876,969; its urban / metropolitan population is 902,112 of which 488,251 are males and 413,861 are females. City: Gurgaon State: Haryana Government: Municipal Corporation Urban Agglomeration: Gurgaon Metropolitan
Government: Municipal Corporation
Urban Agglomeration: Gurgaon Metropolitan
Gurgaon District Population Census 2011-2022, Haryana
https://www.census2011.co.in/census/district/225-gurgaon.html
Population Census of Gurgaon District in 2011 is 1,514,432. Literacy rate of Gurgaon is 84.70 percent. Sex Ratio for Gurgaon district is 854 per 1000 male.
HARYANA - Census of India Website
censusindia.gov.in
https://www.censusindia.gov.in/2011census/dchb/DCHB_A/06/0618_PART_A_DCHB_GURGAON.pdf
Census of India 2011. R R P R R R ... Gurgaon district has witnessed a phenomenal growth in all spheres of developments, ... Population, its distribution 33 (v) Brief analysis of PCA data based on insets tables 1 to 35 35 (vi) Brief analysis of the Village Directory and Town Directory data based on inset tables
Gurugram | Haryana, India, & Population | Britannica
https://www.britannica.com/place/Gurugram
Gurugram, formerly Gurgaon, also called Hidayatpur, city, southeastern Haryana state, northwestern India. It is situated between Delhi (northeast) and Rewari (southwest), to which it is connected by road and rail. High-rise residential buildings in Gurugram, Haryana, India. Gurugram was traditionally an agricultural trade centre.
Gurgaon Tehsil Population Gurgaon, Haryana, List of
censusindia2011.com
https://www.censusindia2011.com/haryana/gurgaon/gurgaon-population.html
34 rows · Gurgaon is a Tehsil located in Gurgaon district of Haryana. It is one of 5 Tehsils of … Female Population: 448844 (45.93%) Number of Households: 225460 Male Population: 528493 (54.07%) Population: 977337
Female Population: 448844 (45.93%)
Number of Households: 225460
Male Population: 528493 (54.07%)
Gurugram (Gurgaon) City Map - Maps of India
mapsofindia.com
https://www.mapsofindia.com/maps/haryana/gurgaon.htm
Area and population of Gurgaon. According to the census conducted in 2011, the population of the city of Gurgaon is 1,514,085. The total area of the Gurgaon district is 1253.07 sq km. State: Haryana Zone: West Zone, North Zone, East Zone, South Zone Ward: 35
Zone: West Zone, North Zone, East Zone, South Zone
Ward: 35
Gurgaon: what life is like in the Indian city built by
https://www.theguardian.com/sustainable-business/2016/jul/04/gurgaon-life-city-built-private-companies-india-intel-google
Jul 04, 2016 · According to census data, Gurgaon's population doubled between 2001 and 2011, from 876,000 to more than 1.5 million. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,652 |
#ifndef TWOBLUECUBES_CATCH_META_HPP_INCLUDED
#define TWOBLUECUBES_CATCH_META_HPP_INCLUDED
#include <type_traits>
namespace Catch {
template<typename T>
struct always_false : std::false_type {};
template <typename> struct true_given : std::true_type {};
struct is_callable_tester {
template <typename Fun, typename... Args>
true_given<decltype(std::declval<Fun>()(std::declval<Args>()...))> static test(int);
template <typename...>
std::false_type static test(...);
};
template <typename T>
struct is_callable;
template <typename Fun, typename... Args>
struct is_callable<Fun(Args...)> : decltype(is_callable_tester::test<Fun, Args...>(0)) {};
#if defined(__cpp_lib_is_invocable) && __cpp_lib_is_invocable >= 201703
// std::result_of is deprecated in C++17 and removed in C++20. Hence, it is
// replaced with std::invoke_result here. Also *_t format is preferred over
// typename *::type format.
template <typename Func, typename U>
using FunctionReturnType = std::remove_reference_t<std::remove_cv_t<std::invoke_result_t<Func, U>>>;
#else
template <typename Func, typename U>
using FunctionReturnType = typename std::remove_reference<typename std::remove_cv<typename std::result_of<Func(U)>::type>::type>::type;
#endif
} // namespace Catch
namespace mpl_{
struct na;
}
#endif // TWOBLUECUBES_CATCH_META_HPP_INCLUDED
| {
"redpajama_set_name": "RedPajamaGithub"
} | 970 |
Q: Byte level length description I have a protocol that requires a length field up to 32-bits, and it must be
generated at runtime to describe how many bytes are in a given packet.
The code below is kind of ugly but I am wondering if this can be refactored to
be slightly more efficient or easily understandable. The problem is that the
code will only generate enough bytes to describe the length of the packet, so
less than 255 bytes = 1 byte of length, less than 65535 = 2 bytes of length
etc...
{
extern char byte_stream[];
int bytes = offset_in_packet;
int n = length_of_packet;
/* Under 4 billion, so this can be represented in 32 bits. */
int t;
/* 32-bit number used for temporary storage. */
/* These are the bytes we will break up n into. */
unsigned char first, second, third, fourth;
t = n & 0xFF000000;
/* We have used AND to "mask out" the first byte of the number. */
/* The only bits which can be on in t are the first 8 bits. */
first = t >> 24;
if (t) {
printf("byte 1: 0x%02x\n",first );
byte_stream[bytes] = first; bytes++;
write_zeros = 1;
}
/* Now we shift t so that it is between 0 and 255. This is the first, highest byte of n. */
t = n & 0x00FF0000;
second = t >> 16;
if (t || write_zeros) {
printf("byte 2: 0x%02x\n", second );
byte_stream[bytes] = second; bytes++;
write_zeros = 1;
}
t = n & 0x0000FF00;
third = t >> 8;
if ( t || write_zeros) {
printf("byte 3: 0x%02x\n", third );
byte_stream[bytes] = third; bytes++;
write_zeros = 1;
}
t = n & 0x000000FF;
fourth = t;
if (t || write_zeros) {
printf("byte 4: 0x%02x\n", fourth);
byte_stream[bytes] = fourth; bytes++;
}
}
A: You should really use a fixed-width field for your length.
*
*When the program on the receiving end has to read the length field of your packet, how does it know where the length stops?
*If the length of a packet can potentially reach 4 GB, does a 1-3 byte overhead really matter?
*Do you see how complex your code has already become?
A: Try this loop:
{
extern char byte_stream[];
int bytes = offset_in_packet;
int n = length_of_packet; /* Under 4 billion, so this can be represented in 32 bits. */
int t; /* 32-bit number used for temporary storage. */
int i;
unsigned char curByte;
for (i = 0; i < 4; i++) {
t = n & (0xFF000000 >> (i * 16));
curByte = t >> (24 - (i * 8));
if (t || write_zeros) {
printf("byte %d: 0x%02x\n", i, curByte );
byte_stream[bytes] = curByte;
bytes++;
write_zeros = 1;
}
}
}
A: I'm not sure I understand your question. What exactly are you trying to count? If I understand correctly you're trying to find the Most Significant non-zero byte.
You're probably better off using a loop like this:
int i;
int write_zeros = 0;
for (i = 3; i >=0 ; --i) {
t = (n >> (8 * i)) & 0xff;
if (t || write_zeros) {
write_zeros = 1;
printf ("byte %d : 0x%02x\n", 4-i, t);
byte_stream[bytes++] = t;
}
}
A: Really you're only doing four calculations, so readability seems way more important here than efficiency. My approach to make something like this more readable is to
*
*Extract common code to a function
*Put similar calculations together to make the patterns more obvious
*Get rid of the intermediate variable print_zeroes and be explicit about the cases in which you output bytes even if they're zero (i.e. the preceding byte was non-zero)
I've changed the random code block into a function and changed a few variables (underscores are giving me trouble in the markdown preview screen). I've also assumed that bytes is being passed in, and that whoever is passing it in will pass us a pointer so we can modify it.
Here's the code:
/* append byte b to stream, increment index */
/* really needs to check length of stream before appending */
void output( int i, unsigned char b, char stream[], int *index )
{
printf("byte %d: 0x%02x\n", i, b);
stream[(*index)++] = b;
}
void answer( char bytestream[], unsigned int *bytes, unsigned int n)
{
/* mask out four bytes from word n */
first = (n & 0xFF000000) >> 24;
second = (n & 0x00FF0000) >> 16;
third = (n & 0x0000FF00) >> 8;
fourth = (n & 0x000000FF) >> 0;
/* conditionally output each byte starting with the */
/* first non-zero byte */
if (first)
output( 1, first, bytestream, bytes);
if (first || second)
output( 2, second, bytestream, bytes);
if (first || second || third)
output( 3, third, bytestream, bytes);
if (first || second || third || fourth)
output( 4, fourth, bytestream, bytes);
}
Ever so slightly more efficient, and maybe easier to understand would be this modification to the last four if statements:
if (n>0x00FFFFFF)
output( 1, first, bytestream, bytes);
if (n>0x0000FFFF)
output( 2, second, bytestream, bytes);
if (n>0x000000FF)
output( 3, third, bytestream, bytes);
if (1)
output( 4, fourth, bytestream, bytes);
I agree, however, that compressing this field makes the receiving state machine overly complicated. But if you can't change the protocol, this code is much easier to read.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,348 |
Q: Why doesn't my recursive longest increasing subsequence function work for large inputs? I wrote it to work on a single test case at a time.
It either takes too long on online judges or returns wrong answers
Source: The problem I used to test it on
It works perfectly for small cases:
#include <iostream>
#include <algorithm>
#include <vector>
int LIS[100000];
void LS (int *arr , int n)
{
if (n == 0)
{
LIS[0] = 1;
return;
}
if (LIS[n])
{
return;
}
int i = 0;
int max = 0;
while (i < n)
{
if (arr[i] < arr[n])
{
LS(arr,i);
if (LIS[i] + 1 > max)
{
max = 1 + LIS[i];
}
}
++i;
}
LIS[n] = max;
}
int main()
{
int n;
std::cin >> n;
int arr[n];
for(int i = 0 ; i < n ; ++i) std::cin >> arr[i];
LS(arr,n-1);
std::sort (LIS , LIS+n);
std::cout << "\n" << LIS[n-1] << "\n";
}
A: You said it works perfectly small cases.. than maybe it is stack overflow..
A function call consume stack memory..
If recursive call depth is too deep, stack memory runs out, and crash..
A: When you read int n value from std::cin, you have to dynamically allocate memory for your array arr, so you should first declare int *arr, then get the user input the same way as you're doing it now, and then allocate memory using arr = new int[n].
When it comes to the long time it takes to compute next values using recursive function, you should think about remaking the function to use tail recursion, which is much closer to iteration. You can check the difference by writing two programs for counting Fibonacci numbers - one recursive and another one iterative, then check how long it takes to compute ~50th value using both methods.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,332 |
/*
Theme: Albeo
RTL: Yoav Farhi yoav@automattic.com
*/
body { direction:rtl; unicode-bidi:embed; font-family: sans-serif; }
/* font reset */
.p-con p, .post-page p, .widget_tag_cloud, .menu li, h1, h2, h3, h4, .header h1, .title, .p-head h1 , .p-head h2, .p-head h3,.post-page h1 , .post-page h2 {font-family: sans-serif;}
.p-con blockquote p {font-family: sans-serif;}
/* Header
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.header { float: right;}
/*ie7 only */
*:first-child+html .header {float:none; margin:0 auto;}
.header h1 { float: right; }
.header h1 a, .header h1 a:visited, .header .LogoText h1 a:hover {background-position:right center; padding-left: 0px; padding-right: 45px; display:inline-block; }
.header .rss { right:auto; left: 0px;}
.header .rss li { background-position: right 2px; padding-left:0; padding-right: 20px; margin-left: 5px; margin-right:0; }
/*ie 6+7 only */
.header .rss li {*float:right;}
.menu { float: right; }
.menu li { float: right; }
#main-menu ul ul ul {
right: 100%;
}
/* Side Left
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.sl-a { float: right; }
.sl-t { float: right; }
.sl-b { float: right; }
.sl { float: right; }
/* Post Single
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.p-con ol { padding: 10px 25px 10px 0;}
.p-con ul li { padding: 2px 15px 2px 0; background-position: right top !important; }
.p-con blockquote li { padding: 2px 13px 2px 0; }
.p-det li { margin-left: 5px; margin-right: 0; }
.p-det .p-cat { background-position: right 2px !important; padding-right: 20px; padding-left: 0; }
.p-det .p-com { background-position: right 2px !important; padding-right: 14px; padding-left: 0;}
.p-tag { background-position: right 1px !important; padding-right: 18px; padding-left: 0;}
.sticky { background: #D7ECF3; padding: 15px; }
/* Post Pages
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.
.post-page ol { padding: 10px 25px 10px 0; }
.post-page ul li { padding: 2px 13px 2px 0; background-position: right top; }
.post-page blockquote li { padding: 2px 13px 2px 0;}
/* Images
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
img.aligncenter, .aligncenter, .aligncenter img, img.centered { display: block; margin-left: auto; margin-right: auto; }
img.alignright { padding: 1px; margin: 0 0 5px 15px; display: inline; border: solid 5px #f2f0ea; }
img.alignleft { padding: 1px; margin: 0 15px 5px 0; display: inline; border: solid 5px #f2f0ea; }
.alignright { float: right; }
.alignleft { float: left }
img.wp-smiley { margin:0; border:0; }
/* Navigation
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.nav { text-align: left;}
.nav .left a { float: right; background: url(images/nav-right.png) no-repeat right 1px; padding-right: 15px; padding-left:0; }
.nav .right a { float: left; background: url(images/nav-left.png) no-repeat left 1px; padding-left: 15px; padding-right:0; }
.nav a { cursor: hand;}
/* Side Right
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.sr-a { float: left;}
.sr-t { float: right;}
.sr-b { float: right;}
.sr { float: right; }
.sr h3 { background-position: right 4px !important; padding-left: 0; padding-right: 18px;}
.search h3 { padding-right: 0px;}
.search input { padding: 5px 30px 10px 12px; }
.categ li { padding: 2px 17px 3px 0; background: url(images/categ-arrow-rtl.png) no-repeat right 7px !important;}
.widget li { padding: 2px 17px 3px 0; background: url(images/categ-arrow-rtl.png) no-repeat right 7px;}
.widget_flickrRSS li { margin: 0px 0 0px 8px; }
/* Recent
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.recent .tabs { float: right; }
.recent .tabs li { float: right; }
#r-com li a { background-position: right 4px !important; padding-right: 13px; padding-left: 0;}
/* Comment List
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.com-list blockquote { margin-left: 0; margin-right: 1em;}
.com-con { padding: 10px 20px 10px 10px; }
.com-con .avatar { float:left; }
/* Comment Form
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
#respond label {
clear: left;
}
#respond input[type="text"] {
float: right;
margin-right: auto;
margin-left: 6px;
}
#respond textarea {
margin-right: 0;
}
/* Footer
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
.footer { float: right;}
.footer p { padding-right:20px; padding-left: 0; }
.previous a {
background: transparent url(images/nav-left.png) no-repeat scroll left .25em;
padding-left: 15px;
padding-right:0;
}
.next a {
background: transparent url(images/nav-right.png) no-repeat scroll right .25em;
padding-right: 15px;
padding-left:0;
}
/* Fonts */
h1, h2, h3, h4, .widget_tag_cloud, #r-tags { font-family: Arial, sans-serif !important; }
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,160 |
{"url":"http:\/\/pldml.icm.edu.pl\/pldml\/element\/bwmeta1.element.bwnjournal-article-doi-10_4064-fm170-3-1?q=bwmeta1.element.bwnjournal-number-fm-2001-170-3;0&qt=CHILDREN-STATELESS","text":"Pe\u0142notekstowe zasoby PLDML oraz innych baz dziedzinowych s\u0105 ju\u017c dost\u0119pne w nowej Bibliotece Nauki.\nZapraszamy na https:\/\/bibliotekanauki.pl\n\nPL EN\n\nPreferencje\nJ\u0119zyk\nWidoczny [Schowaj] Abstrakt\nLiczba wynik\u00f3w\n\u2022 # Artyku\u0142 - szczeg\u00f3\u0142y\n\n## Fundamenta Mathematicae\n\n2001 | 170 | 3 | 219-229\n\n## On strong measure zero subsets of $^{\u03ba}2$\n\nEN\n\n### Abstrakty\n\nEN\nWe study the generalized Cantor space $^{\u03ba}2$ and the generalized Baire space $^{\u03ba}\u03ba$ as analogues of the classical Cantor and Baire spaces. We equip $^{\u03ba}\u03ba$ with the topology where a basic neighborhood of a point \u03b7 is the set {\u03bd: (\u2200j < i)(\u03bd(j) = \u03b7(j))}, where i < \u03ba.\nWe define the concept of a strong measure zero set of $^{\u03ba}2$. We prove for successor $\u03ba = \u03ba^{<\u03ba}$ that the ideal of strong measure zero sets of $^{\u03ba}2$ is $\ud835\udd1f_{\u03ba}$-additive, where ${\ud835\udd1f}_{\u03ba}$ is the size of the smallest unbounded family in $^{\u03ba}\u03ba$, and that the generalized Borel conjecture for $^{\u03ba}2$ is false. Moreover, for regular uncountable \u03ba, the family of subsets of $^{\u03ba}2$ with the property of Baire is not closed under the Suslin operation.\nThese results answer problems posed in [2].\n\n219-229\n\nwydano\n2001\n\n### Tw\u00f3rcy\n\nautor\n\u2022 Department of Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, Helsinki, Finland\nautor\n\u2022 Institute of Mathematics, Hebrew University, Jerusalem, Israel","date":"2023-02-04 02:46:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.687878429889679, \"perplexity\": 1735.4555759182645}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500080.82\/warc\/CC-MAIN-20230204012622-20230204042622-00069.warc.gz\"}"} | null | null |
The Theatre program is part of the Department of Theatre and Film in the BGSU College of Arts and Sciences.
In the theatre program, students are exposed to every aspect of theatre: literature, history, theory, acting, directing, design and technical theatre. The Department of Theatre and Film hosts numerous campus and community performances each year with students actively involved in all elements of production and performance.
Quality classroom experiences
Students receive a solid liberal arts foundation and develop refined analytical and communication skills, in addition to the skills and techniques required for their specific area of interest in theatre. A variety of production opportunities provides practical experience in theatre. Productions are presented in the Donnell Theatre and the Eva Marie Saint Theatre, both located on campus in the Wolfe Center for the Arts.
Stand Out in courses like
Script Analysis
Theatre History and Literature
Principles of Acting
Scene Construction
Theatre majors may earn a Bachelor of Arts or a Bachelor of Arts in Communication. The Bachelor of Arts degree requires a minor. The Bachelor of Arts in Communication does not require a minor, but students choose a specialization in acting/directing, design/technical theatre, musical theatre or youth theatre. Admission to the musical theatre program is through audition only. In addition to the liberal arts curriculum, theatre majors complete a series of three seminar courses on professional development and career preparation culminating with a senior portfolio review and audition.
Our alumni have gone on to produce, direct and perform in award-winning productions, such as "Wicked" and "The Book of Mormon."
The department annually awards a number of competitive scholarships to qualified undergraduate students and has various student employment opportunities, which allow students to develop their craft while receiving financial compensation.
Students may apply for the BGSU exchange program with the University of Wales, the largest undergraduate theatre program in Great Britain and home of the prestigious Performance Research Centre. The London Experience, an intensive immersion in the London theatre scene, offers students another education abroad opportunity.
Go Far in your career
Radio and Television Production
Stage Lighting and Sound
Average Starting Salary & BGSU Placement Rate
$29,739* - Theatre graduates average starting salary according to the National Association of Colleges and Employers
96.55% of Theatre graduates report they're employed, in graduate school or starting a business within six months of graduation
(BGSU data compiled from students who completed the related questions on the graduation survey.)
Theatre Program Requirements
Sample Course Requirements
Map out your Theatre path
Arts Village Learning Community
Upon completion of the baccalaureate degree, students in theatre will be able to:
Analyze diverse performance texts from various historical periods and cultural backgrounds in order to make effective aesthetic decisions as a theatre scholar/artist;
Use performance as the site and process for critical, cultural, and historical understandings;
Research and communicate ideas and feelings in written, visual, and/or oral forms in order to articulate a conceptual and critical approach to theatrical production;
Work collaboratively to solve specific production requirements as actor, director, designer, and/or technician;
Present skills and knowledge as a theatre scholar/artist in a professional format.
Bowling Green State University [BGSU] is accredited by the Higher Learning Commission. BGSU has been accredited by the Higher Learning Commission since 01/01/1916. The most recent reaffirmation of accreditation was received in 2012 - 2013. Questions should be directed to the Office of Institutional Effectiveness.
The Theatre program is accredited by the National Association of Schools of Theatre (NAST) and is in good standing.
More information on accreditation
Bowling Green State University programs leading to licensure, certification and/or endorsement, whether delivered online, face-to-face or in a blended format, satisfy the academic requirements for those credentials set forth by the State of Ohio.
Requirements for licensure, certification and/or endorsement eligibility vary greatly from one profession to another and from state to state. The theatre program does not lead to professional licensure.
More information on professional licensure
Under the Higher Education Act Title IV disclosure requirements, an institution must provide current and prospective students with information about each of its programs that prepares students for gainful employment in a recognized occupation.
The theatre program is not a recognized occupation that requires a Gainful Employment disclosure.
Acting/Directing
Design/Technical Theatre
* Job placement and salary information was compiled by the Office of Academic Assessment through the Graduation Survey from AY2015-2018. The data are gathered around the time of Commencement and a follow-up survey six months post Commencement. For the salary question, data for programs with fewer than fifteen responses are not included. Salaries for those programs are from the National Association of Colleges and Employers Summer 2019 Survey. For questions regarding the data, contact assessment@bgsu.edu.
We're looking for students like you! close
Get the latest news, updates, and more from BGSU
Undergraduate Graduate | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,288 |
You are here: Home > Resources & Training > Rothschild - Rothschild Schools Partnership
Rothschild - Rothschild Schools Partnership
Good for society
100% of reading partner pupils progressed one national curriculum level on completion of the programme.
100% of year 10 students achieved better GCSE grades than was predicted of them, and outperformed their non-mentored peers by one grade.
The Education Award, Finalist, 2016
Children from one of Britain's poorest areas are improving their prospects of finding work thanks to a partnership between investment bank Rothschild and schools in deprived East London schools.
Support for schools and young people forms the backbone of Rothschild's community programme. The investment bank's long-term partnerships aim to develop the confidence and aspirations of young people, and equip them with the skills and knowledge required to make informed decisions about their futures.
This is done primarily through mentoring, work experience, reading support and employability day programmes.
Top Tips from Rothschild:
Make sure you have senior support for the project from the start to ensure it doesn't slip down the list of priorities.
Work collaboratively with the schools or colleges you are partnering with. Remember they have lots of expertise.
Be clear about how you will measure the project and evaluate its success beforehand.
In addition, Rothschild supports partner schools on a strategic level through its partners in leadership programme, where employees coach teachers in management positions, as well as through provision of effective school governors.
The bank found that many young people found it hard to move into employment after school because:
They lacked awareness of career options,
Had mismatched their aspirations with available choices,
Had limited parental support or limited reading abilities, especially for children of non-English-speaking parents.
Since the programme's launch, all pupils have progressed at least one national curriculum level. They also all out-performed their predicted GCSE grades.
Why they did it
Rothschild is a small international investment and private bank. Active support of local community has always been close to the heart of Rothschild; NM Rothschild and his descendants helped to provide social housing, hospitals, educational resources and advice to the immigrant communities of East London, always in response to needs identified by those communities. 'Take local advice' was their maxim.
The partnership with the East London schools is part of the bank's 'Investing in Young People's Futures' strategy by which it hopes to contribute to the development of students from economically deprived backgrounds so they are prepared for their future lives. To achieve this Rothschild has established long-term close working partnerships with local schools.
This financial year, Rothschild invested £34,000 schools and education programmes. The Bank and its staff also volunteered 1,000 hours to support education programmes.
Along project's progress is monitored by a board comprising of senior bankers from various divisions of the business and is chaired by a Managing Director and Partner. The committee oversees Rothschild's activities in the community on behalf of all its stakeholders, and reports to the Board annually.
The Community Champion network comprises of representatives from all divisions in the Bank who act as programme managers with community partners to make sure the programmes run smoothly.
Giving everyone their say
Communicating via special events or the intranet, Rothschild engages with its staff from senior executives to rank and file of the organisation. Even while planning the strategy, all staff were surveyed for their views and focus groups were set up.
Bank employees, from managing directors to the 'shop floor', can get involved in an variety of volunteering opportunities.
Since 2014, Rothschild has extended the programme to its interns. The interns reported that it helped build their network within the Bank and raised their profile with senior staff members.
What Rothschild's CEO said:
"We seek to invest in long term partnerships with local schools and community organisations in disadvantaged areas close to our offices in ways which have a measurable impact on the lives of others. We create opportunities for young people to become socially mobile, broadening their horizons and equipping them with the skills they need to make informed decisions about further education and beyond, so they're able to succeed in a competitive market on an equal playing field. We believe that our community programme has helped with our new hire recruitment and staff retention, and has added value to client relationships." - Nigel Higgins, Chief Executive Officer, Rothschild
Rothschild's HR team indicates that graduates mention their attraction to the company's community initiatives during the recruitment process.
The banks' clients report that Rothschild's education programmes confirm its reputation for integrity and philanthropic interests.
Responsible Business Awards
- Private group -
North East Education Showcase
HR director dinner: Education – Developing the essential skills young people need to succeed
Education in Focus - Responsible Business in Action
Responsible Business Awards 2019 Winners Booklet: Strength in Numbers
How businesses are helping the UK prepare for emergencies
On firm ground: Yorkshire Water's green vision for sustainable land management
The Environmental Sustainability Award: JCDecaux
Cisco - Winner of the UPS International Disaster Relief and Resilience Award
Good Work for All: Influencing Team Heathrow companies | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,638 |
\section{}
\bibliographystyle{elsarticle-num}
\section{Introduction}\label{sec:introduction}
Knowledge is crucial to understanding natural language.
When reading, in addition to linguistic knowledge of the vocabulary and grammar of a language, readers need to have knowledge about the structure of texts, knowledge about the subject, and background or commonsense knowledge about the world in order to comprehend the text.
For example, when a user says ``I am hungry'' to a chatbot at 1:00 pm, the chatbot should be able to understand that the user may want to have lunch rather than breakfast and recommend some nearby restaurants.
This requires the chatbot to understand the complex commonsense knowledge about user's states and potential consequent activities (i.e., being hungry can motivate the user to eat) and the implications of location and time (i.e., compared with breakfast, lunch is more likely to appear at 1:00 pm. Thus the chatbot should recommend some real food rather than just a cup of coffee).
Commonsense reasoning has long been a challenging problem in the artificial intelligence field.
As discussed in \cite{liu2004conceptnet}, commonsense knowledge refers to ``millions of basic facts and understanding possessed by most people.''
Unlike factual knowledge like ``London is the capital of UK,'' which is always true, commonsense knowledge is often not inevitably true and only reflects a kind of contextual preference.
For example, in most cases, rocks are not used for eating, but some birds do eat rocks to digest.
Such kind of knowledge is also called factoids~\cite{GordonDS10a,GordonS10b}.
To effectively represent such preference-like commonsense knowledge, selectional preference~\cite{resnik1997selectional} was proposed, which was traditionally defined on top of single dependency connections (e.g., nsubj, dobj, and amod).
Given a word and a dependency relation, humans have preferences for which words are likely to be connected.
For instance, when seeing the verb ``sing,'' it is highly plausible that its object is ``a song,'' and when seeing the noun ``air,'' it is highly plausible that its modifier is ``fresh.''
However, such selectional preference can only represent commonsense inside an event or state (e.g., which event/state is more likely to happen) and cannot represent commonsense between events/states.
One such example is discussed by~\cite{Wilks1975IAU} and similar examples are frequently observed the Winogrande Schema Challenge~\cite{levesque2011winograd}:
\begin{itemize}
\item The \texttt{soldiers} fired at the \texttt{women}, and we saw several of \texttt{them} fall.
\end{itemize}
To resolve the pronoun ``\texttt{them}'' in the above example, Wilks argued that machines need to access the {\it partial information} (in MaCarthy's phrase) ``{\it hurt things tending to fall down},'' which can be translated into the following form: (hurt, \texttt{X}) $\xrightarrow{\rm ResultIn}$ (\texttt{X}, fall).
In history, many efforts have been devoted to acquiring commonsense knowledge in the form of multi-relational factoids.
For example, the Cyc project initiated in the 1980s~\cite{researchCyc} and ConceptNet (originated from Open Mind Common Sense, OMCS) initiated in 2002~\cite{liu2004conceptnet}, tried to use experts or ordinary people to annotate commonsense knowledge collectively.
However, as aforementioned, two properties of commonsense knowledge determine that we cannot acquire all commonsense knowledge with such approaches.
First, the scale of commonsense knowledge could be enormous and it is infeasible to perform crowd-sourcing for commonsense knowledge acquisition in such a huge scale.
Second, commonsense knowledge is often a kind of preference rather than fixed fact, and thus it is not suitable to represent commonsense knowledge with fixed triplets (e.g., $\langle$``rock'', \texttt{NotUsedFor}, ``eat''$\rangle$) as used in Cyc and ConceptNet.
Recently, pre-trained language representation models (e.g., BERT~\cite{devlin2018bert} and RoBERTa~\cite{DBLP:journals/corr/abs-1907-11692}) have been developed to acquire rich human knowledge implicitly and have demonstrated promising results on many downstream tasks.
However, as shown in LAMA~\cite{DBLP:conf/emnlp/PetroniRRLBWM19} and TransOMCS~\cite{DBLP:conf/ijcai/ZhangKSR20}, even though these models are good at capturing factual knowledge about named entities, they still struggle at capturing commonsense knowledge, especially those complex commonsense knowledge between eventualities (a linguistic term covering activities, states, and events after~\cite{bach1986algebra}, e.g., ``I am hurt'').
One possible explanation is that compared with tokens or named entities, the distribution of eventualities is generally much more sparse.
More importantly, as discussed by ~\cite{liu2004conceptnet}, much commonsense knowledge, which is trivial for humans, is typically not discussed in our daily language at all.
As a result, even though these deep pre-trained language representation models are good at acquiring knowledge from textual data, they could not effectively acquire or reason commonsense knowledge they rarely or never see in the form of word sequences.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{image/ASER_demo_new.pdf}
\caption{ASER Demonstration. Eventualities are connected with weighted directed edges. Each eventuality is a dependency graph.}
\label{fig:ASER-demo}
\end{figure}
To explore a scalable way of acquiring commonsense knowledge, in this paper, we propose an approach to constructing a large-scale weighted eventuality knowledge graph, ASER (Activities, States, Events, and their Relations), by extending the traditional definition of selectional preference to higher-order selectional preference over eventualities.
The eventualities (i.e., nodes of ASER) are extracted using selected dependency patterns.
The edges are based on discourse relations (e.g., \texttt{Result}) in discourse analysis.
As shown in Figure~\ref{fig:ASER-demo}, both nodes and edges are associated with frequency-based weights to reflect higher-order selectional preferences given a specific linguistic (either dependency and discourse) pattern.
As discussed by~\cite{resnik1997selectional,zhang2019sp-10k}, such frequency distribution can serve as a good fit for humans' selectional preference, which is indeed the commonsense knowledge.
An example is shown in Figure~\ref{fig:ASER-demo}. In ASER, ``I eat plate'' and ``I eat fork'' never appear in ASER while ``I eat pizza'' appears 57 times. We can infer that ``plate'' and ``fork'' are not subjects that can be eaten while ``pizza'' is.
Similarly, the frequencies of edges can be used to reflect higher-order selectional preference between eventualities.
For example, by observing that $\langle$``{\it Person} be hungry''-\texttt{Result}-``{\it Person} eat''$\rangle$ appear at least 12 times while $\langle$``{\it Person} be hungry''-\texttt{Reason}-``{\it Person} eat''$\rangle$ appears only once, we can know that ``{\it Person} be hungry'' is more likely to result in rather than be caused by ``{\it Person} eat.''
We argue that the higher-order selectional preference in ASER can be scalable and effective to represent previously defined commonsense knowledge types in ConceptNet~\cite{liu2004conceptnet} and potentially many other types of commonsense knowledge.
To build such a large-scale eventuality knowledge graph, we first leverage unsupervised algorithms and existing tools (e.g., dependency/discourse parsing) to extract eventualities and their relations from raw documents.
For the eventuality extraction, considering that the English language's syntax is relatively fixed and consistent across domains and topics,
instead of defining complex triggers and role structures of events, we use syntactic patterns to extract all possible eventualities.
We do not distinguish between semantic senses or categories of particular triggers or arguments in eventualities but treat all extracted words with their dependency relations as hyperedge in a graph to define an eventuality as a primitive semantic unit in our knowledge graph.
For eventuality relations extraction, we adopt an end-to-end discourse parser~\cite{DBLP:conf/conll/WangL15} to determine the discourse relations between eventuality spans automatically and then create edges based on the predicted relation.
Compared with previous commonsense knowledge acquisition methods, acquiring selectional preference knowledge with linguistic patterns and discourse relation prediction models is much cheaper and scalable. Thus, it can be used to extract large-scale selectional preference knowledge from the unlabeled corpus.
After that, to overcome the challenge that a large portion of the commonsense knowledge is rarely expressed in textual corpus and motivated by the observation~\cite{zacks2001event} that human beings often conceptualize the events to a more abstract level such that they can be applied to new events, we propose to leverage existing conceptualization techniques ~\cite{SongWWLC11,SongWW15} to automatically generalize the knowledge we observed and extracted to those unseen eventualities.
As a result, we create ASER, which contains \textbf{438,648,952} unique eventualities and \textbf{648,514,465} edges.
Table~\ref{tab:size_comparison} provides a size comparison between three variations of ASER\footnote{ASER (core) includes all extracted eventualities that appear more than once, ASER (full) includes all extracted eventualities, and ASER (concept) includes all conceptualized eventualities.} (i.e., core, full, and concept) and existing eventuality-related (or simply verb-centric) knowledge bases.
Essentially, they are not large enough as modern knowledge graphs and inadequate for capturing the richness and complexity of eventualities and their relations.
FrameNet~\cite{framenet} is considered the earliest knowledge base defining events and their relations. It provides annotations about relations among about 1,000 human-defined eventuality frames, which contain 27,691 eventualities.
However, given the fine-grained definition of frames, the scale of the annotations is limited.
ACE~\cite{NIST05} (and its follow-up evaluation TAC-KBP~\cite{aguilar2014comparison}) reduces the number of event types and annotates more examples in each of event types.
PropBank~\cite{palmer2005proposition} and NomBank~\cite{meyers2004nombank} build frames over syntactic parse trees, and focus on annotating popular verbs and nouns.
TimeBank focuses only on temporal relations between verbs~\cite{pustejovsky2003timebank}.
While the aforementioned knowledge bases are annotated by domain experts,
OMCS/ConceptNet\footnote{Following the original definition, we only select the four relations (``HasPrerequisite'', ``HasFirstSubevent'', ``HasSubEvent'', and ``HasLastSubEvent'') that involve eventualities.}~\cite{liu2004conceptnet}, Event2Mind~\cite{Event2Mind}, ProPora~\cite{proparNaacl2018}, ATOMIC~\cite{Maarten2019Atomic}, ATOMIC-2020~\cite{DBLP:journals/corr/abs-2010-05953}, and GLUECOSE~\cite{DBLP:conf/emnlp/MostafazadehKMB20} leveraged crowdsourcing platforms or the general public to annotate commonsense knowledge about eventualities, in particular the relations among them.
Furthermore, KnowlyWood~\cite{TandonMDW15KnowlyWood} uses semantic parsing to extract activities (verb+object) from movie/TV scenes and novels to build four types of relations (parent, previous, next, similarity) between activities using inference rules.
Compared with all these eventuality-related KGs, ASER is larger by one or more orders of magnitude in terms of the numbers of eventualities and relations it contains.
\begin{table}[t]
\centering
\caption{ Size comparison of ASER and existing eventuality-related resources. \# Eventuality, \# Relation, and \# Relation Types are the number of eventualities, relations between these eventualities, and relation types. For KGs containing knowledge about both entity and eventualities, we report the statistics about the eventualities subset. ASER (core) filters out eventualities that appear only once and thus has better accuracy while ASER (full) can cover more knowledge. ASER (concept) runs conceptualization to aggregate diverse relations from a much cleaner ASER, resulting in a much denser commonsense knowledge graph.}
{\footnotesize
\begin{tabular}{l|c|c|c}
\toprule
& \# Eventuality & \# Relation & \# Relation Types \\
\midrule
FrameNet & 27,691 & 1,709 & 7 \\
ACE & 3,290 & 0 & 0 \\
PropBank & 112,917 & 0 & 0 \\
NomBank & 114,576 & 0 & 0 \\
TimeBank & 7,571 & 8,242 & 1 \\
OMCS (Only include edges about eventualities) & 74,989 & 116,097 & 4\\
Event2Mind & 24,716 & 57,097 & 3\\
ProPora & 2,406 & 16,269 & 1 \\
ATOMIC & 309,515 & 877,108 & 9 \\
ATOMIC-2020 & 638,128 & 1,331,113& 23 \\
GLUECOSE & 286,753 & 304,099 & 10 \\
Knowlywood & 964,758 & 2,644,415 & 4 \\
\midrule
ASER (core) & 52,940,258 & 52,296,498 & 14 \\
ASER (full) & 438,648,952 & 648,514,465 & 14 \\
\midrule
ASER (concept) & 15,640,017 & 224,213,142 & 14 \\
\bottomrule
\end{tabular}
}
\label{tab:size_comparison}
\end{table}
In summary, our contributions are as follows.
\begin{enumerate}
\item \textbf{Represent commonsense knowledge with higher-order selectional preference}: We extend the original definition of selectional preference to higher-order selectional preference between eventualities. By showing that we can cheaply acquire selectional preference knowledge from the unlabeled corpus and convert such knowledge into commonsense knowledge in the format of other commonsense knowledge bases such as ConceptNet and ATOMIC
\item \textbf{Definition of ASER}: We define a brand new knowledge graph (KG) where the primitive units of semantics are eventualities. We organize our KG as a relational graph of hyperedges.
Each eventuality instance is a hyperedge connecting several vertices, which are words. A relation between two eventualities in our KG represents one of the 14 relation types defined in PDTB~\cite{prasad2007penn} or a co-occurrence relation.
\item \textbf{Scalable Extraction of ASER}: We perform eventuality extraction over large-scale corpora. We design several high-quality patterns based on dependency parsing results to extract all eventualities that match these patterns and then apply a discourse parsing system to extract the eventuality relations. In the end, we leverage a conceptualization module to generalize the extracted knowledge to unseen eventualities.
\item \textbf{Inference over ASER:} We also provide several ways of commonsense inference over ASER.
We show that both eventuality and relation retrieval over one-hop or multi-hop relations can be modeled as conditional probability inference problems.
\item \textbf{Evaluation and Applications of ASER:} We conduct an extensive evaluation to demonstrate the quality of extracted eventuality knowledge and the transferability from such linguistic-based knowledge to commonsense knowledge.
\end{enumerate}
The rest of the paper is organized as follows. In Section~\ref{sec:related_work}, we introduce related works. In Section~\ref{sec:design_principles}, \ref{sec:aser-concepts}, \ref{sec:aser-construction}, and \ref{sec:statistics}, we introduce the details about the design principles, definition, construction, and statistics of ASER. After that, we present extensive evaluations to demonstrate the quality of extracted knowledge in Section~\ref{sec:intrinsic-evaluation}. In Section~\ref{sec:inference}, we present various kinds of inference methods ASER can support. Last but not least, we demonstrate the transferability from SP knowledge to commonsense knowledge defined in ConceptNet~\cite{liu2004conceptnet} and ATOMIC~\cite{Maarten2019Atomic} in Section~\ref{sec:transferability}. The applications of ASER in downstream tasks are discussed in Section~\ref{sec:extrinsic-evaluation}. In the end, we conclude this paper with Section~\ref{sec:conclusion}.
\section{Related Works}\label{sec:related_work}
In his conceptual semantics theory, Ray Jackendoff, a Rumelhart Prize\footnote{The David E. Rumelhart Prize is funded for contributions to the theoretical foundations of human cognition.} winner, describes semantic meaning as ``a finite set of mental primitives and a finite set of principles of mental combination~\cite{Jackendoff}''. The primitive units of semantic meanings include {\it Thing} (or {\it Object}), {\it Activity},\footnote{In his original book, he called it {\it Action}. But given the other definitions and terminologies we adopted~\cite{ALEXANDER1978,bach1986algebra}, it means {\it Activity}.} {\it State}, {\it Event}, {\it Place}, {\it Path}, {\it Property}, {\it Amount}, etc.
Understanding the semantics related to the world requires the understanding of these units and their relations.
Traditionally, linguists and domain experts built knowledge graphs (KGs)\footnote{Traditionally, people used the term ``knowledge base'' to describe the database containing human knowledge. In 2012, Google released its knowledge graph where vertices and edges in a knowledge base are emphasized. We discuss in the context of the knowledge graph, as our knowledge is also constructed as a complex graph. For more information about terminologies, please refer to~\cite{DBLP:conf/i-semantics/EhrlingerW16}.} to formalize these units and enumerate categories (or senses) and relations of them.
Typical KGs include WordNet~\cite{WordNet} for words, FrameNet~\cite{framenet} for events, and Cyc~\cite{researchCyc} and ConceptNet~~\cite{liu2004conceptnet} for commonsense knowledge.
However, their small scales restricted their usage in real-world applications.
Nowadays, with the growth of Web contents, computational power, and the availability of crowdsourcing platforms, many modern and large-scale KGs, such as Freebase~\cite{freebase}, KnowItAll~\cite{knowitall}, TextRunner~\cite{BankoCSBE07}, YAGO~\cite{YAGO,HoffartSBW13}, BabelNet~\cite{NavigliP12}, DBpedia~\cite{auer2007dbpedia}, NELL~\cite{NELL}, Probase~\cite{wu2011taxonomy}, and Google Knowledge Vault~\cite{dong2014knowledge}, have been built based on semi-automatic mechanisms.
Most of these KGs are designed and constructed based on facts about {\it Things} or {\it Objects}, such as instances and their concepts, named entities and their categories, as well as their properties and relations.
On top of them, a lot of semantic understanding problems such as question answering~\cite{BerantCFL13} can be supported by grounding natural language texts on knowledge graphs, e.g., asking a bot for {the nearest restaurants for lunch}.
Nevertheless, these KGs may fall short in circumstances that require not only fact knowledge about {\it Things} or {\it Objects}, but also the commonsense knowledge about {\it Activities}, {\it States}, and {\it Events}.
Consider the aforementioned utterance that a human would talk to the bot at 1 PM: {``I am hungry''}, which may also imply one's need for a restaurant recommendation. This, however, will not be possible unless the bot can identify that the consequence of being hungry would be {``having lunch''} at noon.
In this paper, we propose to leverage higher-order selectional preference to discover and store commonsense knowledge about {\it Activities} (or process, e.g., ``I sleep''), {\it States} (e.g., ''I am hungry''), {\it Events} (e.g., ``I make a call''), and their {\it Relations} (e.g., ``I am hungry'' may result in ``I have lunch''), for which we call ASER.
In fact, {\it Activities}, {\it States}, and {\it Events}, which are expressed by verb-related clauses, are all eventualities following the commonly adopted terminology and categorization proposed by Mourelatos~\cite{ALEXANDER1978} and Bach~\cite{bach1986algebra}.
Previous literature on eventualities mostly focuses on extracting eventualities from text with pre-defined event schemas, which enumerates triggers with senses and arguments with roles, defined in FrameNet~\cite{framenet} or ACE~\cite{NIST05}.
However, as the pre-defined event ontology is often domain-specific and small (e.g., ACE contains 33 event types), the extracted events cannot cover all commonsense.
Different from them, instead of using a small event ontology, we use patterns over the dependency graphs, which could contain multiple words and dependency edges, to extract eventualities.
Any events that satisfy the pre-defined patterns will be extracted, and thus we achieve much broader coverage.
Besides the eventuality extraction, extracting relations between eventualities is another vital research problem in the NLP community.
For example, HieVe~\cite{DBLP:conf/lrec/GlavasSMK14} focuses on extracting super-sub event relations, and TimeBank~\cite{pustejovsky2003timebank} focuses on the temporal relations.
These works typically focus on identifying implicit relations, which is a very challenging task, and the state-of-the-art models can only achieve 59.5 F1~\cite{DBLP:conf/emnlp/WangCZR20} and 75.5 F1~\cite{DBLP:conf/emnlp/HanNP19} on the HieVe and TimeBank datasets, respectively.
As a result, current models are still not ready to be used to extract high-quality relations between events.
As an alternative, we discard the implicit relations and only focus on explicit discourse relations between events. By doing so, we sacrifice the recall but make sure the high-quality of the collected knowledge.
For example, the used discourse parser proposed by
\cite{DBLP:conf/conll/WangL15}
can guarantee 90.14\% F1 on explicit discourse relation classification.
Simultaneously, we try to scan a huge corpus to guarantee the resulting knowledge graph's overall coverage.
\section{Design Principles}\label{sec:design_principles}
As aforementioned, ASER is a large-scale eventuality-based knowledge graph.
Here by eventuality, we mean {\it Activities}, {\it States}, and {\it Events}, which are defined based on the commonly adopted terminology and categorization proposed by Mourelatos~\cite{ALEXANDER1978} and Bach~\cite{bach1986algebra}:
\begin{itemize}
\item {\bf Activity: } An activity is also called a process~\cite{ALEXANDER1978,bach1986algebra}. Both activity and event are occurrences (actions) described by active verbs. An example is ``The coffee machine is brewing coffee.''
\item {\bf State:} A state is usually described by a stative verb and cannot be qualified as actions.
A typical state expression is ``The coffee machine is ready for brewing coffee''.
\item {\bf Event:} An event is defined as an occurrence that is inherently countable~\cite{ALEXANDER1978}. For example, ``The coffee machine brews a cup of coffee once more'' is an event
because it admits a countable noun ``a cup'' and cardinal count adverbials ``once,'' while ``The coffee machine brews coffee'' is not an event with an imperfective aspect which is not countable.
\end{itemize}
Unlike the previous works~\cite{DBLP:journals/coling/SiegelM00,DBLP:conf/emnlp/ZhangCWSR20}, we do not distinguish activities (or processes), states, and events. Instead, we use dependency patterns to represent all the eventualities that can be activities, states, and events and also discourse relations~\cite{prasad2007penn} such as \textit{COMPARISON.Contrast} and \textit{CONTINGENCY.Cause} as the relation types between eventualities based on the following two design principles.
\begin{figure}
\centering
\includegraphics[width = 0.7\linewidth]{image/Principle_demo.png}
\caption{Principle Demonstration. When we fix the grammar, humans' preference over the linguistic description are reflecting the commonsense. The examples are based on ones used in~\cite{katz1963structure}.}
\label{fig:principle_demo}
\end{figure}
\subsection{The Lower Bound of a Semantic Theory}
As discussed by the lower bound of a semantic theory~\cite{katz1963structure}, understanding human language requires both the knowledge about the language (i.e., grammar) and knowledge about the world.
As a result, if we fix the grammar structure of the linguistic descriptions, their difference will be the semantics.
An example is shown in Figure~\ref{fig:principle_demo}. There are three sentences that share the same grammar structure but describe different events, which may have different reasons, effect, and sub-events.
Give that the previous context is ``It is dangerous,'' humans normally will prefer the second sentence to appear in this context because lion is a dangerous animal.
And such preference can reflect the commonsense knowledge we are after.
Motivated by this, we propose to use dependency patterns to categorize eventualities and discourse relations as the relations between eventualities.
As a result, the frequency about eventualities and edges can be naturally used to represent humans' preference when the grammar structure is fixed.
\begin{table*}[t]
\small
\centering
\subtable[dobj]{
\begin{tabular}{c|c}
\toprule
SP Pair & Plausibility \\
\midrule
(eat, meal) & 10.00 \\
(close, door) & 8.50 \\
(convince, people) & 7.75 \\
(touch, food) & 5.50 \\
(hate, investment) & 4.00 \\
(confront, impulse) & 2.78 \\
(eat, mail) & 0.00 \\
\bottomrule
\end{tabular}
}
\subtable[nsubj]{
\begin{tabular}{c|c}
\toprule
SP Pair & Plausibility \\
\midrule
(singer, sing) & 10.00 \\
(law, permit) & 7.78 \\
(women, pray) & 5.83 \\
(realm, remain) & 3.06 \\
(victim, contain) & 2.22 \\
(bar, act) & 1.39 \\
(textbook, eat) & 0.00 \\
\bottomrule
\end{tabular}
}
\subtable[amod]{
\begin{tabular}{c|c}
\toprule
SP Pair & Plausibility \\
\midrule
(fresh, air) & 9.77 \\
(new, method) & 8.89 \\
(young, people) & 6.82 \\
(medium, number) & 4.09\\
(immediate, food) & 2.50\\
(eager, price) & 1.36 \\
(secret, wind) & 0.75 \\
\bottomrule
\end{tabular}
}
\subtable[dobj\_amod]{
\begin{tabular}{c|c}
\toprule
SP Pair & Plausibility \\
\midrule
(lift, heavy \textit{object}) & 9.17 \\
(design, new \textit{object}) & 8.00 \\
(recall, previous \textit{object}) & 7.05 \\
(attack, small \textit{object}) & 5.23 \\
(drag, drunk \textit{object}) & 4.25 \\
(inform, weird \textit{object}) & 3.64 \\
(earn, rubber \textit{object}) & 0.63 \\
\bottomrule
\end{tabular}
}
\subtable[nsubj\_amod]{
\begin{tabular}{c|c}
\toprule
SP Pair & Plausibility \\
\midrule
(friendly \textit{subject}, smile) & 10.00 \\
(evil \textit{subject}, attack) & 9.00 \\
(recent \textit{subject}, demonstrate) & 6.00\\
(random \textit{subject}, bear) & 4.00\\
(happy \textit{subject}, steal) & 2.25 \\
(stable \textit{subject}, understand) & 1.75 \\
(sunny \textit{subject}, make) & 0.56 \\
\bottomrule
\end{tabular}
}
\caption{Examples of first-order and second-order selectional preference and their plausibility ratings provided by human annotators~\cite{zhang2019sp-10k}. \textit{Object} and \textit{subject} are place holders to help understand the second-hop selectional preference relations.} \label{tab:SP10Kdemo}
\end{table*}
Historically, such grammar-based semantics is called selectional preference~\cite{resnik1993selection}, which is a relaxation of selectional restrictions~\cite{katz1963structure}.
Initially, the research on selectional preference focuses on the IsA hierarchy in WordNet~\cite{WordNet} and verb-object dependency relations.
Later on, the idea of selectional preference was extended to verb-subject dependency relations.
Several first-order selectional preference examples are as follows.
\begin{itemize}
\item \textsf{SP}(Cat, \texttt{IsA}, Animal) $>$ \textsf{SP}(Cat, \texttt{IsA}, Plant)
\item \textsf{SP}(Eat, \texttt{dobj}, Food) $>$ \textsf{SP}(Eat, \texttt{dobj}, Rock)
\item \textsf{SP}(Sing, \texttt{nsubj}, Singer) $>$ \textsf{SP}(Sing, \texttt{nsubj}, House)
\end{itemize}
Recently, to represent more complex commonsense knowledge, the principle of selectional preference was extended to the second-order~\cite{zhang2019sp-10k}.
The motivation is that humans tend to have a strong preference over the property of certain verbs' subjects and objects.
For example, we can formalize the commonsense that the subject of eat is more likely to be ``hungry'' rather than ``tasty'' with the following second-order selectional preference:
\begin{itemize}
\item \textsf{SP}(Eat, \texttt{Nsubj-amod}, Hungry) $>$ \textsf{SP}(Eat, \texttt{Nsubj-amod}, Tasty)
\end{itemize}
More examples are shown in Table~\ref{tab:SP10Kdemo}. Higher plausibility scores indicate that the annotators have a stronger preference for the combination.
For the first-order selectional preference, people most likely to select ``meal'' rather than ``mail'' as the object of ``eat.''
Similarly, we can see that ``heavy'' is a common property of the \textit{object} of ``lift.''
As shown by the experiments in ~\cite{zhang2019sp-10k}, such selectional preference knowledge is crucial for solving commonsense reasoning tasks such as Winograd Schema Challenge~\cite{levesque2011winograd}.
In this work, we further extend the idea of selectional preference to discourse relations between eventualities, which is denoted as higher-order selectional preference over eventualities.
\subsection{The Need of Aggregating ``Partial Information'' in Commonsense Reasoning}
As discussed by~\cite{Wilks1975IAU}, to effectively represent the selectional preference over linguistic relations and use that knowledge for language inference, we need to do aggregation over the ``partial information,'' which may ``not be invariably true'' but ``tends to be of a very high degree of generality indeed''~\cite{liu2004conceptnet}.
For example, Wilks used the following sentence as an example.
\begin{itemize}
\item The \texttt{soldiers} fired at the \texttt{women}, and we saw several of \texttt{them} fall.
\end{itemize}
We know that \textit{them} should refer to \textit{women} rather than \textit{soldiers} because we have the partial information that ``hurt things tending to fall down.''
Formally, it can be translated into the following form:
\begin{itemize}
\item (hurt, \texttt{X}) $\xrightarrow{\rm ResultIn}$ (\texttt{X}, fall).
\end{itemize}
There are many ways to find such representations of knowledge, e.g., first or even second-order logic.
However, existing logic-based or semantic frame-based methods such as combinatory categorial grammar~\cite{steedman2011combinatory} or semantic role labeling~\cite{framenet,kingsbury2002treebank} require large amounts of annotation.
Moreover, the semantic roles defined in labeled frames~\cite{framenet,kingsbury2002treebank} are too coarse-grained to support fine-grained conceptual reasoning.
An efficient way of acquiring such partial information is to do the aggregation over collected selectional preference about the instance-level eventualities and their conceptualizations.
We have shown that using such higher-order selectional preference, we can solve a subset of Winograd Schema Challenge (WSC)~\cite{levesque2011winograd} with 70\% accuracy~\cite{zhang2019sp-10k}.
For example, to solve the WSC example:
\begin{itemize}
\item The \texttt{fish} ate the \texttt{worm}. \texttt{It} was tasty.
\item The \texttt{fish} ate the \texttt{worm}. \texttt{It} was hungry.
\end{itemize}
\noindent we can merge all subjects and object, and get the following frequency information:
\begin{itemize}
\item \textsf{Frequency}('\texttt{X} eats \texttt{Y}', \texttt{co-occur}, '\texttt{X} is hungry') = 18 and \\ \textsf{Frequency}('\texttt{X} eats \texttt{Y}', \texttt{co-occur}, '\texttt{Y} is hungry') = 1;
\item \textsf{Frequency}('\texttt{X} eats \texttt{Y}', \texttt{co-occur}, '\texttt{X} is tasty') = 0 and \\ \textsf{Frequency}('\texttt{X} eats \texttt{Y}', \texttt{co-occur}, '\texttt{Y} is tasty') = 7.
\end{itemize}
These numbers reflect the aforementioned second-order selectional preferences based on which we can solve the questions.
Although such aggregation has been shown to be useful for the Winograd Schema Challenge, the collected partial information can be too coarse.
It only aggregates all information to be \texttt{X} or \texttt{Y}.
However, in real-world applications, we also need to know the following question for fine-grained concepts other than humans
\begin{itemize}
\item \textsf{Frequency}('\texttt{Company} acquires \texttt{Startups}', \texttt{ResultIn}, '\texttt{Stock} increases')=?
\end{itemize}
Therefore, a principled way of performing the conceptualization of instances and partial concept information aggregation is needed.
Thus, we propose to leverage another existing knowledge base, Probase~\cite{wu2011taxonomy}, to perform conceptualization~\cite{SongWWLC11,SongWW15} over entities that Probase can recognize.
For example, after observing that both ``having a cat'' and ``having a dog'' can cause ``being happy,'' and with the help of Probase, we can conceptualize and aggregate ``having a cat'' and ``having a dog'' to be ``having a pet,'' we can then conclude that ``having a pet'' can cause ``being happy.''
One thing worth mentioning is that the main methodology of ASER is that after the aggregation, the more heavily weighted (i.e., frequent) eventualities or edges make more sense than the less heavily weighted ones.
As a result, when we conduct the conceptualization, we do not need to consider the context because other eventualities exist, and after the aggregation, the more heavily weighted ones will still make more sense. For example, given the eventuality ``I eat apple,'' we do not need to worry about which one of ``fruit'' and ``company'' we should conceptualize ``apple'' to because we will see many other eventualities related to fruit such as ``I eat banana'' and ``I eat orange.'' In the end, the overall weight of ``I eat fruit'' will still be much higher than ``I eat company.''
\section{Overview of ASER}\label{sec:aser-concepts}
ASER is a hybrid graph combining a hypergraph $\{{\mathcal V}, {\mathcal E}\}$ where each hyperedge is constructed over vertices, and a traditional graph $\{{\mathcal E}, {\mathcal R}\}$ where each edge is built among eventualities.
For example, $E_h$=\texttt{(I, am, hungry)} and $E_t$=\texttt{(I, eat, anything)} are eventualities, where we omit the internal dependency structures for brevity.
They have a relation $\langle E_h, \texttt{Result}, E_t \rangle$, where \texttt{Result} is the relation type.
We devise the formal definition of ASER as below.
\begin{definition}
{\bf ASER KG} is a hybrid graph ${\mathcal H}$ of eventualities $E$'s. Each {\bf eventuality} $E$ is a hyperedge linking to a set of vertices $v$'s.
Each {vertex} $v$ is a {\bf word} in the vocabulary.
We define $v\in {\mathcal V}$ in the vertex set and $E \in {\mathcal E}$ in the hyperedge set.
${\mathcal E} \subseteq {\mathcal P}({\mathcal V})\setminus \{\emptyset\}$ is a subset of the power set of ${\mathcal V}$.
We also define a {\bf relation} $R_{i,j}\in {\mathcal R}$ between two eventualities $E_i$ and $E_j$, where ${\mathcal R}$ is the relation set.
Each relation has a {\bf type} $T\in {\mathcal T}$ where ${\mathcal T}$ is the type set.
Overall, we have ASER KG ${\mathcal H}=\{{\mathcal V}, {\mathcal E}, {\mathcal R}, {\mathcal T}\}$.
\end{definition}
\subsection{Eventuality}\label{sec:eventuality-pattern}
Unlike named entities or concepts, which are noun phrases, eventualities are usually expressed as verb phrases, which are more complicated in structure.
Our definition of eventualities is built upon the following two assumptions:
(1) Syntactic English patterns are relatively fixed and consistent; (2) The eventuality's semantic meaning is determined by the words it contains.
To avoid the extracted eventualities being too sparse, we use words fitting specific patterns rather than a whole sentence to represent an eventuality.
Also, to make sure the extracted eventualities have complete semantics,
we retain all necessary words extracted by patterns rather than those simple verbs or verb-object pairs in sentences.
The selected patterns are shown in Table~\ref{tab:eventuality-pattern}.
For example, for the eventuality \texttt{(dog, bark)}, we have a relation \texttt{nsubj} between the two words to indicate that there is a subject-of-a-verb relation in between.
We now formally define an eventuality as follows.
\begin{definition}
An eventuality $E$ is a hyperedge linking multiple words $\{v_1, \ldots, v_N \}$, where $N$ is the number of words in eventuality $E$. Here, $v_1, \ldots, v_N\in {\mathcal V}$ are all in the vocabulary. A pair of words in $E$ $(v_i,v_j)$ may follow a syntactic relation $e_{i,j}$.
The weight of $E$, denoted as $w_{E}^{(e)}$, is defined by the frequencies of appearance in the whole corpora.
\end{definition}
\begin{table}[t]
\small
\centering
\caption{Selected eventuality patterns (``v'' stands for normal verbs other than ``be'', ``be'' stands for ``be'' verbs, ``n'' stands for nouns, ``a'' stands for adjectives, and ``p'' stands for prepositions.), Code (to save space, we create a unique code for each pattern and will use that in the rest of this paper), and the corresponding examples.
} \label{tab:eventuality-pattern}
{
\begin{tabular}{p{0.41\textwidth}|c|p{0.41\textwidth}}
\toprule
Pattern & Code & Example \\
\midrule
$n_1$-nsubj-$v_1$ &s-v& ``The dog barks'' \\
$n_1$-nsubj-$v_1$-dobj-$n_2$ &s-v-o& ``I love you'' \\
$n_1$-nsubj-$v_1$-xcomp-$a$ &s-v-a& ``He felt ill'' \\
$n_1$-nsubj-$v_1$-xcomp-$v_2$ &s-v-v&``I want to go'' \\
$n_1$-nsubj-($v_1$-iobj-$n_2$)-dobj-$n_3$ &s-v-o-o& ``You give me the book''\\
$n_1$-nsubj-$v_1$-xcomp-$v_2$-dobj-$n_2$ &s-v-v-o&``I want to eat the apple'' \\
$n_1$-nsubj-($v_1$-dobj-$n_2$)-xcomp-$v_2$-dobj-$n_3$& s-v-o-v-o & ``I ask you to help us''\\
$n_1$-nsubj-($v_1$-dobj-$n_2$)-xcomp-($v_2$-iobj-$n_3$)-dobj-$n_4$& s-v-o-v-o-o & ``president urges the congress to make her citizen''\\
$n_1$-nsubj-$a_1$-cop-$be$ &s-be-a& ``The dog is cute'' \\
$n_1$-nsubj-$n_2$-cop-$be$ &s-be-o& ``He is a boy'' \\
$n_1$-nsubj-$v_1$-xcomp-$n_2$-cop-$be$ &s-v-be-o& ``I want to be a hero''\\
$n_1$-nsubj-$v_1$-xcomp-$a_1$-cop-$be$ &s-v-be-a& ``I want to be slim''\\
$n_1$-nsubj-($v_1$-iobj-$n_2$)-xcomp-$n_3$-cop-$be$& s-v-o-be-o & ``I want her to be hero''\\
$n_1$-nsubj-($v_1$-iobj-$n_2$)-xcomp-$a_1$-cop-$be$& s-v-o-be-a & ``I want her to be happy''\\
$there$-expl-$be$-nsubj-$n_1$& there-be-o & ``There is an apple''\\
$n_1$-nsubjpass-$v_1$ &spass-v& ``The bill is paid''\\
$n_1$-nsubjpass-$v_1$-dobj-$n_2$ & spass-v-o & ``He is served water''\\
$n_1$-nsubjpass-$v_1$-xcomp-$v_2$-dobj-$n_2$& spass-v-v-o & ``He is asked to help us''\\
\bottomrule
\end{tabular}
}
\end{table}
We use patterns from dependency parsing to extract eventualities $E$'s from unstructured large-scale corpora.
Here $e_{i,j}$ is one of the relations that dependency parsing may return.
Although the recall is sacrificed in this way, our patterns are of high precision, and we use massive corpora to extract as many eventualities as possible.
This strategy is also shared with many other modern KGs~\cite{knowitall,BankoCSBE07,NELL,wu2011taxonomy}.
\begin{table}[t]
\small
\centering
\caption{Eventuality relation types between two eventualities $E_h$ and $E_t$ and explanations.} \label{tab:relation-def}
{\
\begin{tabular}{p{0.24\textwidth}|p{0.68\textwidth}}
\toprule
Relation & Explanation \\
\midrule
$\langle E_h, \texttt{Precedence}, E_t \rangle$ & $E_h$ happens before $E_t$. \\
$\langle E_h, \texttt{Succession}, E_t \rangle$ & $E_h$ happens after $E_t$. \\
$\langle E_h, \texttt{Synchronous}, E_t \rangle$ & $E_h$ happens at the same time as $E_t$. \\ \midrule
$\langle E_h, \texttt{Reason}, E_t \rangle$ & $E_h$ happens because $E_t$ happens. \\
$\langle E_h, \texttt{Result}, E_t \rangle$ & If $E_h$ happens, it will result in the happening of $E_t$. \\
$\langle E_h, \texttt{Condition}, E_t \rangle$ & Only when $E_t$ happens, $E_h$ can happen. \\ \midrule
$\langle E_h, \texttt{Contrast}, E_t \rangle$ & $E_h$ and $E_t$ share a predicate or property and have significant difference on that property. \\
$\langle E_h, \texttt{Concession}, E_t \rangle$ & $E_h$ should result in the happening of $E^\prime$, but $E_t$ indicates the opposite of $E^\prime$ happens. \\ \midrule
$\langle E_h, \texttt{Conjunction}, E_t \rangle$ & $E_h$ and $E_t$ both happen. \\
$\langle E_h, \texttt{Instantiation}, E_t \rangle$ & $E_t$ is a more detailed description of $E_h$. \\
$\langle E_h, \texttt{Restatement}, E_t \rangle$ & $E_t$ restates the semantics meaning of $E_h$.\\
$\langle E_h, \texttt{Alternative}, E_t \rangle$ & $E_h$ and $E_t$ are alternative situations of each other. \\
$\langle E_h, \texttt{ChosenAlternative}, E_t \rangle$ & $E_h$ and $E_t$ are alternative situations of each other, but the subject prefers $E_h$. \\
$\langle E_h, \texttt{Exception}, E_t \rangle$ & $E_t$ is an exception of $E_h$. \\ \midrule
$\langle E_h, \texttt{Co-Occurrence}, E_t \rangle$ & $E_h$ and $E_t$ appear in the same sentence. \\
\bottomrule
\end{tabular}
}
\end{table}
\subsection{Eventuality Relation}
For relations among eventualities, as introduced in Section~\ref{sec:introduction}, we follow PDTB's~\cite{prasad2007penn} definition of relations between sentences or clauses but simplify them to eventualities.
Following the CoNLL 2015 discourse parsing shared task~\cite{xue2015conll}, we select 14 discourse relation types and an additional co-occurrence relation to build our knowledge graph.
\begin{definition}\label{def:relations}
A relation $R$ between a pair of eventualities $E_h$ and $E_t$ has one of the following types $T \in {\mathcal T}$ and all types can be grouped into five categories:
{\bf Temporal} (including {Precedence}, {Succession}, and {Synchronous}),
{\bf Contingency} (including {Reason}, {Result}, and {Condition}),
{\bf Comparison} (including {Contrast} and {Concession}),
{\bf Expansion} (including {Conjunction}, {Instantiation}, {Restatement}, {Alternative}, {ChosenAlternative}, and {Exception}), and
{\bf Co-Occurrence}.
The detailed definitions of these relation types are shown in Table~\ref{tab:relation-def}.
The weight of $R = \langle E_h, T, E_t \rangle$, which is denoted as $w_{R}^{(r)}$, is defined by the sum of weights of $\langle E_h, T, E_t \rangle$ that appear in the whole corpora.
\end{definition}
\subsection{ASER Conceptualization}
As aforementioned, to overcome the challenge that trivial commonsense is often omitted in humans' communication, we propose to leverage the conceptualization to generalize the knowledge about observed eventualities to unseen ones.
For each eventuality $E \in {\mathcal E}$, whose weight is $w_{E}^{(e)}$, and we can conceptualize $E$ to $E^\prime$ with confidence $w_{E, E^\prime}^{(c)}$, we will get a new conceptualized eventuality ${\bf E}^\prime$ with the weight $w_{E^\prime}^{(c)} = w_E^{(e)} \cdot w_{E, E^\prime}^{(c)}$.
Similarly, assume that an edge $R \in {\mathcal R}$ is $ \langle E_h, T, E_t \rangle$ and its weight is $w_{R}^{(r)}$, and $E_h$ and $E_t$ can be conceptualized to $E_h^\prime$ and $E_t^\prime$ with the confidence $w_{E_h, E_h^\prime}^{(c)}$ and $w_{E_t, E_t^\prime}^{(c)}$, respectively.
We can then get a new conceptualized edge $\langle E_h^\prime, T, E_t^\prime \rangle$ with the weight $w_{R}^{(r)} \cdot w_{E_h, E_h^\prime}^{(c)} \cdot w_{E_t, E_t^\prime}^{(c)}$.
Details about how to leverage an external hypernym knowledge base to get the conceptualized eventualities and determine the confidence scores are presented in Section~\ref{sec:aser-construction}.
\subsection{KG Storage}
In total, we use the following three tables of the SQLite database to store ASER.
\begin{itemize}
\item \textit{Eventuality}: As aforementioned, all eventualities in ASER are dependency graphs, where vertices are the words and edges are dependency relations.
We generate unique ``eids'' for eventualities by hashing their words, pos-tags, and dependencies and store eventualities in an \textit{Eventuality} table with SQLite database where ``eids'' is the key, and patterns, verb(s), skeleton words, words, pos-tags, dependencies, and frequencies are the other attribute columns.
\item \textit{Concept}: To effectively distinguish the eventualities before and after the conceptualization, we store eventualities created by the conceptualization step in another \textit{Concept} table and denote the id as ``cid.''
As the dependency edges are inherited from the original eventualities, we only hash the conceptualized words to generate the ``cids.'' For each conceptualized eventuality, we store its ``cid,'' pattern, frequency, and ``eids'' of the original eventualities.
\item \textit{Relations}: We store the relations between eventualities in the \textit{Relations} table. For each pair of eventualities (i.e., $E_h$ and $E_t$), if there is at least an edge between them, we will create an instance and generate a ``rid'' for them by hashing the concatenation of their ``eids.'' For the storage efficiency and retrieval feasibility, we store all edges and the associated weights between $E_h$ and $E_t$ as well as the eventuality ids of $E_h$ and $E_t$ in that instance.
\end{itemize}
\section{ASER Construction}\label{sec:aser-construction}
In this section, we introduce the ASER construction details.
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\linewidth]{image/ASER_pipeline.pdf}
\caption{ASER construction framework. The extraction and the conceptualization process are shown in the orange dash-dotted and green dashed boxes, respectively. The blue database is Probase and three gray databases are the resulted ASER.}
\label{fig:framework}
\end{figure}
\subsection{System Overview}\label{sec:system-overview}
The overall framework of our extraction system is shown in Figure~\ref{fig:framework}.
After collecting the raw corpora, we first preprocess the texts with the dependency parser.
Then we perform eventuality extraction with pattern matching.
We collect sentences and adjacent sentence pairs that contain more than two eventualities into an instance collection.
After that, we extract discourse relations from these candidate instances with the help of an explicit discourse parser~\cite{DBLP:conf/conll/WangL15}.
Considering that the discourse parser's discourse argument span might not be identical to the extracted eventualities, we apply token-based Simpson's similarity between the arguments spans and eventualities to determine whether the discourse arguments are enough to represent the meaning of the extracted eventualities. We only keep the extraction results with the Simpson's similarity larger than 0.8.
After the initial ASER construction, we leverage the \textit{IS-A} relations between nouns and named entities from Probase~\cite{wu2011taxonomy} to conduct the conceptualization.
In the end, we aggregate relations between conceptualized eventualities by retrieving head and tail eventualities from the conceptualized eventuality database and the eventuality relation database.
In the following sub-sections, we will introduce each part of the system separately.
\subsection{Corpora}
To ensure the broad coverage of ASER, we select corpora from different resources (reviews, news, forums, social media, movie subtitles, e-books) as the raw data. The details of these datasets are as follows.
$\bullet$ Yelp: Yelp is a social media platform where users can write reviews for businesses (e.g., restaurants). The latest release of the Yelp dataset\footnote{\url{https://www.yelp.com/dataset/challenge}} contains over five million reviews.
$\bullet$ New York Times (NYT): The NYT~\cite{nyt} corpus contains over 1.8 million news articles from the NYT throughout 20 years (1987 - 2007).
$\bullet$ Wiki: Wikipedia is one of the largest free knowledge datasets. To build ASER, we select the English version of Wikipedia.\footnote{\url{https://dumps.wikimedia.org/enwiki/}}
$\bullet$ Reddit: Reddit is one of the largest online forums. In this work, we select the anonymized post records\footnote{\url{https://www.reddit.com/r/datasets/comments/3bxlg7}} over one period month.
$\bullet$ Movie Subtitles: The movie subtitles corpus was collected by~\cite{lison2016opensubtitles2016}, and we select the English subset, which contains subtitles for more than 310K movies.
$\bullet$ E-books: The last resource we include is the free English electronic books from Project Gutenberg.\footnote{\url{https://www.gutenberg.org/}}
We merge these resources as a whole to perform the knowledge extraction. The detailed statistics are presented in Table~\ref{tab:corpora-statistics}.
\begin{table}[!t]
\small
\centering
\caption{Statistics of used corpora. (M means millions and G means Gigabytes.)} \label{tab:corpora-statistics}
{
\begin{tabular}{c|c|c|c|c}
\toprule
Name & \# Sentences & \# Tokens & Corpus Size & Category \\
\midrule
YELP & 54.5 M & 838.8 M & 2.5G & Reviews \\ %
NYT & 49.8 M & 1,179.4 M & 3G & News \\ %
Wiki & 110.6 M & 2,435.4 M & 13G & Knowledge \\
Reddit & 253.6 M & 3,371.3 M & 21G & Forum \\
Subtitles & 444.6 M & 3,229.4 M & 13G & Movie Scripts \\ %
E-books & 210.6 M & 3,610.0 M & 21G & Stories \\ %
\midrule
Overall & 1,123.7 M & 14,664.2 M & 73.5G & -\\
\bottomrule
\end{tabular}
}
\end{table}
\subsection{Preprocessing}\label{sec:preprocessing}
For each document, we aim to extract eventualities, relations between eventuality, conceptualized eventualities, and relations between conceptualized eventualities.
Based on the consideration of the text parsing complexity and quality, we parse each paragraph\footnote{As the discourse parser extracts discourse relations by the constituency tree of a sentence or trees of adjacent sentences, parsing sentences one by one would miss or misclassify some discourse relations.} instead of a whole document with the CoreNLP tool\footnote{\url{https://stanfordnlp.github.io/CoreNLP}} to acquire the lemmatized tokens, pos-tags, named entities, the dependency graph, and the constituency tree.
Before parsing, we replace URLs with a special token $\langle$\textit{URL}$\rangle$ and drop tables in Reddit data.
\subsection{Eventuality Extraction}\label{sec:eventuality-extraction}
\begin{algorithm}[t]\caption{Eventuality Extraction with One Pattern $p$}\label{algorithm:eventuality-extraction}
\textbf{INPUT:} Parsed dependency graph ${\mathcal D}$, center verb $v$, positive dependency edges ${\mathcal P}_p^{(pos)}$, optional edges ${\mathcal P}_p^{(opt)}$, and negative edges ${\mathcal P}_p^{(neg)}$.\\
\textbf{OUTPUT:} extracted eventuality $E$.
\begin{algorithmic}[1]
\State Initialize eventuality edge list ${\mathcal D}^\prime$.
\State Set the center verb $v$ as the $v_1$ in the pattern $p$
\For{each connection $d$ (a relation and the associated word) in positive dependency edges ${\mathcal P}_p^{(pos)}$}
\If{find $d$ in ${\mathcal D}$}
\State Append $d$ in ${\mathcal D}^\prime$.
\Else
\State Return \textit{null}.
\EndIf
\EndFor
\For{each connection $d$ in optional dependency edges ${\mathcal P}_p^{(opt)}$}
\If{find $d$ in ${\mathcal D}$}
\State Append $d$ in ${\mathcal D}^\prime$.
\EndIf
\EndFor
\For{each connection $d$ in negative dependency edges ${\mathcal P}_p^{(neg)}$}
\If{find $d$ in ${\mathcal D}$}
\State Return \textit{null}.
\EndIf
\EndFor
\State Build eventuality instance $E$ from ${\mathcal D}^\prime$.
\State Return $E$.
\end{algorithmic}
\end{algorithm}
To ensure that all the extracted eventualities are semantically complete without being too complicated, we design 18 patterns to extract the eventualities via pattern matching.
Each of the patterns contains three kinds of dependency edges: positive dependency edges, optional dependency edges, and negative dependency edges.
All the positives edges are shown in Table~\ref{tab:eventuality-pattern}.
Six more dependency relations (\texttt{advmod}, \texttt{amod}, \texttt{nummod}, \texttt{aux}, \texttt{compound}, and \texttt{neg}) are optional dependency edges that can associate with any of the selected patterns.
We omit all optional edges in the table because they are the same for all patterns.
All other dependency edges are considered negative dependency edges, designed to ensure all the extracted eventualities are semantically complete and all the patterns are exclusive with each other.
Take sentence ``I have a book'' as an example, we will only select $\langle$``I,'' ``have,'' ``book''$\rangle$ rather than $\langle$``I,'' ``have''$\rangle$ as the valid eventuality, because ``have''-dobj-``book'' is a negative dependency edge for pattern ``s-v.''
To extract eventualities from sentence $s$, considering that $s$ may contain multiple eventualities, we first split it into simple clauses based on the constituency tree. To do so, besides the commonly used \textit{SBAR} node, we also follow previous discourse parsing systems~\cite{DBLP:conf/conll/WangL15} to use a connective classifier to detect possible separators.
As a result, we split sentences based on both the subordinate conjunctions and connectives.
After that, for each verb $v$ in sentence $s$, we find the dependency graph ${\mathcal D}$ of the simple clause that contains $v$.
We then try to match ${\mathcal D}$ with all patterns one by one.
For each pattern, we put the verb $v$ as the starting point (i.e., $v_1$ in the pattern) and then try to find all the positive dependency edges. If we can find all the positive dependency edges around the center verb, these match edges and words linked by these edges are considered as potential edges and words of a valid eventuality.
Next, other edges and words are added via optional dependency edges.
In the end, we will check if any negative dependency edge can be found in the dependency graph. If not, we will keep current edges and words as a valid eventuality. Otherwise, we will disqualify it.
The pseudo-code of the eventuality extraction algorithm is in Algorithm~\ref{algorithm:eventuality-extraction}.
The time complexity of eventuality extraction is $\mathcal{O}(|{\mathcal S}| \cdot \overline{|{\mathcal D}|} \cdot \overline{|{\mathcal V}^{(v)}|})$ where $|{\mathcal S}|$ is the number of sentences, $\overline{|{\mathcal D}|}$ is the average number of dependency edges in a dependency parse tree, and $\overline{|{\mathcal V}^{(v)}|}$ is the average number of verbs in a sentence.
\subsection{Eventuality Relation Extraction}\label{sec:relation_extraction}
\begin{algorithm}[t]\caption{Eventuality Relation Extraction}\label{algorithm:relation-extraction}
\textbf{INPUT:} Parsed constituency trees ${\mathcal K}_1$ and ${\mathcal K}_2$ from adjacent sentences.\\
\textbf{OUTPUT:} Extracted relations ${\mathcal R}$.
\begin{algorithmic}[1]
\State Initialize relation list ${\mathcal R}$ as empty.
\State Extract possible connectives ${\mathcal C}$ by a connective extractor given ${\mathcal K}_1$ and ${\mathcal K}_2$.
\For {each possible connective $c \in {\mathcal C}$}
\If {two arguments of $c$ in the same sentence}
\State Extract $A_1$ and $A_2$ by a SS arguments extractor given $c$ and the sentence.
\Else
\State Extract $A_1$ by a PS argument1 extractor given $c$ and ${\mathcal K}_1$.
\State Extract $A_2$ by a PS argument2 extractor given $c$ and ${\mathcal K}_2$.
\EndIf
\If {$A_1$ is not \textit{null} and $A_2$ is not \textit{null}}
\State Classify the relation $y$ by a explicit relation classifier given $c$, ${\mathcal K}_1$, and ${\mathcal K}_2$
\State Find eventualities ${\mathcal E}_h$ that are extracted from $A_1$.
\State Find eventualities ${\mathcal E}_t$ that are extracted from $A_2$.
\State Set weight $w$ as $1 / (|{\mathcal E}_h| \cdot |{\mathcal E}_t|)$.
\For{each eventuality $E_h$ in ${\mathcal E}_h$}
\For{each eventuality $E_t$ in ${\mathcal E}_t$}
\State Build relation instance $R = \langle E_h, y, E_t \rangle$ with a weight $w$.
\State Append $R$ in ${\mathcal R}$.
\EndFor
\EndFor
\EndIf
\EndFor
\State Return ${\mathcal R}$.
\end{algorithmic}
\end{algorithm}
We then introduce how to extract the relations between eventualities.
Specifically, we employ an end-to-end discourse parser to extract the discourse relations.
The discourse parser's job is to parse a piece of text into a set of discourse relations between two adjacent or non-adjacent discourse units.
Take the sentence ``I have a story book, but it is not interesting.'' as an example. Ideally, a good discourse parser extracts ``I have a story book'' as arg1, ``it is not interesting'' as arg2, ``but'' as the connective, and annotate the relation as ``Contrast.''
In our current pipeline, we use the state-of-the-art discourse parser~\cite{DBLP:conf/conll/WangL15}, which is pretrained on the CoNLL 2015 Shared Task data (PDTB)~\cite{xue2015conll}. From CoNLL 2015 results,\footnote{\url{https://www.cs.brandeis.edu/~clp/conll15st/results.html}} we can find out that this discourse parser can achieve 90.00\% and 90.79\% F1 scores on the test data from PDTB and the blind test data from Wikinews respectively on the explicit relation classification, but performance drops to 42.72\% and 34.45\% on the implicit relation classification. Hence, to guarantee the extraction quality, we only consider the explicit discourse relations.
In explicit discourse parsing, there are two situations: both arguments are in the same sentence or not.
As statistics shows that less than 0.1\% cases that arguments are located in non-adjacent sentences in the explicit part, we simply assume arguments of which argument1 is located in the same sentence (SS) and the previous sentence (PS).
Specifically, the explicit discourse parser is consist of five components:
(1) connective extractor to identify whether a word is a possible connective,
(2) arg1 position classifier to decide whether the arg1 is located in the same sentence as the connective $c$ or the previous sentence of $c$;
(3) SS argument extractor to extract the spans of two arguments in the same sentence;
(4) PS argument extractor to extract the spans of two arguments in adjacent sentences;
(5) explicit relation classifier to classify the relation type of $c$.
Extractors in this system are essentially binary classifiers to identify whether a word is a connective or a part of any argument.
The pseudo-code of eventuality relation extraction algorithm is shown in Algorithm~\ref{algorithm:relation-extraction}.
As the extracted arguments might not be identical as the extracted eventualities, we use the Simpson's similarity to determine whether the discourse relations between arguments can be assigned to the extracted eventualities:
\begin{align}
w_{A,E}^{(sim)} = \text{Simpson}(A, E) = \frac{| {\mathcal W}_A \cup {\mathcal W}_E |}{\text{min}\{|{\mathcal W}_A|, |{\mathcal W}_E|\}},
\end{align}
where $A$ is an argument, $E$ is an eventuality, ${\mathcal W}_A$ and ${\mathcal W}_E$ are token sets of $A$ and $E$, $|\cdot|$ is the size of a token set.
If the similarity $\text{Simpson}(A, E) \geq 0.8$, we consider the argument-level relations relevant to $A$ can be assigned to the eventuality $E$ with a weight $w_{A,E}^{(sim)}$, which is inversely proportional to the size of all matched eventualities $|{\mathcal E}|$.
\subsection{Enriching ASER with Conceptualization}\label{sec:conceptualization}
We then introduce the conceptualization details. For each noun or pronoun in the extracted eventualities, we will try to conceptualized it to a higher level with the following steps. If it is a named entity, we will conceptualized it to the corresponding NER tags. Specifically, we include the 13 NER types: ``{\it Time},'' ``{\it Date},'' ``{\it Duration},'' ``{\it Money},'' ``{\it Percent},'' ``{\it Number},'' ``{\it Country},'' ``{\it State or Province},'' ``{\it City},'' ``{\it Nationality},'' ``{\it Person},'' ``{\it Religion},'' ``{\it URL}.''
If it is a personal pronoun (e.g., ``I'', ``you'', or ``they''),
we will conceptualize it to ``{\it PersonX}.''\footnote{If there are multiple people in the same edge, we will distinguish them with ``{\it PersonX}'' and ``{\it PersonY}'' etc.}
As all aforementioned conceptualization is designed by experts, we set the conceptualization probability to be 1.
If it is a regular noun, we will try to conceptualize it with Probase~\cite{wu2011taxonomy}.
Specifically, for each noun, we will retrieve its top-five hypernyms (i.e., concepts) and the associated probability from Probase.
Given an eventuality $E$ with $m$ tokens ${t_1, t_2, \cdots, t_m}$ to be mapped into concept tokens, we conceptualize it to a conceptualized eventuality $C$ with the probability:
\begin{equation}
\text{Pr}(C|E) = \prod_{i=1}^m \text{Pr}(t_i^{(c)} | t_i^{(e)}).
\end{equation}
Here $t_i^{(c)}$ is the corresponding token-level concept for token $t_i^{(e)}$. And $\text{Pr}(t_i^{(c)} | t_i^{(e)})$ is the likelihood for $\langle t_i^{(e)}, \texttt{IS-A}, t_i^{(c)} \rangle$ provided by Probase or 1.0 if $t_i^{(e)}$ can be conceptualized with rules.
For each conceptualized eventuality $C$, we would have a list of eventualities ${\mathcal E}_{C}$ that can be conceptualized to it. We can then compute the overall weight of $C$ with Eq.~\ref{eq:concept_weight}, where $w_{E}^{(e)}$ is the weight of $E$.
\begin{equation}
w_{C}^{(c)} = \sum_{E \in {\mathcal E}_{C}} \text{Pr}(C|E) \cdot w_{E}^{(e)}. \label{eq:concept_weight}
\end{equation}
\begin{figure}[!t]
\centering
\includegraphics[width=0.95\linewidth]{image/concept_demo.pdf}
\caption{Demonstration of the conceptualized ASER. The eventualities are conceptualized and connected with weighted edges. Each concept contains it's projections to specific eventualities.}
\label{fig:concept_demo}
\end{figure}
We then introduce how to construct the edges between a conceptualized eventuality $C$ and an original eventuality $E$.
For any $E^\prime \in {\mathcal E}_{C}$, if there is an edge $\langle E^\prime, T, E \rangle$ or $\langle E, T, E^\prime \rangle$, we can then construct a new edge $\langle C, T, E \rangle$ or $\langle E, T, C \rangle$ with the weight based on Eq.~\ref{eq:event_concept_edge1} or Eq.~\ref{eq:event_concept_edge2}, respectively, where $w_{R}^{(r)}$ means of weight of the relation $R$.
\begin{align}
w_{\langle C, T, E \rangle}^{(r)} &= \sum_{E^\prime \in {\mathcal E}_{C}} {\text{Pr}(C|E^\prime) \cdot w_{\langle E^\prime, T, E \rangle}^{(r)}}, \label{eq:event_concept_edge1} \\
w_{\langle E, T, C \rangle}^{(r)} &= \sum_{E^\prime \in {\mathcal E}_{C}} {w_{\langle E, T, E^\prime \rangle}^{(r)} \cdot \text{Pr}(C|E^\prime)}. \label{eq:event_concept_edge2}
\end{align}
As each conceptualized eventuality is correlated with a set of original eventualities, we need to aggregate the edges between the original eventualities to build the connections between the conceptualized ones.
Formally, given two conceptualized eventualities $C_h$ and $C_t$, we first retrieve all related original edges $\{\langle E_h, T, E_t \rangle | E_h \in {\mathcal E}_{C_h}, E_t \in {\mathcal E}_{C_t} \}$. Then we calculate the weight $\text{Pr}(C_h|E_h) \cdot w_{\langle E_h, T, E_t \rangle}^{(r)} \cdot \text{Pr}(C_t|E_t)$ for each related edge. Finally, we aggregate all weights to construct the weight as Eq.~\ref{eq:concept_edge} for the edge between $C_h$ and $C_t$ associated with the relation type $T$.
\begin{equation}
w_{\langle C_h, T, C_t \rangle}^{(r)} = \sum_{E_h \in {\mathcal E}_{C_h}} \sum_{E_t \in {\mathcal E}_{C_t}} {\text{Pr}(C_h|E_h) \cdot w_{\langle E_h, T, E_t \rangle}^{(r)} \cdot \text{Pr}(C_t|E_t)} \label{eq:concept_edge}
\end{equation}
An illustration of the conceptualized ASER is shown in Figure~\ref{fig:concept_demo}.
We can get the conceptualized eventuality ``\textit{PersonX} be hungry'' from ``I am hungry,'' ``they are hungry,'' and other extracted eventualities with $Pr(C|E)=1.000$ because their subjects (pronouns or names) are mapped to the token-level concept ``\textit{PersonX}.''
As a comparison, ``\textit{PersonX} order \textit{Meat}'' is not a deterministic eventuality: it can be conceptualized from ``I order chicken'' with $\text{Pr}(\textit{PersonX}\text{ order }\textit{Meat}|\text{I order chicken}) = \text{Pr}(\textit{PersonX}|\text{I}) \cdot \text{Pr}(\textit{Meat}|\text{chicken}) = 1.000 \cdot 0.069 = 0.069$, ``I order pork rib'' with $\text{Pr}(\textit{PersonX}\text{ order }\textit{Meat}|\text{I order pork rib}) = \text{Pr}(\textit{PersonX}|\text{I}) \cdot \text{Pr}(\textit{Meat}|\text{pork rib}) = 1.000 \cdot 0.120 = 0.120$, or other extracted ones.
Based on Eq.~\ref{eq:concept_weight}, after aggregating all weights together, we can get the concept weights for ``\textit{PersonX} be hungry'' and ``\textit{PersonX} order \textit{Meat}'' are 1389.000 and 27.705, respectively.
As for the relations between the two conceptualized eventualities,
we find $w^{(r)}_{\langle \text{I am hungry}, \texttt{Result}, \text{I order orange chicken} \rangle} = 0.125$ and $w^{(r)}_{\langle \text{I am too hungry}, \texttt{Result}, \text{I order the fried chicken} \rangle} = 1.000$, so the relation weight in the concept-level is calculated as follows:
\begin{align*}
& w^{(r)}_{\langle \textit{PersonX} \text{ be hungry}, \texttt{Result}, \textit{PersonX}\text{ order }\textit{Meat} \rangle} \\
&= \text{Pr}(\textit{PersonX} \text{ be hungry} | \text{I am hungry}) \cdot 0.125 \cdot \text{Pr}(\textit{PersonX} \text{ order } \textit{Meat} | \text{I order orange chicken}) \\
& \quad + \text{Pr}(\textit{PersonX} \text{ be hungry} | \text{I am too hungry}) \cdot 1.000 \cdot \text{Pr}(\textit{PersonX} \text{ order } \textit{Meat} | \text{I order the fried chicken}) \\
&= 1.000 \cdot 0.125 \cdot 0.069 + 1.000 \cdot 1.000 \cdot 0.069 \\
&= 0.077.
\end{align*}
Similarly, we can calculate all weights among conceptualized eventualities associated with different relation types.
One thing worth mentioning is that the relation weights depend not only on the relation weights in the extracted knowledge bases but also on the conceptualization probabilities.
\subsection{ASER Building Example}
\begin{figure}[!t]
\centering
\includegraphics[width=0.98\linewidth]{image/ASER_case.pdf}
\caption{ASER building example. The eventuality extraction, the relation extraction, and the conceptualization process are shown in green, violet, and orange colors, respectively.
For the clear representation, for each conceptualized eventuality, we only show the skeleton words and hide the optional ones.
We only show the two conceptualized eventualities with the highest probabilities for each extracted eventuality. As a demonstration, the weights of eventualities and relations are calculated from this example instead of the whole KG.
}
\label{fig:aser_building_example}
\end{figure}
At the end of this section, we use an example to demonstrate the whole extraction pipeline.
As shown in Figure~\ref{fig:aser_building_example}, given a sentence ``My army will find your boat. In the meantime, I'm sure we could find you suitable accommodations.,''\footnote{This case comes from Movie Subtitles.} our system will first detect the possible connective ``meantime'' and split this text into four simple clauses: ``My army will find your boat,'' ``In the,'' ``I'm sure,'' and ``we could find you suitable accommodations'' with the constituency parsing.
After that, our system will leverage the patterns designed in Table~\ref{tab:eventuality-pattern} to extract eventualities from the raw text by Algorithm~\ref{algorithm:eventuality-extraction}.
Simultaneously, two arguments ``My army will find your boat'' and ``we could find you suitable accommodations'' are extracted by argument extractors.
The discourse parsing system predicts the corresponding discourse relation as \texttt{Synchronous}.
As the first and last extracted eventualities can perfectly match the extracted arguments, we then create an edge $\langle$ ``my army will find your boat,'' \texttt{Synchronous}, ``we could find you suitable accommodations'' $\rangle$.
We also create an edge $\langle$ ``I am sure,'' \texttt{Co-Occurrence}, ``we could find you suitable accommodations'' $\rangle$ because the two eventualities appear in the same sentence.
After extracting the original eventualities and edges, we then try to expand it with the conceptualization.\footnote{In the real system, we first extract the original ASER, and then apply the conceptualization step over the whole KG. The presented single sentence example is just for the demonstration.}
For example, ``I am sure'' can be directly conceptualized as ``\textit{PersonX} be sure'' directly because ``I'' is a personal pronoun.
As both of the other two eventualities contain regular nouns (i.e., ``army''), these eventualities can be conceptualized to multiple eventualities.
After checking Probase, we find out that ``army'' can be conceptualized to ``\textit{Institution}'' and ``\textit{Organization}'' with the weights 0.058 and 0.038, ``boat'' can be conceptualized to ``\textit{Vehicle}'' and ``\textit{Item}'' with the weights 0.059 and 0.049, ``accommodation'' can be conceptualized to ``\textit{Service}'' and ``\textit{Facility}'' with the weights 0.056 and 0.019, respectively.
We show the two most likely results for each original eventuality (if it has multiple possible conceptualization results) in Figure~\ref{fig:aser_building_example}.
In the end, we can construct edges between conceptualized eventualities, where the weights are the product of conceptualization probabilities, e.g., $\langle$ ``\textit{Institution} find boats,'' \texttt{Synchronous}, ``\textit{PersonX} find \textit{PersonY} \textit{Service}'' $\rangle$ with the weight $0.058 \times 0.056 = 0.003$,
$\langle$ ``\textit{PersonX} be sure,'' \texttt{Co-Occurrence}, ``\textit{PersonX} find \textit{PersonY} \textit{Facility}'' $\rangle$ with the weight $1.000 \times 0.019 = 0.019$.
\section{ASER Statistics}\label{sec:statistics}
\begin{table}[!ht]
\small
\centering
\caption{Statistics of the eventuality extraction. \# Eventuality and \# Unique mean the total number and the unique number of extracted eventualities using corresponding patterns or conceptualized eventualities from them. }\label{tab:eventuality-statistics}
{
\begin{tabular}{c|c|c|c|c|c}
\toprule
\multirow{2}{*}{Pattern} & \multicolumn{2}{c|}{ASER (full)} & \multicolumn{2}{c|}{ASER (core)} & {ASER (concept)} \\
& \# Eventuality & \# Unique & \# Eventuality & \# Unique & \# Unique \\
\midrule
s-v & 351,082,855 & 100,645,728 & 260,663,083 & 14,337,769 & 1,022,415 \\
s-v-o & 284,103,317 & 159,948,356 & 139,031,585 & 18,100,360 & 8,252,653 \\
s-v-a & 11,546,768 & 6,149,584 & 5,951,980 & 752,468 & 139,087 \\
s-v-v & 24,549,946 & 11,129,566 & 14,624,526 & 1,591,424 & 216,413 \\
s-v-o-o & 6,154,685 & 3,789,253 & 2,765,728 & 460,526 & 514,084 \\
s-v-v-o & 29,445,708 & 18,659,717 & 12,720,497 & 2,187,577 & 1,482,783 \\
s-v-o-v-o & 3,863,478 & 2,674,229 & 1,462,883 & 288,326 & 522,613 \\
s-v-o-v-o-o & 91,532 & 59,290 & 40,428 & 8,499 & 18,461 \\
s-be-a & 79,235,136 & 29,845,112 & 52,068,570 & 3,733,978 & 465,747 \\
s-be-o & 98,411,474 & 53,503,410 & 49,979,659 & 6,337,042 & 2,312,209 \\
s-v-be-a & 1,927,990 & 982,438 & 1,035,864 & 123,263 & 29,738 \\
s-v-be-o & 2,322,890 & 1,574,896 & 909,250 & 184,298 & 139,239 \\
s-v-o-be-a & 277,087 & 191,973 & 100,917 & 18,793 & 6,151 \\
s-v-o-be-o & 307,031 & 231,289 & 95,815 & 22,411 & 32,796 \\
there-be-o & 16,021,849 & 6,642,438 & 10,013,628 & 953,041 & 39,500 \\
spass-v & 61,524,872 & 38,270,144 & 25,935,769 & 3,498,516 & 276,817 \\
spass-v-o & 5,519,982 & 4,129,709 & 1,677,244 & 330,229 & 154,410 \\
spass-v-v-o & 257,004 & 221,820 & 46,475 & 11,738 & 14,901 \\
\midrule
Overall & 976,643,604 & 438,648,952 & 579,123,901 & 52,940,258 & 15,640,017 \\
\bottomrule
\end{tabular}
}
\vspace{-0.1in}
\end{table}
\begin{figure}[!ht]
\centering
\subfigure[Extracted eventualities]{\label{fig:dist_eventualities}
\includegraphics[width=0.43\linewidth]{image/eventuality_rank.png}
}
\subfigure[Conceptualized eventualities]{\label{fig:dist_concepts}
\includegraphics[width=0.43\linewidth]{image/concept_rank.png}
}
\caption{Eventuality distributions.
}
\end{figure}
\begin{table}[!t]
\small
\centering
\caption{Statistics of the eventuality relation extraction. }\label{tab:relation-statistics}
{
\begin{tabular}{c|c|c|c}
\toprule
{Relation} & {ASER (full)} & {ASER (core)} & {ASER (concept)}\\
\midrule
Precedence & 14,058,213 & 1,790,016 & 4,798,015 \\
Succession & 4,939,291 & 663,183 & 1,963,820 \\
Synchronous & 19,464,898 & 3,123,042 & 8,013,943 \\
Reason & 9,775,829 & 2,205,076 & 6,439,128 \\
Result & 16,153,925 & 2,012,311 & 6,718,666 \\
Condition & 18,052,484 & 3,160,271 & 8,063,967 \\
Contrast & 59,333,901 & 8,655,661 & 24,978,311 \\
Concession & 5,684,395 & 477,155 & 1,499,276 \\
Conjunction & 82,121,343 & 13,978,907 & 45,597,200 \\
Instantiation & 1,278,381 & 18,496 & 93,266 \\
Restatement & 1,304,095 & 65,753 & 242,301 \\
Alternative & 3,539,892 & 583,174 & 123,883 \\
ChosenAlternative & 647,228 & 35,406 & 1,843,140 \\
Exception & 106,000 & 20,155 & 93,412 \\
Co-Occurrence & 412,054,590 & 49,232,161 & 113,744,814 \\
\midrule
Overall & 648,514,465 & 86,020,767 & 224,213,142 \\
\bottomrule
\end{tabular}
}
\end{table}
In total, we collect 976,643,604 eventualities from the raw documents. We filter those low-frequency eventualities that only appear once and retain 52,940,258 unique eventualities in ASER (core). From Table~\ref{tab:eventuality-statistics}, we can find the ``s-v'' and ``s-v-o'' are the most frequent patterns.
On the other hand, even though those complex patterns appear relatively less frequently, thanks to the large scale of ASER, they still appear thousands to millions of times.
The original eventuality distribution is presented in Figure~\ref{fig:dist_eventualities}.
In general, the distribution follows Zipf's law, where only a small number of eventualities appear many times while the majority of eventualities appear only a few times.
To better illustrate the distribution of eventualities, we also show several representative eventualities along with their weights, and we have two observations.
First, eventualities which can be used in general cases, like ``I think (7,501,444)'' and
``I know'' (4,267,911) appear much more times than other eventualities.
Second, eventualities in ASER are more closely related to our daily life like ``I sleep (18,347)'' or ``food is tasty (1,828)'' rather than domain-specific ones such as ``I learn python (16).''
To achieve the balance between the quality and quantity of conceptualization results, we apply the conceptualization over eventualities whose frequencies are no less than five.
As shown in Table~\ref{tab:eventuality-statistics}, after the conceptualization, we get 15,640,017 more unique eventualities. It is obvious that patterns with more nouns (e.g., ``s-v-o,'' ``s-v-v-o,'' ``s-be-o'') dominate the conceptualized eventualities. The reason is that the conceptualization is only designed for nouns, and each noun phrase would be replaced with a general noun phase if such hypernym relation appears in Probase.
For conceptualized eventualities, we can observe a similar distribution in Figure~\ref{fig:dist_concepts}.
The top three conceptualized eventualities are ``{\it PersonX} know'' (16,478,603.0), ``{\it PersonX} think'' (14,117,254.0), and ``{\it PersonX} say'' (12,113,913.0).
Although ``I think'' (7,501,444) appears the most in the raw data (``I know'' appears 3,447,429 times in the raw data), but
``you think'' (1,444,333),
``he thinks'' (314,806),
``they thinks'' (205,432),
``we think'' (196,633),
``it thinks" (174,729),
``she thinks" (142,628),
etc. appear much less than
``you know'' (4,726,264),
``he knows'' (396,013),
``they know'' (260,656),
``we know'' (457,115),
``it knows'' (247,409),
``she knows'' (190,803),
etc., respectively.
Finally, the weight of ``{\it PersonX} know'' exceeds that of ``{\it PersonX} think.''
For relations, as shown in Table~\ref{tab:relation-statistics}, we collect 648,514,465 unique relations from six data resources across different categories.
To reduce noises in parsing and extraction, we also filter out relations that $\sum_{T^\prime \in {\mathcal T}}{w_{\langle E_h, T^\prime, E_t \rangle}^{(r)}} <= 1$ where $E_h$ and $E_t$ are the head eventuality and the tail eventuality.
Furthermore, if the head or the tail is filtered out by eventuality filtering, the relation is also dropped.
Finally, we keep 86,020,767 unique relations in ASER (core), among which there are 36,788,606 relations belonging to 14 discourse relation types depending on the connectives and arguments, like \textit{Conjunction} (e.g., ``and''), \textit{Contrast} (e.g., ``but''), \textit{Condition} (e.g., ``if''), \textit{Synchronous} (e.g., ``meanwhile''), \textit{Reason} (e.g., ``because''), \textit{Result} (e.g., ``so'').
When we filter out more low-frequency eventualities, the number of relations decreases slightly.
For example, when we keep high-frequency eventualities whose frequencies are no less than five, 26.0\% of eventualities (13,766,746) and 61.5\% of relations (88,629,385) are preserved.
We apply the conceptualization over these preserved eventualities and relations based on quantity and quality considerations.
Finally, we obtain 15,640,017 unique conceptualized eventualities and 224,213,142 relations between these conceptualized eventualities.
In total, we have about 26 times more relations between conceptualized eventualities than original eventualities.
To better understand the distributions of extracted and conceptualized knowledge, we show the number of eventualities and edges over different filtering thresholds in Figure~\ref{fig:dist_nodes} and Figure~\ref{fig:dist_links}, respectively.
For the extracted knowledge, the number of eventualities and relations decreases exponentially when the threshold ranges from 5 to 100.
For the conceptualized knowledge, the rate of diminishing is even larger.
When the threshold is less than 10, the conceptualized eventuality size is greater than the original size.
But it is significantly less than the size of extracted eventualities as the threshold is larger than 10.
On the other hand, the number of conceptualized eventuality relations consistently exceeds the original relation size, which results in a denser conceptualized knowledge graph.
\begin{figure}[t]
\centering
\subfigure[Eventualities]{\label{fig:dist_nodes}
\includegraphics[width=0.4\linewidth]{image/dist_node.png}
}
\subfigure[Relations]{\label{fig:dist_links}
\includegraphics[width=0.4\linewidth]{image/dist_link.png}
}
\caption{Distributions of eventualities and relations.}
\end{figure}
\section{Evaluation}\label{sec:intrinsic-evaluation}
In this section, we leverage human annotation to evaluate the quality of ASER from the following perspectives:
\begin{enumerate}
\item \textbf{Eventuality Extraction}: We first evaluate how well the extracted eventualities can represent the original sentence's semantics. For example, if the original sentence is ``The kid goes to study,'' eventuality ``kid-go-to-study'' with pattern ``s-v-v'' can fully represent the semantics, but eventuality ``kid-go'' with the pattern ``s-v'' cannot. We show the percentage of all extracted eventualities that can fully represent the original sentences' core semantic based on different eventuality patterns.
\item \textbf{Lower-order Selectional Preference}: Besides the extraction quality, we also care about how well the eventuality statistics in ASER can reflect human's selectional preference. For example, the frequency of ``I eat food'' should be higher than ``I eat house.'' As such preference appears inside eventualities, we denote them as the lower-order selectional preference.
\item \textbf{Discourse Extraction}: After evaluating the eventualities, we assess how well the extracted edges can correctly represent the discourse relations in the original sentence. For example, assume that the original sentence is ``he went to school while I was still preparing the breakfast.'' and we have successfully extracted two eventualities ``he went to school'' and ``I was preparing breakfast,'' the correct discourse relation between them should be ``Synchronous'' rather than ``Contrast.'' In this evaluation, we report the accuracy based on different discourse relations.
\item \textbf{Higher-order Selectional Preference}: Last but not least, we annotate whether edge frequencies in ASER can reflect the higher-order selectional preference among eventualities or not. For example, the frequency of ``I am hungry''-\texttt{Result}-``I eat food'' should be larger than ``I am hungry''-\texttt{Reason}-``I eat food.''
\end{enumerate}
Evaluation details and result analysis are as follows.
\subsection{Eventuality Extraction}
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{image/eventuality_extraction.png}
\caption{Human annotation of eventuality extraction quality.}
\label{fig:eventuality_extraction}
\end{figure}
To evaluate the correctness of the selected eventuality patterns and the effectiveness of the extraction algorithm, we first employ the Amazon Mechanical Turk platform (MTurk)\footnote{\url{https://www.mturk.com/}} to evaluate the quality of eventuality extraction.
We randomly select 50 extracted eventualities for each eventuality pattern and then provide these extracted eventualities along with their original sentences to the annotators.
For each pair of eventuality and sentences, the annotators are asked to label whether the extracted eventuality phrase can fully and precisely represent the original sentences' semantic meaning.
If so, they should label them with ``Valid.'' Otherwise, they should label it with ``Not Valid.''
For each eventuality, we invite six workers to label, and if at least four of them label it as ``Valid,'' we will consider it valid.
In total, we collected 5,400 annotations. To ensure high annotation quality, we require all the annotators to be the master annotator on the MTurk.
The annotation results are shown in Figure~\ref{fig:eventuality_extraction}.
From the result, we can see that all patterns achieve over 90\% accuracy, which demonstrates the high quality of the selected patterns and the association extraction algorithm.
As introduced in Algorithm~\ref{algorithm:eventuality-extraction}, to guarantee the quality of extracted eventualities, we require the extraction algorithm to be strict and selective.
Specifically, if there is an extra dependency edge not in the positive or possible relations of a corresponding pattern, we will discard the whole sentence.
By doing so, even though we sacrifice the overall recall, we guarantee high accuracy.
Luckily, as our approach is unsupervised, we can remedy the recall problem with a larger-scale corpus.
Among the 18 patterns, we notice that the more complex patterns tend to have relatively lower accuracy.
This makes sense because the more complex an extracted eventuality is, the more likely that some of the words are redundant to the eventuality semantics. Thus the annotator may think that the extracted eventuality is not elegant enough.
\subsection{Low-order Selectional Preference}
\begin{figure*}[tb]
\centering
\subfigure[High Frequency vs. Low Frequency (before).]{\label{fig:low_sp_frequency}
\includegraphics[width=0.47\linewidth]{image/low_sp_frequency.png}
}
\subfigure[Exist vs. Non-exist (before).]{\label{fig:low_sp_exist}
\includegraphics[width=0.47\linewidth]{image/low_sp_exist.png}
}
\subfigure[High Frequency vs. Low Frequency (after). ]{\label{fig:low_sp_concept_frequency}
\includegraphics[width=0.47\linewidth]{image/low_sp_concept_frequency.png}
}
\subfigure[(C) Exist vs. Non-exist (after).]{\label{fig:low_sp_concept_exist}
\includegraphics[width=0.47\linewidth]{image/low_sp_concept_exist.png}
}
\caption{Human annotation of lower selectional preference in ASER. Experiments on the eventualities before and after the conceptualization are denoted with (before) and (after), respectively. The green color indicates the number of eventuality pairs that the more frequent eventuality makes more sense. The purple color indicates the number of eventualities pairs the less frequent eventuality makes more sense. }
\label{fig:low_sp}
\end{figure*}
To evaluate whether the eventuality frequencies in ASER can reflect human's low-order selectional preference, we first compare the plausibility of more frequent eventualities versus less frequent ones.
For each eventuality pattern, we randomly select 50 eventuality pairs such that they only have one-word difference but with a significant frequency difference. Specifically, we require the frequency of the high frequency one to be larger than five, which is the medium frequency of all eventualities, and the frequency of the high frequency one must be at least five times larger than the frequency of the low frequency one.
For each eventuality, we invite six annotators from MTruk to ask them which one of the eventuality seems more plausible to them. If more annotators agree that the more frequent one makes more sense, we will label that pair as a positive correlation. On the other hand, if more annotators agree that the less frequent makes more sense, we will label that pair as a negative correlation. If the voting draws, we will label it as similar.
As this evaluation fails to consider the eventualities with the frequency zero (i.e., they do not exist in ASER) and whether an eventuality exist or not is also a good preference indicator, we add another evaluation to prove that.
For each eventuality pattern, we randomly select 50 eventualities. Then for each of the eventualities, we randomly select a negative example by randomly changing a word inside the eventuality with another word of the same postag label such that their grammar structure is the same.
We also conduct filtering to guarantee the negative examples do not appear in ASER.
Last but not least, to show the influence of the conceptualization, we conduct the aforementioned two experiments on both the original ASER before the conceptualization and the final one after the conceptualization.\footnote{For the experiment on the ASER after the conceptualization, we only sample the conceptualized eventualities and ignore the original ones.}
We present the annotation results in Figure~\ref{fig:low_sp}
The green color indicates the number of eventuality pairs that the more frequent eventuality makes more sense, and the purple color indicates the number of eventualities pairs the less frequent eventuality makes more sense.
From the result in Figure~\ref{fig:low_sp} (a), we can see that more than 70\% of the eventuality pairs as positively correlated, which is consistent with the previous study on the correlation between frequency and selectional preference~\cite{zhang2019sp-10k}.
At the same time, we also observe that about 30\% of the less frequent eventualities are also quite plausible, which is mainly because the frequency of an eventuality is also severely influenced by the rareness of the words inside the eventuality. For example, the eventuality ``I eat avocado'' appears much less than ``I eat apple'' because avocado is much rarer than apple rather than ``I eat apple'' makes more sense than ``eat avocado.''
The results in Figure~\ref{fig:low_sp} (b) help prove that the low-frequent eventualities still contain rich low-order selectional preference because, for more than 90\% of the pairs, the randomly extracted pairs in ASER makes more sense than those out of ASER.
Furthermore, the experimental results in Figure~\ref{fig:low_sp} (c) and (d) show that even though the conceptualization process significantly improves the coverage of ASER, it would not hurt the overall quality. This is mainly because, during the conceptualization step, we carefully design the new weights based on the original weight and the confidence scores provided by Probase~\cite{wu2011taxonomy}.
\subsection{Relation Extraction}
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{image/edge_extraction.png}
\caption{Human annotation of discourse relation extraction quality.}
\label{fig:edge_extraction}
\end{figure}
Besides the eventuality extraction, we also care about the extraction quality of the discourse relations between eventualities.
For each relation type, we randomly select 50 edges and the corresponding sentences. We generate a question for each pair of them by asking the annotators if they think the extracted discourse relation can represent the correct relation in the original sentence.
If so, they should label it as ``Valid.'' Otherwise, they should label it as ``Not Valid.''
Similar to the eventuality extraction experiment, we invite six annotators for each edge. If more than four of them agree that the extracted relation is ``Valid,'' we will consider it to be ``Valid.''
From the results in Figure~\ref{fig:edge_extraction} we can see that the overall accuracy is about 80\%, which is consistent with the reported performance of the used discourse relations extraction system~\cite{DBLP:conf/conll/WangL15}. Besides that, we also notice that the model performance varies on different relation types.
For example, the model tends to perform well on simple types such as ``Reason'' and ``Alternative'' because the popular connectives (i.e., ``because'' and ``or'') are less ambiguous.
As a comparison, when the connective is more ambiguous (e.g., ``while'' for ``Synchronous''), the overall performance will drop.
\subsection{Higher-order Selectional Preference}
\begin{figure*}[tb]
\centering
\subfigure[High Frequency vs. Low Frequency (before).]{\label{fig:high_sp_frequency}
\includegraphics[width=0.47\linewidth]{image/high_sp_frequency.png}
}
\subfigure[Exist vs. Non-exist (before).]{\label{fig:high_sp_exist}
\includegraphics[width=0.47\linewidth]{image/high_sp_exist.png}
}
\subfigure[High Frequency vs. Low Frequency (after).]{\label{fig:high_sp_concept_frequency}
\includegraphics[width=0.47\linewidth]{image/high_sp_concept_frequency.png}
}
\subfigure[Exist vs. Non-exist (after).]{\label{fig:high_sp_concept_exist}
\includegraphics[width=0.47\linewidth]{image/high_sp_concept_exist.png}
}
\caption{Human annotation of the higher selectional preference in ASER. Experiments on the eventualities before and after the conceptualization are denoted with (before) and (after), respectively. Green color indicates the number of edge pairs that the more frequent edge makes more sense, and purple color indicates the number of edge pairs the less frequent edge makes more sense. }
\label{fig:high_sp}
\end{figure*}
Finally, we evaluate whether the edge frequency in ASER can be used to reflect human's high-order selectional preference about eventualities.
Similar to the evaluation on the lower-order selectional preference, we conduct two experiments (i.e., (1) High frequency vs. Low Frequency; (2) Exist vs. None-exist) on ASER before and after the conceptualization.\footnote{For the experiment on the ASER after the conceptualization, we only sample the conceptualized eventualities and ignore the original ones.}
For the ``High frequency vs. Low Frequency'' experiment, we randomly sample 50 edge pairs for each relation type such that the two edges in each pair share the same head eventuality, relation type, but different tail eventuality (e.g., $\langle$``I am hungry'', \texttt{Result}, ``I eat food''$\rangle$ versus $\langle$``I am hungry'', \texttt{Result}, ``I exercise''$\rangle$).
More importantly, the two sampled edges should have significantly different frequencies.
Specifically, we require the frequency of the high frequency one to be larger than five, and the frequency of the high frequency one must be at least five times larger than the frequency of the low frequency one.
For the ``Exist vs. None-exist'' experiment, for each relation type, we first randomly sample 50 edges. Then for each edge, we randomly replace a single word of the tail eventuality such that the new tail is very similar to the original one but the created edge does not exist in ASER.
The annotations results are presented in Figure~\ref{fig:high_sp}.
In general, we can make similar observations as the lower SP that the correlation is more significant when we compare the existing and non-existing edges.
Besides that, the experiments on the conceptualized ASER help demonstrate that the conceptualization module will not influence the overall quality of edge frequencies.
\section{Inference over ASER}~\label{sec:inference}
In this section, we first introduce two kinds of inferences (eventuality retrieval and relation retrieval) based on ASER. For each of them, inferences over both one-hop and multi-hop are provided. Complexities of these two retrieval algorithms are $\mathcal{O}(n^k)$, where $n$ is the number of average adjacent eventualities per eventuality and $k$ is the number of hops.
In this section, we show how to conduct these inferences over one-hop and two-hop as the demonstration. ASER is composed of eventualities and concepts. In line with the settings, we conduct case studies over extracted sub-graphs of eventuality and concept graph.
After that, we investigate the rule and meta-path based inferences on ASER. For the rule based inference, we leverage AMIE+ \cite{DBLP:journals/vldb/GalarragaTHS15}, a rule mining system on ontological knowledge bases (KBs), to discover closed and connected Horn rules on ASER. For the meta-path based inference, we obtain the frequent meta-paths using statistical methods and perform case studies by instantiating the meta-paths in both eventuality and concept graphs.
\subsection{Eventuality Retrieval}
The eventuality retrieval inference is defined as follows. Given a head eventuality\footnote{ASER also supports the prediction of head eventualities given tail eventualities and relations. We omit it in this section for a clear presentation.} $E_h$ and a relation list ${\mathcal L}$ = ($T_1, T_2, ..., T_k$), find related eventualities and their associated probabilities such that for each eventuality $E_t$ we can find a path, which contains all the relations in ${\mathcal L}$ in order from $E_h$ to $E_t$.
\subsubsection{One-hop Inference}
\begin{table}[]
\centering
\small
\begin{tabular}{l|c|l|c}
\toprule
Head & Relation& Tail & Probability\\
\midrule
You drink alcohol & $\texttt{Synchronous}$& You drown & 0.50 \\
I drink coffee & $\texttt{Result}$ & I calm down & 0.33 \\
You are an employee & $\texttt{Contrast}$ & You get fired & 0.50 \\
I am programmer & $\texttt{Result}$ & I have free time & 1.00 \\
You go to restaurant & $\texttt{Precedence}$ & You get sick & 0.50 \\
I am frightened & $\texttt{Reason}$ & Dog barks & 0.80 \\
I order chicken & $\texttt{Concession}$ & I am a vegan & 1.00 \\
It is my birthday & $\texttt{Result}$ & We go to zoo & 0.20 \\
It is a cat & $\texttt{Condition}$ & It is a tiger & 0.67 \\
The surgery goes well & $\texttt{Result}$ & There is no complication & 0.50 \\
\bottomrule
\end{tabular}
\caption{Cases of one-hop eventuality inference in the eventuality graph. In the tables of the case study of eventualities, the words in eventualities are stored as lemmas in KBs. However, to clarify the examples, we manually correct the grammar mistakes.}
\label{tab:event_one_hop_node}
\end{table}
\begin{table}[]
\centering
\small
\begin{tabular}{l|c|l|c}
\toprule
Head& Relation& Tail & Probability\\
\midrule
\textit{Company} be \textit{Stakeholder-Group} & $\texttt{Condition}$ & \textit{PersonX} be successful & 0.53 \\
\textit{PersonX} have \textit{Issue} & $\texttt{Reason}$ & \textit{PersonX} be proud & 0.52 \\
\textit{PersonX} get \textit{Symptom} & $\texttt{Synchronous}$ & \textit{PersonX} be \textit{Vulnerable-Group} & 0.50 \\
\textit{PersonX} be \textit{Emotion} & $\texttt{Succession}$ & \textit{PersonX} marry & 0.51 \\
\textit{AnimalX} bark & $\texttt{Result}$ & \textit{AnimalX} kill \textit{AnimalY} & 0.33 \\
\textit{PersonX} be \textit{Predator} & $\texttt{Result}$ & \textit{PersonX} tease \textit{PersonY} & 0.25 \\
\textit{PersonX} do \textit{Academic-Misconduct} & $\texttt{Contrast}$ & \textit{PersonX} tell \textit{Institute} & 0.52 \\
\textit{PersonX} play \textit{Sport} & $\texttt{Reason}$ & \textit{PersonX} love \textit{Activity} & 0.27 \\
\textit{PersonX} hurt \textit{Insect} & $\texttt{Condition}$ & \textit{PersonX} help \textit{Insect} & 0.83 \\
\textit{PersonX} have \textit{Social-Medium} & $\texttt{Result}$ & \textit{PersonX} post it & 0.72 \\
\bottomrule
\end{tabular
\caption{Cases of one-hop eventuality inference in the concept graph. The concepts are marked as $\textit{italic}$ texts.}
\label{tab:concept_one_hop_node}
\end{table}
For the one-hop inference, we assume the target relation is $T_1$. We then define the probability of any potential tail node $E_t$ as:
\begin{equation}\label{eq:one-hop-eventuality-retrieval}
\text{Pr}(E_t| E_h, T_1) =
\frac{w_{\langle E_h, T_1, E_t \rangle}^{(r)}}{\sum_{E_t^\prime, s.t., (E_h,T_1,E_t^\prime)\in {\mathcal R}}{w_{\langle E_h, T_1, E_t^\prime \rangle}^{(r)}}},
\end{equation}
where $w_{\langle E_h, T_1, E_t \rangle}^{(r)}$ is the relation weight, which is defined in Definition 3. If no node is connected with $E_h$ via $T_1$, $\text{Pr}(E^\prime | E_h, T_1)$ will be 0 for any $E^\prime \in {\mathcal E}$.
Several interesting inference examples are observed. In Table \ref{tab:event_one_hop_node}, we list the reasonable examples of one-hop eventuality inference in eventuality graph. We also list some of them as follows for discussion:
\begin{itemize}
\item $\langle$``I drink coffee'', $\texttt{Reason}$, ``I enjoy the flavor''$\rangle$
\item $\langle$``You go to restaurant'', $\texttt{Precedence}$, ``You got sick''$\rangle$
\item $\langle$``It is a cat'', $\texttt{Condition}$, ``It is a tiger''$\rangle$
\end{itemize}
It is observed that ``I enjoy the (coffee) flavor'' is likely to be the reason for ``I drink coffee.'' It is also common that if you eat in an unhygienic restaurant, you would probably get sick after you go to the restaurant. Given the fact that the tiger is the largest cat species, it is reasonable to say that if ``it is a tiger,'' ``it is a cat.''
The following examples in Table \ref{tab:concept_one_hop_node} show the results of one-hop eventuality inference in concept graph.
\begin{itemize}
\item $\langle$``\textit{Company} be \textit{Stakeholder-Group}'', $\texttt{Condition}$, ``\textit{PersonX} be successful''$\rangle$
\item $\langle$``\textit{PersonX} hurt \textit{Insect}'', $\texttt{Condition}$, ``\textit{PersonX} help \textit{Insect}''$\rangle$
\item $\langle$``\textit{PersonX} be \textit{Emotion}'', $\texttt{Succession}$, ``\textit{PersonX} marry''$\rangle$
\end{itemize}
For instance, if someone is successful, his/her company is likely a big corporation of stakeholders. The second one shows a situation that if an unprofessional person helps insects out of good wills, he/she probably hurts them in reverse. We could also infer from the last case that people tend to be emotional when they get married.
\subsubsection{Two-hop Inference}
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{l|c|l|c|l|c}
\toprule
Head& Relation1 & Middle &Relation2& Tail & Probability\\
\midrule
I go to school & $\texttt{Reason}$ & [I admire] & $\texttt{Synchronous}$& I am grown up & 0.50 \\
I go to bed & $\texttt{Conjunction}$ & [I sleep early] & $\texttt{Result}$ & I am healthy & 0.86 \\
We have dinner & $\texttt{Conjunction}$ & [Food is very good] & $\texttt{Contrast}$ & Service is not & 0.95 \\
You go to restaurant & $\texttt{Condition}$ & [They do something right] & $\texttt{Reason}$ & There is a line-up & 0.50 \\
We have lunch & $\texttt{Conjunction}$ & [We really hit it off] & $\texttt{Contrast}$ & She has a boyfriend at time & 0.50 \\
You drink alcohol & $\texttt{Contrast}$ & [You are fine] & $\texttt{Contrast}$ & You have no work &0.75 \\
I am a vegan & $\texttt{Result}$ & [I do not eat fish] & $\texttt{Contrast}$ & We are hungry & 0.73 \\
I go to bar & $\texttt{Precedence}$ & [Our table is ready] & $\texttt{Result}$ & We take seats & 0.35\\
I go to restaurant & $\texttt{Reason}$ & [I have a coupon] & $\texttt{Contrast}$ & It is expired & 0.36 \\
I go to gym & $\texttt{Precedence}$ & [I go on a date] & $\texttt{Contrast}$ & We have nothing in common & 0.25 \\
\bottomrule
\end{tabular} }
\caption{Cases of two-hop eventuality inference in the eventuality graph. In the table, we provide a typical example of middle nodes (embraced by brackets) to create a scenario for better understanding.}
\label{tab:event_two_hop_node}
\end{table}
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{l|c|l|c|l|c}
\toprule
Head& Relation1 & Middle &Relation2& Tail & Probability\\
\midrule
\textit{PersonX} wait for \textit{PersonY} & $\texttt{Precedence}$ & [\textit{PersonX} be tired] & $\texttt{Result}$ & \textit{PersonX} go to sleep & 0.50 \\
\textit{PersonX} hate \textit{Animal} & $\texttt{Contrast}$ & [\textit{PersonX} be harmless] & $\texttt{Contrast}$ & \textit{PersonX} be \textit{Symptom} & 0.40\\
\textit{PersonX} be cranky & $\texttt{Synchronous}$ & [\textit{PersonX} be hungry] & $\texttt{Result}$ & \textit{PersonX} order \textit{Meat} & 0.23 \\
\textit{PersonX} be \textit{Artist} & $\texttt{Contrast}$ & [\textit{PersonX} play \textit{Sport}] & $\texttt{Reason}$ & \textit{PersonX} be strong & 0.33 \\
\textit{PersonX} regret & $\texttt{Condition}$ & [\textit{PersonX} despise \textit{PersonY}] & $\texttt{Reason}$ & \textit{PersonY} be \textit{Performer} & 0.20 \\
\textit{PersonX} pull gun & $\texttt{Reason}$ & [\textit{PersonX} startle] & $\texttt{Synchronous}$ & \textit{Domestic-Animal} bark & 0.50 \\
\textit{Predator} take down \textit{Animal} & $\texttt{Reason}$ & [It be \textit{Predator}] & $\texttt{Synchronous}$ & \textit{PersonX} shoot & 0.32 \\
\textit{PersonX} be \textit{Academic-Title} & $\texttt{Result}$ & [\textit{PersonX} be right] & $\texttt{Contrast}$ & \textit{PersonY} doubt it & 0.28 \\
\textit{PersonX} hear it & $\texttt{Synchronous}$ & [\textit{PersonY} play \textit{Musical-Instrument}] & $\texttt{Synchronous}$ & \textit{PersonY} be blue & 0.65 \\
\textit{PersonX} be \textit{Artist} & $\texttt{Condition}$ & [\textit{PersonX} strike \textit{PersonY}] & $\texttt{Synchronous}$ & \textit{PersonY} interview \textit{PersonX} & 0.40 \\
\bottomrule
\end{tabular}}
\caption{Cases of two-hop eventuality inference in the concept graph. In the table, we provide a typical example of middle nodes (embraced by brackets) to create a scenario for better understanding. The concepts are marked as $\textit{italic}$ texts.}
\label{tab:concept_two_hop_node}
\end{table}
On top of Eq.~(\ref{eq:one-hop-eventuality-retrieval}), it is easy for us to define the probability of $E_t$ on two-hop setting. Assume the two relations are $T_1$ and $T_2$ in order. We can define the probability as follows:
\begin{equation}\label{eq:two-hop-eventuality-retrieval}
\text{Pr}(E_t| E_h, T_1, T_2) =
\sum_{E_m\in {\mathcal E}_m}{\text{Pr}(E_m|E_h, T_1) \text{Pr}(E_t|E_m, T_2)},
\end{equation}
where ${\mathcal E}_m$ is the set of intermediate node $E_m$ such that $(E_h,T_1,E_m)$ and $(E_m, T_2, E_t) \in {\mathcal R}$.
We list the intuitive examples of two-hop eventuality inference in eventuality and concept graph in Table \ref{tab:event_two_hop_node} and Table \ref{tab:concept_two_hop_node}. To better understand the two relations between the head node and the tail node, a typical middle node embraced by brackets is provided. In the eventuality graph, three examples in Table \ref{tab:event_two_hop_node} are given for further explanation.
\begin{itemize}
\item $\langle$``I go to bed'', $\texttt{Conjunction}$, [``I sleep early''], $\texttt{Result}$, ``I am healthy''$\rangle$
\item $\langle$``We have lunch'', $\texttt{Conjunction}$, [``We really hit it off''], $\texttt{Contrast}$, ``She has a boyfriend at time''$\rangle$
\item $\langle$``I go to restaurant'', $\texttt{Reason}$, [``I have a coupon''], $\texttt{Contrast}$, ``It is expired''$\rangle$
\end{itemize}
The first example illustrates that ``I go bed'' and ``I sleep early'' tend to result in ``I am healthy.'' The second one describes a common social situation that I have lunch with a girl and we really hit it off. But she has a boyfriend at that time. Also, it is inferred that the reason why I go to that restaurant is that I have a coupon. However, I find out that the coupon is expired.
Leveraging the same method, we perform two-hop eventuality inference in the concept graph and the results are presented in Table \ref{tab:concept_two_hop_node}.
\begin{itemize}
\item $\langle$``\textit{PersonX} wait for \textit{PersonY}'', $\texttt{Precedence}$, [``\textit{PersonX} be tired''], $\texttt{Result}$, ``\textit{PersonX} go to sleep''$\rangle$
\item $\langle$ ``\textit{PersonX} be cranky'', $\texttt{Synchronous}$, [``\textit{PersonX} be hungry''], $\texttt{Result}$, ``\textit{PersonX} order \textit{Meat}'' $\rangle$
\item $\langle$ ``\textit{PersonX} be \textit{Artist}'', $\texttt{Condition}$, [``\textit{PersonX} strike \textit{PersonY}''], $\texttt{Synchronous}$, ``\textit{PersonY} interview \textit{PersonX}'' $\rangle$
\end{itemize}
An interesting example shows that someone is waiting for his/her friend for such a long time that he/she is tired and decides to go to sleep. We also observe that the result of someone is cranky and hungry is most likely to be that he/she orders meats. The last one shows that The artifacts of an artist $\textit{PersonX}$ strikes $\textit{PersonY}$ and it happens at the same time as $\textit{PersonY}$ interviews with $\textit{PersonX}$.
\subsection{Relation Retrieval}
\begin{table}[]
\centering
\small
\begin{tabular}{c|l|l|c}
\toprule
Relation & Head& Tail & Probability\\
\midrule
$\texttt{Result}$ & You drink alcohol & You have to pee & 1.00 \\
$\texttt{Result}$ & I drink coffee & I order a cappuccino & 0.50 \\
$\texttt{Alternative}$ & You are a employee & You will be fired & 0.50 \\
$\texttt{Contrast}$ & I eat meat & I am not a steak lover & 1.00 \\
$\texttt{Precedence}$ & You go to sleep & you wake up & 1.00 \\
$\texttt{Contrast}$ & I go to school & I drop out & 0.50\\
$\texttt{Reason}$ & I am not picky & I go to restaurant & 0.43\\
$\texttt{Contrast}$ & I love to cook & I go to restaurant & 0.57\\
$\texttt{Precedence}$ & He waves his hat & The train stops & 1.00\\
$\texttt{Concession}$ & I go to gym & I am tired & 0.83\\
\bottomrule
\end{tabular
\caption{Cases of one-hop relation inference in the eventuality graph. }
\label{tab:event_one_hop_relation}
\end{table}
\begin{table}[]
\centering
\small
\begin{tabular}{c|c|c|c}
\toprule
Relation & Head& Tail & Probability\\
\midrule
$\texttt{Condition}$ & \textit{Company} be \textit{Stakeholder-Group} & \textit{PersonX} do \textit{Local-Ad} & 0.10 \\
$\texttt{Contrast}$ & \textit{PersonX} call \textit{Agency} & It take \textit{Duration} & 0.18 \\
$\texttt{Reason}$ & \textit{PersonX} be \textit{Public-Figure} & \textit{PersonX} be professional & 0.30 \\
$\texttt{Synchronous}$& \textit{Animal} bite & \textit{Animal} be frightened & 0.23\\
$\texttt{Precedence}$ & \textit{PersonX} be \textit{Vulnerable-Group} & \textit{PersonX} quit \textit{Activity} & 0.88 \\
$\texttt{Synchronous}$ & It be \textit{Domestic-Animal} & It be \textit{Mammal} & 0.77 \\
$\texttt{Contrast}$ & \textit{Bird} catch \textit{Animal} & \textit{Animal} get cheese & 0.67 \\
$\texttt{Synchronous}$ & \textit{PersonX} whistle & \textit{Animal} bark & 1.00 \\
$\texttt{Condition}$ & \textit{PersonX} give lecture & \textit{PersonX} be \textit{Academic-Title} & 0.21 \\
$\texttt{Result}$ & \textit{PersonX} play \textit{Sport} & \textit{PersonX} be fit & 0.87 \\
\bottomrule
\end{tabular
\caption{Cases of one-hop relation inference in the concept graph. The concepts are marked as $\textit{italic}$ texts.}
\label{tab:concept_one_hop_relation}
\end{table}
The relation retrieval inference is defined as follows. Given two nodes $E_h$ and $E_t$, find all relation lists and their probabilities such that for each relation list ${\mathcal L}$ = ($T_1, T_2, ..., T_k$), we can find a path from $E_h$ to $E_t$, which contains all the relations in ${\mathcal L}$ in order.
\subsubsection{One-hop Inference}
Assuming that the path length is one, we define the probability of one relation $R = \langle E_h, T, E_t \rangle$ given $E_h$ and $E_t$ as:
\begin{equation}\label{eq:one-hop-plausibility}
\text{Pr}(R | E_h, E_t) = \text{Pr}(T| E_h, E_t) =
\frac{w_{\langle E_h, T, E_t \rangle}^{(r)}}{\sum_{T^\prime \in {\mathcal T}}{w_{\langle E_h, T^\prime, E_t \rangle}^{(r)}}},
\end{equation}
where ${\mathcal T}$ is the relation type set.
In Table \ref{tab:event_one_hop_relation} and \ref{tab:concept_one_hop_relation}, we perform one-hop relation inference in eventuality and concept graph separately. The relations between head nodes and tail nodes are retrieved to present the commonsense in daily life. In Table \ref{tab:event_one_hop_relation}, the eventuality relations are mined to show frequent patterns in eventuality graph.
\begin{itemize}
\item $\langle$ $\texttt{Result}$, ``I drink coffee'', ``I order cappuccino''$\rangle$
\item $\langle$ $\texttt{Contrast}$, ``I love to cook'', ``I go to restaurant'' $\rangle$
\item $\langle$ $\texttt{Concession}$, ``I go to gym'', ``I am tired'' $\rangle$
\end{itemize}
For example, ``You drink alcohol'' usually leads to ``You have to pee''. Another intriguing case illustrates that if someone loves to cook, he/she tends not to go to the restaurant regularly. The last one describes a diligent and determined person who decides to go to the gym, although he/she is tired.
In the concept graph, the same process is used and the results are stored in Table \ref{tab:concept_one_hop_relation}.
\begin{itemize}
\item $\langle$ $\texttt{Contrast}$, ``\textit{Bird} catch \textit{Animal}'', ``\textit{Animal} get cheese'' $\rangle$
\item $\langle$ $\texttt{Condition}$, ``\textit{PersonX} give lecture'', ``\textit{PersonX} be \textit{Academic-Title}'' $\rangle$
\item $\langle$ $\texttt{Result}$, ``\textit{PersonX} play \textit{Sport}'', ``\textit{PersonX} be fit'' $\rangle$
\end{itemize}
We learn from the first example that if birds do not catch these animals (e.g., rats), they would probably get cheese. The second one shows that if someone has an academic title (e.g., professor), he/she will deliver a lecture. The third example tells that the result of $\textit{PersonX}$ plays sport is he/she is fit.
\subsubsection{Two-hop Inference}
Similarly, given two nodes $E_h$ and $E_t$, we define the probability of a two-hop connection ($T_1$, $T_2$) between them as follows:
\begin{align}
\text{Pr}(T_1, T_2 | E_h, E_t) &= \sum_{E_m\in {\mathcal E}_m} \text{Pr}(T_1, T_2, E_m | E_h, E_t) \nonumber\\
&=\sum_{E_m\in {\mathcal E}_m} \text{Pr}(T_1 | E_h) \text{Pr}(E_m | T_1, E_2 | E_m, E_t),
\end{align}
where $\text{Pr}(T | E_h)$ is the probability of a relation type $T$ given a head eventuality $E_h$, which is defined as follows:
\begin{equation}\label{eq:relation-probability}
\text{Pr}(T | E_h)=
\frac{\sum_{E_t, s.t., (E_h, T, E_t) \in {\mathcal R}}{w_{\langle E_h, T^\prime, E_t \rangle}^{(r)}}}{\sum_{T^\prime\in {\mathcal T}} \sum_{E_t, s.t., (E_h, T^\prime, E_t) \in {\mathcal T}}{w_{\langle E_h, T^\prime, E_t \rangle}^{(r)}}}.
\end{equation}
The two-hop relations are inferred from the eventuality and concept graph in Table \ref{tab:event_two_hop_relation} and \ref{tab:concept_two_hop_relation}. Some reasonable results are listed below. In line with two-hop eventuality inference, we give a typical middle node embraced by brackets to show the circumstance more clearly.
\begin{itemize}
\item $\langle$ $\texttt{Synchronous}$, $\texttt{Conjunction}$, ``We have breakfast'', [``Our room is ready''], ``The front desk staff is friendly'' $\rangle$
\item $\langle$ $\texttt{Synchronous}$, $\texttt{Reason}$, ``I sit on chair'', [``I get my hair washed''], ``Stylist tells me'' $\rangle$
\item $\langle$ $\texttt{Reason}$, $\texttt{Result}$, ``I go to supermarket'', [``I have a coupon''], ``The price is great'' $\rangle$
\end{itemize}
In the eventuality graph, we find that some tourists visit a hotel. The guests have breakfast when their room is cleaned and ready. Meanwhile, they find the staff at the front desk is friendly and nice. The second example shows a common thing at the haircut salons. ``I sit on chair'' to get my hair washed because the stylist tells me to do so before haircut. We also find that someone goes to a supermarket because he/she has a coupon to lower the prices of grocery.
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{c|c|l|l|l|c}
\toprule
Relation1 & Relation2 & Head & Middle & Tail & Probability\\
\midrule
$\texttt{Conjunction}$ & $\texttt{Synchronous}$& I go to bed & [I would sleep] & I heal & 0.25 \\
$\texttt{Result}$ & $\texttt{Reason}$ & I go to bed & [I fall right to sleep] & I am drunk & 0.30 \\
$\texttt{Synchronous}$& $\texttt{Conjunction}$ & We have breakfast & [Our room is ready] & The front desk staffs are friendly & 0.12 \\
$\texttt{Contrast}$ & $\texttt{Precedence}$ & You are an employee & [You get fired] & The contract ends & 0.11 \\
$\texttt{Reason}$ & $\texttt{Conjunction}$ & I drink coffee & [I have severe ADHD] & I do not get any help & 0.33 \\
$\texttt{Synchronous}$& $\texttt{Reason}$ & I sit on chair & [I get my hair wash] & Stylist tell me & 0.25 \\
$\texttt{Contrast}$ & $\texttt{Reason}$ & I am a vegan & [I eat meat] & It tastes good & 0.33\\
$\texttt{Reason}$ & $\texttt{Result}$ & I go to supermarket & [I have a coupon] & The price is great & 0.23\\
$\texttt{Reason}$ & $\texttt{Contrast}$ & I go to restaurant & [The service is great] & The food is mediocre & 0.49\\
$\texttt{Contrast}$ & $\texttt{Contrast}$ & The surgery goes well & [She is in a coma] & She is stabilized & 0.53\\
\bottomrule
\end{tabular}}
\caption{Cases of two-hop relation inference in the eventuality graph. In the table, we provide a typical example of middle nodes (embraced by brackets) to create a scenario for better understanding.}
\label{tab:event_two_hop_relation}
\end{table}
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{c|c|l|l|l|c}
\toprule
Relation1 & Relation2 & Head & Middle & Tail & Probability\\
\midrule
$\texttt{Precedence}$ & $\texttt{Precedence}$ & \textit{PersonX} be \textit{Stakeholder} & [\textit{PersonX} tell \textit{PersonX}] & \textit{PersonX} sign \textit{Document} & 0.85 \\
$\texttt{Precedence}$ & $\texttt{Precedence}$ & \textit{PersonX} wait for \textit{PersonX} & [\textit{PersonX} send \textit{Information}] & \textit{PersonX} drag \textit{PersonX} away & 0.40 \\
$\texttt{Result}$ & $\texttt{Precedence}$ & \textit{PersonX} be \textit{Vulnerable-Population} & [\textit{PersonX} be homeless] & \textit{Organization} help \textit{PersonX} & 0.73 \\
$\texttt{Precedence}$ & $\texttt{Result}$ & \textit{Predator} catch \textit{Herbivore} & [\textit{Predator} eat \textit{Meat}] & \textit{Mammal} live & 0.79\\
$\texttt{Synchronous}$ & $\texttt{Result}$ & \textit{PersonX} be thirsty & [\textit{PersonX} be hungry] & \textit{PersonX} order \textit{Meat} & 0.56\\
$\texttt{Contrast}$ & $\texttt{Condition}$ & \textit{PersonX} be \textit{Musician} & [\textit{PersonX} play \textit{Sport}] & \textit{PersonX} be tall & 0.42\\
$\texttt{Result}$ & $\texttt{Synchronous}$ & \textit{PersonX} excite & [\textit{PersonX} imagine \textit{Emotion}] & \textit{PersonX} answer \textit{Electronic-Device} & 0.58\\
$\texttt{Reason}$ & $\texttt{Contrast}$ & \textit{PersonX} eat \textit{Animal-Product} & [\textit{PersonX} enjoy it] & It be spicy & 0.25\\
$\texttt{Synchronous}$ & $\texttt{Condition}$ & \textit{PersonX} hear it & [\textit{PersonX} play \textit{Musical-Instrument}] & \textit{PersonX} be \textit{Extracurricular-Activity} & 0.16\\
$\texttt{Alternative}$ & $\texttt{Result}$ & \textit{PersonX} eat \textit{Meat} & [\textit{PersonX} be vegetarian] & \textit{PersonX} starve & 0.50\\
\bottomrule
\end{tabular}}
\caption{Cases of two-hop relation inference in the concept graph. In the table, we provide a typical example of middle nodes (embraced by brackets) to create a scenario for better understanding. The concepts are marked as $\textit{italic}$ texts.}
\label{tab:concept_two_hop_relation}
\end{table}
In Table \ref{tab:concept_two_hop_relation}, the two-hop relations among head, middle, and tail nodes are extracted to show some insights behind the relation inference.
\begin{itemize}
\item $\langle$ $\texttt{Precedence}$, $\texttt{Precedence}$, ``\textit{PersonX} wait for \textit{PersonY}'', [``\textit{PersonY} send \textit{Information}''], ``\textit{PersonZ} drag \textit{PersonX} away'' $\rangle$
\item $\langle$ $\texttt{Result}$, $\texttt{Synchronous}$, ``\textit{PersonX} excite'', [``\textit{PersonX} imagine \textit{Emotion}''], ``\textit{PersonX} answer \textit{Electronic-Device}'' $\rangle$
\item $\langle$ $\texttt{Synchronous}$, $\texttt{Condition}$, ``\textit{PersonX} hear it'', [``\textit{PersonX} play \textit{Musical-Instrument}''], ``\textit{PersonX} be \textit{Extracurricular-Activity}'' $\rangle$
\end{itemize}
In the first example, $\textit{PersonX}$ waits for $\textit{PersonY}$ before $\textit{PersonY}$ sends information (e.g., a letter) to inform $\textit{PersonX}$ not wait for him. $\textit{PersonX}$ is reluctant to leave until his/her friends drag him/her away. In the latter example, the result of a person is excited is that he/she imagines the situation and answers the phone. The third case shows that $\textit{PersonX}$ hears that $\textit{PersonY}$ plays instrument since $\textit{PersonY}$ is having extracurricular activities.
\subsection{Rule Mining}
\begin{table}[]
\centering
\small
\resizebox{\textwidth}{!}{
\begin{tabular}{l|l}
\toprule
Rule & $\langle E_b \xrightarrow{\texttt{Concession}} E_f \rangle \land \langle E_a \xrightarrow{\texttt{Result}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Contrast}} E_b \rangle$ \\
\hline
Instances & $\langle$ I do not know $\rightarrow$ I guess $\rangle \land \langle$ I believe $\rightarrow$ I guess $\rangle \Rightarrow \langle$ I believe $\rightarrow$ I do not know $\rangle$ \\
& $\langle$ I am not sure $\rightarrow$ I guess $\rangle \land \langle$ I hope so $\rightarrow$ I guess $\rangle \Rightarrow \langle$ I hope so $\rightarrow$ I am not sure $\rangle$ \\
& $\langle$ I understand $\rightarrow$ I can not speak $\rangle \land \langle$ I am not a lawyer $\rightarrow$ I can not speak $\rangle \Rightarrow \langle$ I am not a lawyer $\rightarrow$ I understand $\rangle$\\
\midrule
Rule & $\langle E_f \xrightarrow{\texttt{Contrast}} E_b \rangle \land \langle E_a \xrightarrow{\texttt{Instantiation}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Contrast}} E_b \rangle$\\
\hline
Instances & $\langle$ I remember $\rightarrow$ I could not find it $\rangle \land \langle$ I get $\rightarrow$ I remember $\rangle \Rightarrow \langle$ I get $\rightarrow$ I could not find it $\rangle$\\
& $\langle$ I would say $\rightarrow$ I might be wrong $\rangle \land \langle$ I hope $\rightarrow$ I would say $\rangle \Rightarrow \langle$ I hope $\rightarrow$ I might be wrong $\rangle$\\
& $\langle$ It have been suggested $\rightarrow$ This is unlikely $\rangle \land \langle$ It is possible $\rightarrow$ It have been suggested $\rangle \Rightarrow \langle$ It is possible $\rightarrow$ This is unlikely $\rangle$\\
\midrule
Rule & $\langle E_e \xrightarrow{\texttt{ChosenAlternative}} E_b \rangle \land \langle E_a \xrightarrow{\texttt{ChosenAlternative}} E_e \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{ChosenAlternative}} E_b \rangle$ \\
\hline
Instances & $\langle$ I will not go $\rightarrow$ You come here $\rangle \land \langle$ I want to see $\rightarrow$ I will not go $\rangle \Rightarrow \langle$ I want to see $\rightarrow$ You come here $\rangle$\\
& $\langle$ I want $\rightarrow$ It is $\rangle \land \langle$ I wish $\rightarrow$ I want $\rangle \Rightarrow \langle$ I wish $\rightarrow$ It is $\rangle$\\
& $\langle$ I want $\rightarrow$ I get $\rangle \land \langle$ I do not get that $\rightarrow$ I want $\rangle \Rightarrow \langle$ I do not get that $\rightarrow$ I get $\rangle$\\
\midrule
Rule & $\langle E_a \xrightarrow{\texttt{Reason}} E_e \rangle \land \langle E_e \xrightarrow{\texttt{Restatement}} E_b \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Reason}} E_b \rangle$ \\
\hline
Instances & $\langle$ I have ever see $\rightarrow$ I know $\rangle \land \langle$ I know $\rightarrow$ They are $\rangle \Rightarrow \langle$ I have ever see $\rightarrow$ They are $\rangle$\\
& $\langle$ I am curious $\rightarrow$ I think $\rangle \land \langle$ I think $\rightarrow$ It seems $\rangle \Rightarrow \langle$ I am curious $\rightarrow$ It seems $\rangle$\\
& $\langle$ It is not $\rightarrow$ You are lying $\rangle \land \langle$ You are lying $\rightarrow$ I do not believe you $\rangle \Rightarrow \langle$ It is not $\rightarrow$ I do not believe you $\rangle$\\
\midrule
Rule & $\langle E_a \xrightarrow{\texttt{Concession}} E_f \rangle \land \langle E_b \xrightarrow{\texttt{Reason}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Contrast}} E_b \rangle$ \\
\hline
Instances & $\langle$ I have no clue $\rightarrow$ I hope $\rangle \land \langle$ It be $\rightarrow$ I hope $\rangle \Rightarrow \langle$ I have no clue $\rightarrow$ It is $\rangle$\\
& $\langle$ I reckon $\rightarrow$ I do not know $\rangle \land \langle$ I can not talk about it $\rightarrow$ I do not know $\rangle \Rightarrow \langle$ I reckon $\rightarrow$ I can not talk about it $\rangle$\\
& $\langle$ You do not understand it $\rightarrow$ You are admitted $\rangle \land \langle$ That is $\rightarrow$ You are admitted $\rangle \Rightarrow \langle$ You do not understand it $\rightarrow$ That is $\rangle$\\
\midrule
Rule & $\langle E_b \xrightarrow{\texttt{Alternative}} E_f \rangle \land \langle E_a \xrightarrow{\texttt{Result}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Contrast}} E_b \rangle$ \\
\hline
Instances & $\langle$ I am going $\rightarrow$ I am not going $\rangle \land \langle$ I do not care $\rightarrow$ I am not going $\rangle \Rightarrow \langle$ I do not care $\rightarrow$ I am going $\rangle$\\
& $\langle$ You do $\rightarrow$ I do $\rangle \land \langle$ I suppose $\rightarrow$ I do $\rangle \Rightarrow \langle$ I suppose $\rightarrow$ You do $\rangle$\\
& $\langle$ I reckon $\rightarrow$ I guess $\rangle \land \langle$ I wonder $\rightarrow$ I guess $\rangle \Rightarrow \langle$ I wonder $\rightarrow$ I reckon $\rangle$\\
\midrule
Rule & $\langle E_a \xrightarrow{\texttt{Reason}} E_f \rangle \land \langle E_b \xrightarrow{\texttt{Succession}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Reason}} E_b \rangle$ \\
\hline
Instances & $\langle$ I ask $\rightarrow$ I am not sure $\rangle \land \langle $ I do not know $\rightarrow$ I am not sure $\rangle \Rightarrow \langle$ I ask $\rightarrow$ I do not know $\rangle$\\
& $\langle$ We are lucky $\rightarrow$ We notice $\rangle \land \langle$ We order $\rightarrow$ We notice $\rangle \Rightarrow \langle$ We are lucky $\rightarrow$ We order $\rangle$\\
& $\langle$ I remember it $\rightarrow$ I see it $\rangle \land \langle$ I realize $\rightarrow$ I see it $\rangle \Rightarrow \langle$ I remember it $\rightarrow$ I realize $\rangle$\\
\midrule
Rule & $\langle E_a \xrightarrow{\texttt{Concession}} E_f \rangle \land \langle E_b \xrightarrow{\texttt{Precedence}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Contrast}} E_b \rangle$ \\
\hline
Instances & $\langle$ I am unconscious $\rightarrow$ I wake up $\rangle \land \langle $ I see $\rightarrow$ I wake up $\rangle \Rightarrow \langle$ I am unconscious $\rightarrow$ I see $\rangle$\\
& $\langle$ I swear $\rightarrow$ I guess $\rangle \land \langle$ I do not know $\rightarrow$ I guess $\rangle \Rightarrow \langle$ I swear $\rightarrow$ I do not know $\rangle$\\
& $\langle$ I can not believe $\rightarrow$ It is great $\rangle \land \langle$ I think $\rightarrow$ It is great $\rangle \Rightarrow \langle$ I can not believe $\rightarrow$ I think $\rangle$\\
\midrule
Rule & $\langle E_a \xrightarrow{\texttt{Alternative}} E_e \rangle \land \langle E_e \xrightarrow{\texttt{Exception}} E_b \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Exception}} E_b \rangle$ \\
\hline
Instances & $\langle$ It is not $\rightarrow$ It is wrong $\rangle \land \langle$ It is wrong $\rightarrow$ It is $\rangle \Rightarrow \langle$ It is not $\rightarrow$ It is $\rangle$\\
& $\langle$ I really want $\rightarrow$ I think $\rangle \land \langle$ I think $\rightarrow$ I know $\rangle \Rightarrow \langle$ I really want $\rightarrow$ I know $\rangle$\\
& $\langle$ It is not $\rightarrow$ I suppose $\rangle \land \langle$ I suppose $\rightarrow$ You know $\rangle \Rightarrow \langle$ It is not $\rightarrow$ You know $\rangle$\\
\midrule
Rule & $\langle E_a \xrightarrow{\texttt{ChosenAlternative}} E_f \rangle \land E_b \langle \xrightarrow{\texttt{ChosenAlternative}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Restatement}} E_b \rangle$ \\
\hline
Instances & $\langle$ I am hoping $\rightarrow$ We get $\rangle \land \langle$ I think $\rightarrow$ We get $\rangle \Rightarrow \langle$ I am hoping $\rightarrow$ I think $\rangle$\\
& $\langle$ I suppose $\rightarrow$ He is $\rangle \land \langle$ I think $\rightarrow$ He is $\rangle \Rightarrow \langle$ I suppose $\rightarrow$ I think $\rangle$\\
& $\langle$ I am glad $\rightarrow$ I think $\rangle \land \langle$ The food is good $\rightarrow$ I think $\rangle \Rightarrow \langle$ I am glad $\rightarrow$ The food is good $\rangle$\\
\bottomrule
\end{tabular}}
\caption{Cases of AMIE+ rule mining in the eventuality graph. For the simplicity of formatting, we represent $\langle E_h, T, E_t \rangle$ triples as $\langle E_h \xrightarrow{\texttt{T}} E_t \rangle$.}
\label{tab:amie_aser}
\end{table}
AMIE+ \cite{DBLP:journals/vldb/GalarragaTHS15} aims at mining close and connected Horn Rules in the form of
$$\langle E_a, T_1, E_b \rangle \land \langle E_b, T_2, E_c \rangle \Rightarrow \langle E_a, T_3, E_b \rangle$$
where $E_a, E_b, E_c$ and $T_1, T_2, T_3$ are eventuality variables and relation variables, respectively. As demonstrated through experiments on different KBs, AMIE+ provides an effective approach for the investigation and inference over KB from a logic-rule perspective. We therefore apply AMIE+ on ASER to probe whether it preserves logical properties among multi-hop eventualities and relations. To make the notation consistent with AMIE+, we denote a fact triple with variables at head and/or tail eventuality positions, such as $\langle E_a, T_1, E_b \rangle$, as an \textit{atom}. A rule of interests
comprise of a body of a set of atoms $B_1, B_2, \cdots, B_n$ (n = 2 in this case), and a head of a single atom $\langle E_h, T, E_t \rangle$. We abbreviate the rule as $\Vec{B} \, \Rightarrow \, \langle E_h, T, E_t \rangle$
The Algorithm of AMIE+ adopts an iterative procedure: starting with a queue of all possible items (rule of size 1), it dequeues a rule in each iteration, and outputs/grows/prunes the rule with certain criterion (listed below), and enqueues the grown new rules back. Throughout the iteration, AMIE+ sets the following significance criterion:
\textit{Head coverage}:
$$hc(\Vec{B} \, \Rightarrow \, \langle E_h, T, E_t \rangle) := \frac{supp(\Vec{B} \, \Rightarrow \, \langle E_h, T, E_t \rangle)}{size(T)},$$
where $$supp(\Vec{B} \, \Rightarrow \, \langle E_h, T, E_t \rangle) := \#(E_h, E_t): \exists \, z_1, \cdots, z_m: \Vec{B} \land \langle E_h, T, E_t \rangle: $$ denotes the support, i.e., the number of correct prediction yielded with the rule in the current KB, and $size(T)$ denotes the number of facts with $T$ as relations.
\textit{Standard confidence}:
$$conf(\Vec{B} \, \Rightarrow \, \langle E_h, T, E_t \rangle) := \frac{supp(\Vec{B} \, \Rightarrow \, \langle E_h, T, E_t \rangle)}{\#(E_h,E_t): \exists \, z_1, \cdots, z_m : \Vec{B}},$$
where $\#(E_h,E_t): \exists z_1, \cdots, z_m : \Vec{B}$ denotes all possible predictions of the rule.
\textit{PCA confidence}:
$$conf_{pca}(\Vec{B} \, \Rightarrow \, \langle E_h, T, E_t \rangle) := \frac{supp(\Vec{B} \, \Rightarrow \, \langle E_h, T, E_t \rangle)}{\#(E_h,E_t): \exists \, z_1, \cdots, z_m : \Vec{B} \, \land \langle E_h, T, E^{\prime}_t \rangle},$$
where $\#(E_h,E_t): \exists \, z_1, \cdots, z_m : \Vec{B} \, \land \langle E_h, T, E^{\prime}_t \rangle$ denotes the number of pairs of $(E_h, E_t)$ predicted by corresponding relation body $\Vec{B}$ but with an existing pair of $\langle E_h, T, E^{\prime}_t \rangle$ in KB.
AMIE+ focuses on RDF Knowledge Bases, where an RDF KB could be represented as a set of facts in the form $\langle {Subject}, \texttt{Relation}, {Object} \rangle$. To pair ASER Graph with AMIE+, we extract all $\langle E_1, \texttt{Relation}, E_2 \rangle$ triples from the relation table as our set of facts. To preserve the frequency information, we duplicate each extracted fact for $f$ times where $f$ is the corresponding triple frequency in the relation table. During our experiments, we set the threshold of minimal PCA confidence \textit{minPCA}=0.1 and minimal head coverage \textit{minHC}=0.01, and run AMIE+ on both the eventuality graph and concept graph of ASER.
Some of the mined rules and instantiated cases are shown in Table \ref{tab:amie_aser} and \ref{tab:amie_concept}, respectively.
For the eventuality graph, the rule ``$\langle E_b \xrightarrow{\texttt{Concession}} E_f \rangle \land \langle E_a \xrightarrow{\texttt{Result}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Contrast}} E_b \rangle$'' demonstrates that if the result of some event is opposite to what should happen, then the reason that induces this ``opposite'' result is likely to contrast the original event in some core properties. The last instance ``$\langle$ I understand $\rightarrow$ I can not speak $\rangle \land \langle$ I am not a lawyer $\rightarrow$ I can not speak $\rangle \Rightarrow \langle$ I am not a lawyer $\rightarrow$ I understand $\rangle$'' illustrates this rule with a lawsuit scenario where the ``opposite'' result "I can not speak" is induced by the reason ``I am not a lawyer'', which contrasts the original event ``I understand''.
And the rule ``$\langle E_a \xrightarrow{\texttt{ChosenAlternative}} E_f \rangle \land E_b \langle \xrightarrow{\texttt{ChosenAlternative}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Restatement}} E_b \rangle$'' captures that if the subject consistently prefers two events as alternative to a third event, then it is possible that the first two events have very similar semantic meaning. This is illustrated with its last instance ``$\langle$ I am glad $\rightarrow$ I think $\rangle \land \langle$ The food is good $\rightarrow$ I think $\rangle \Rightarrow \langle$ I am glad $\rightarrow$ The food is good $\rangle$'' where both ``I am glad'' and ``The food is good'' express similar meanings in which the subject consistently prefers to ``I think''.
For the concept graph, the rule ``$\langle E_e \xrightarrow{\texttt{Instantiation}} E_a \rangle \land \langle E_e \xrightarrow{\texttt{Instantiation}} E_b \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Conjunction}} E_b \rangle$'' shows that the concepts that both describe a shared concept in details are likely to both happen. Its first instance ``$\langle$ \textit{PersonX} realize $\rightarrow$ \textit{PersonX} point out $\rangle \land \langle$ \textit{PersonX} realize $\rightarrow$ \textit{PersonX} have \textit{Information} $\rangle \Rightarrow \langle$ \textit{PersonX} point out $\rightarrow$ \textit{PersonX} have \textit{Information}'' demonstrates this through a information-capture process, where the two concepts ``\textit{PersonX} point out'' and ``\textit{PersonX} have \textit{Information} $\rangle$'' that both describe the concept ``\textit{PersonX} realize'' happen together.
The rule ``$\langle E_e \xrightarrow{\texttt{Exception}} E_b \rangle \land \langle E_e \xrightarrow{\texttt{Succession}} E_a \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Contrast}} E_b \rangle$'' shows the association between $\texttt{Exception}$ and $\texttt{Contrast}$ that is bridged via $\texttt{Succession}$. This is demonstrated with its first instance ``$\langle$ \textit{Item} be ready $\rightarrow$ \textit{PersonX} wait $\rangle \land \langle$ \textit{Item} be ready $\rightarrow$ \textit{PersonX} check $\rangle \Rightarrow \langle$ \textit{PersonX} check $\rightarrow$ \textit{PersonX} be wait $\rangle$'' that shows a scenario of a customer checking and waiting for products.
\begin{table}[]
\centering
\small
\resizebox{\textwidth}{!}{
\begin{tabular}{l|l}
\toprule
Rule & $\langle E_e \xrightarrow{\texttt{Restatement}} E_a \rangle \land \langle E_e \xrightarrow{\texttt{Restatement}} E_b \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Conjunction}} E_b \rangle$ \\
\hline
Instances & $\langle$ \textit{PersonX} laugh $\rightarrow$ \textit{PersonX} smile $\rangle \land \langle$ \textit{PersonX} laugh $\rightarrow$ \textit{PersonX} open \textit{Facial-Feature} $\rangle \Rightarrow \langle$ \textit{PersonX} smile $\rightarrow$ \textit{PersonX} open \textit{Facial-Feature} $\rangle$\\
& $\langle$ \textit{PersonX} love it $\rightarrow$ It be good $\rangle \land \langle$ \textit{PersonX} love it $\rightarrow$ It be tasty $\rangle \Rightarrow \langle$ It be good $\rightarrow$ It be tasty $\rangle$\\
& $\langle$ \textit{PersonX} wish $\rightarrow$ \textit{PersonX} need $\rangle \land \langle$ \textit{PersonX} wish $\rightarrow$ \textit{PersonX} need $\rangle \Rightarrow \langle$ \textit{PersonX} need $\rightarrow$ \textit{PersonX} need $\rangle$\\
\midrule
Rule & $\langle E_e \xrightarrow{\texttt{Instantiation}} E_a \rangle \land \langle E_e \xrightarrow{\texttt{Instantiation}} E_b \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Conjunction}} E_b \rangle$ \\
\hline
Instances & $\langle$ \textit{PersonX} realize $\rightarrow$ \textit{PersonX} point out $\rangle \land \langle$ \textit{PersonX} realize $\rightarrow$ PersonX have \textit{Information} $\rangle \Rightarrow \langle$ \textit{PersonX} point out $\rightarrow$ \textit{PersonX} have \textit{Information} $\rangle$\\
& $\langle$ \textit{PersonX} have $\rightarrow$ \textit{PersonX} get $\rangle \land \langle$ \textit{PersonX} have $\rightarrow$ \textit{PersonX} own $\rangle \Rightarrow \langle$ \textit{PersonX} get $\rightarrow$ \textit{PersonX} own $\rangle$\\
& $\langle$ \textit{PersonX} know $\rightarrow$ \textit{PersonX} be sure $\rangle \land \langle$ \textit{PersonX} know $\rightarrow$ \textit{PersonX} remember $\rangle \Rightarrow \langle$ \textit{PersonX} be sure $\rightarrow$ \textit{PersonX} remember $\rangle$\\
\midrule
Rule & $\langle E_e \xrightarrow{\texttt{Concession}} E_b \rangle \land \langle E_e \xrightarrow{\texttt{Restatement}} E_a \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Contrast}} E_b \rangle$ \\
\hline
Instances & $\langle$ \textit{PersonX} order \textit{Dish} $\rightarrow$ \textit{PersonX} be hungry $\rangle \land \langle$ \textit{PersonX} order \textit{Dish} $\rightarrow$ \textit{PersonX} order $\rangle \Rightarrow \langle$ \textit{PersonX} order $\rightarrow$ \textit{PersonX} be hungry $\rangle$\\
& $\langle$ \textit{PersonX} wish $\rightarrow$ \textit{PersonX} doubt $\rangle \land \langle$ \textit{PersonX} wish $\rightarrow$ \textit{PersonX} need $\rangle \Rightarrow \langle$ \textit{PersonX} doubt $\rightarrow$ \textit{PersonX} need $\rangle$\\
& $\langle$ \textit{PersonX} love it $\rightarrow$ \textit{PersonX} hate it $\rangle \land \langle$ \textit{PersonX} love it $\rightarrow$ It be good $\rangle \Rightarrow \langle$ \textit{PersonX} hate it $\rightarrow$ It be good $\rangle$\\
\midrule
Rule & $\langle E_e \xrightarrow{\texttt{Exception}} E_b \rangle \land \langle E_e \xrightarrow{\texttt{Succession}} E_a \rangle \Rightarrow E_a \langle \xrightarrow{\texttt{Contrast}} E_b \rangle$ \\
\hline
Instances & $\langle$ \textit{Item} be ready $\rightarrow$ \textit{PersonX} wait $\rangle \land \langle$ \textit{Item} be ready $\rightarrow$ \textit{PersonX} check $\rangle \Rightarrow \langle$ \textit{PersonX} check $\rightarrow$ \textit{PersonX} be wait $\rangle$\\
& $\langle$ \textit{PersonX} say $\rightarrow$ \textit{PersonX} be sorry $\rangle \land \langle$ \textit{PersonX} say $\rightarrow$ \textit{PersonX} be surprised $\rangle \Rightarrow \langle$ \textit{PersonX} be sorry $\rightarrow$ \textit{PersonX} be surprised $\rangle$\\
& $\langle$ It be $\rightarrow$ \textit{PersonX} guess $\land$ It be $\rightarrow$ It be \textit{factor} $\rangle \Rightarrow \langle$ \textit{PersonX} guess $\rightarrow$ It be \textit{factor} $\rangle$\\
\midrule
Rule & $\langle E_a \xrightarrow{\texttt{Restatement}} E_f \rangle \land \langle E_b \xrightarrow{\texttt{Restatement}} E_f \rangle \Rightarrow \langle E_a \xrightarrow{\texttt{Synchronous}} E_b \rangle$ \\
\hline
Instances & $\langle$ \textit{PersonX} love it $\rightarrow$ It be good $\rangle \land \langle$ \textit{PersonX} feel $\rightarrow$ It be good $\rangle \Rightarrow \langle$ \textit{PersonX} love it $\rightarrow$ \textit{PersonX} feel $\rangle$\\
& $\langle$ It be cool $\rightarrow$ It be good $\rangle \land \langle$ \textit{PersonX} think $\rightarrow$ It be okay $\rangle \Rightarrow \langle$ It be cool $\rightarrow$ It be okay $\rangle$\\
& $\langle$ \textit{PersonX} like it $\rightarrow$ It be good $\rangle \land \langle$ \textit{PersonX} be happy $\rightarrow$ It be good $\rangle \Rightarrow \langle$ \textit{PersonX} like it $\rightarrow$ \textit{PersonX} be happy $\rangle$\\
\bottomrule
\end{tabular}}
\caption{Cases of AMIE+ rule mining in the concept graph. For the simplicity of formatting, we represent $\langle E_h, T, E_t \rangle$ triples as $\langle E_h \xrightarrow{\texttt{T}} E_t \rangle$.}
\label{tab:amie_concept}
\end{table}
\subsection{Meta-path Mining}
ASER is a complex heterogeneous graph that encodes the commonsense knowledge. ASER is composed of two types of nodes (i.e., extracted eventuality and conceptualized eventuality) and 15 types of edges (e.g., $\texttt{Reason}$ and $\texttt{Precedence}$). We leverage meta-path \cite{sun2012metapath} mining which studies the semantic meanings behind paths to tackle the heterogeneity of ASER. A meta-path is a path that consists of a sequence of different relations defined among various node types. Formally, a meta-path $P$ is defined as a path $ E_1 \xrightarrow{T_1} E_2 \xrightarrow{T_2} \cdots \xrightarrow{T_{l-1}} E_l$, in which $T=T_1 \circ T_2 \circ \cdots \circ T_l$ is the composite relation between $N_1$ and $N_l$. Take an example from Table \ref{tab:metapath}, the meta-path ``$E_1 \xrightarrow{\texttt{Conceptualization}} C_1 \xrightarrow{\texttt{ ConceptInstantiation}} E_2$'' defines a composite relation in which the two eventuality $E_1$ and $E_2$ are conceptualized to the same concept $C_1$.
To automatically select the most frequent and influential meta-paths, we first perform a random walk on the hybrid graph. Specifically, 50,000 seed nodes are chosen independently and uniformly from the nodes of ASER. Starting from each seed node, a random walk is used to generate 50 multi-hop paths of different nodes and relations. The nodes in a path are represented by their types rather than their contents. For example, a path ``I drink coffee $\xrightarrow{\texttt{Result}}$ I stay up late'' is converted into a meta-path ``$E_1 \xrightarrow{\texttt{Result}} E_2$''. After collecting meta-paths, we search for the frequent patterns of 2-hop and 3-hop meta-paths. The appearance of metapaths is counted. The 2-hop and 3-hop metapaths are later ranked by their frequencies. The frequent metapaths are selected for the further case study. We list the intriguing instances from these meta-paths in Table \ref{tab:metapath}.
For 2-hop metapaths, the results are very similar to the ones of eventuality/relation retrieval inference. For example, ``$E_1 \xrightarrow{\texttt{Reason}} E_2 \xrightarrow{\texttt{Result}} E_3$'' describes paths following cause and effect relations. A typical instance in daily life is ``I am in pain$\rightarrow$I am alone$\rightarrow$I sit at bar,'' describing a scenario in which a man suffers from loneliness and goes to the bar to numb the pains. In addition to relations among eventualities, the interaction between concepts and eventualities are also discovered by the meta-paths. In the cases of ``$E_1 \xrightarrow{\texttt{Conceptualization}} C_1 \xrightarrow{\texttt{ConceptInstantiation}} E_2$,'' two semantically distinct eventualities are unified in the concept-level. For example, ``He is psychiatrist'' and ``I am attorney'' follows the same pattern, ``$\textit{PersonX}$ be specialist.''
For 3-hop metapaths, the reasoning paths are longer and illustrates the daily life in more details. For example, in the meta-path ``$E_1 \xrightarrow{\texttt{Result}} E_2 \xrightarrow{\texttt{Contrast}} E_3 \xrightarrow{\texttt{Conjunction}} E_4$,'' an instantiated example, ``I have you number$\rightarrow$I call you$\rightarrow$I have a meeting$\rightarrow$I have a presentation,'' shows that a person wants to call his friends with the phone number. However, he has to do a presentation in the coming meeting and decides to call his friend later. As for the hybrid meta-path with extracted and conceptualized eventualities, ``$E_1 \xrightarrow{\texttt{Conjunction}} E_2 \xrightarrow{\texttt{Conceptualization}} C_1 \xrightarrow{\texttt{ConceptInstantiation}} E_3$,'' we find out that someone is sweating because of the hot weather while someone is unfortunately in a coma. Both of them are unified under the concept ``$\textit{PersonX}$ be $\textit{Symptom}$.''
\begin{table}[]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{c|l|l}
\toprule
\#Hop & meta-path & Instances\\
\midrule
\multirow{15}*{2} & \multirow{3}*{$E_1$ $\xrightarrow{\texttt {Conjunction}}$ $E_2$ $\xrightarrow{\texttt {Contrast}}$ $E_3$} & I go to bed $\rightarrow$ I go to sleep $\rightarrow$ I wake up\\
~ & ~ & I have breakfast $\rightarrow$ I have milk $\rightarrow$ I feel sick \\
~ & ~ & I take bus $\rightarrow$ I go to work $\rightarrow$ I go home \\
\cline{2-3}
~ & \multirow{3}*{$E_1$ $\xrightarrow{\texttt{Precedence}}$ $E_2$ $\xrightarrow{\texttt{Precedence}}$ $E_3$} & You go to sleep $\rightarrow$ You wake up $\rightarrow$ You hit the ground \\
~ & ~ & You drink alcohol $\rightarrow$ You go to toilet $\rightarrow$ You have to pee \\
~ & ~ & You go to restaurant $\rightarrow$ You are sick $\rightarrow$ You go to hospital \\
\cline{2-3}
~ & \multirow{3}*{$E_1$ $\xrightarrow{\texttt {Conceptualization}}$ $C_1$ $\xrightarrow{\texttt{ ConceptInstantiation}}$ $E_2$} & He is psychiatrist $\rightarrow$ \textit{PersonX} is \textit{Specialist} $\rightarrow$ I am attorney \\
~ & ~ & I want milk $\rightarrow$ \textit{PersonX} want \textit{Animal-Product} $\rightarrow$ He wants burgers \\
~ & ~ & You make reservation $\rightarrow$ \textit{PersonX} make \textit{Service} $\rightarrow$ He makes statement \\
\cline{2-3}
~ & \multirow{3}*{$E_1$ $\xrightarrow{\texttt{Conjunction}}$ $E_2$ $\xrightarrow{\texttt{Conjunction}}$ $E_3$} & I go to gym $\rightarrow$ I have to wait $\rightarrow$ I go home\\
~ & ~ & I am vegan $\rightarrow$ My wife is vegan $\rightarrow$ I used to eat meat \\
~ & ~ & It is a cat $\rightarrow$ It is fine $\rightarrow$ It is beautiful \\
\cline{2-3}
~ & \multirow{3}*{$E_1$ $\xrightarrow{\texttt{Reason}}$ $E_2$ $\xrightarrow{\texttt{Result}}$ $E_3$} & I go to bar $\rightarrow$ I have many friends $\rightarrow$ I have parties \\
~ & ~ & I go to school $\rightarrow$ We could afford $\rightarrow$ I get my first job\\
~ & ~ & I am in pain $\rightarrow$ I am alone $\rightarrow$ I sit at bar \\
\hline
\multirow{15}*{3} & \multirow{3}*{$E_1$ $\xrightarrow{\texttt{Precedence}}$ $E_2$ $\xrightarrow{\texttt{Conjunction}}$ $E_3$ $\xrightarrow{\texttt{Precedence}}$ $E_4$} & The rain comes down $\rightarrow$ The engine whistles $\rightarrow$ The train starts $\rightarrow$ The train moves on\\
~ & ~ & The moon arises $\rightarrow$ The weather is pleasant $\rightarrow$ The snow ceases $\rightarrow$ The night is still\\
~ & ~ & She sleeps $\rightarrow$ The phone rings $\rightarrow$ We gets home $\rightarrow$ She hangs up the phone\\
\cline{2-3}
~ & \multirow{3}*{$E_1$ $\xrightarrow{\texttt{Conjunction}}$ $E_2$ $\xrightarrow{\texttt{Conceptualization}}$ $C_1$ $\xrightarrow{\texttt{ConceptInstantiation}}$ $E_3$} & I play piano $\rightarrow$ I am musician $\rightarrow$ \textit{PersonX} be \textit{Artist} $\rightarrow$ He is actor\\
~ & ~ & I am chill $\rightarrow$ It is a snake $\rightarrow$ It be \textit{Predator} $\rightarrow$ It is a bear\\
~ & ~ & It is hot $\rightarrow$ I am sweating $\rightarrow$ \textit{PersonX} be \textit{Symptom} $\rightarrow$ She is in a coma\\
\cline{2-3}
~ & \multirow{3}*{$E_1$ $\xrightarrow{\texttt{Condition}}$ $E_2$ $\xrightarrow{\texttt{Reason}}$ $E_3$ $\xrightarrow{\texttt{Conjunction}}$ $E_4$
} & Everyone knows him $\rightarrow$ He comes off the bench $\rightarrow$ He makes his debut for club $\rightarrow$ He scores his first goal\\
~ & ~ & I am healthy $\rightarrow$ I sleep $\rightarrow$ I am exhausted $\rightarrow$ I am cold\\
~ & ~ & We get the check $\rightarrow$ We order dessert $\rightarrow$ I am still hungry $\rightarrow$ We eat everything\\
\cline{2-3}
~ & \multirow{3}*{$E_1$ $\xrightarrow{\texttt{Result}}$ $E_2$ $\xrightarrow{\texttt {Contrast}}$ $E_3$ $\xrightarrow{\texttt{Conjunction}}$ $E_4$
} & I am tired $\rightarrow$ I go to bed $\rightarrow$ The sun is shining $\rightarrow$ The wind blows\\
~ & ~ & There is a storm coming $\rightarrow$ The rain falls $\rightarrow$ The sky is clear $\rightarrow$ The air is warm\\
~ & ~ & I have you number $\rightarrow$ I call you $\rightarrow$ I have a meeting $\rightarrow$ I have a presentation\\
\cline{2-3}
~ & \multirow{3}*{$E_1$ $\xrightarrow{\texttt{Contrast}}$ $E_2$ $\xrightarrow{\texttt{Reason}}$ $E_3$ $\xrightarrow{\texttt{Reason}}$ $E_4$
} & I am a vegan $\rightarrow$ I eat meat $\rightarrow$ I enjoy it $\rightarrow$ It tastes good\\
~ & ~ & The painting is controversial $\rightarrow$ It is a masterpiece $\rightarrow$ It belongs to museum $\rightarrow$ It is valuable\\
~ & ~ & I get over it quickly $\rightarrow$ I go to mall $\rightarrow$ I buy clothes $\rightarrow$ I have a job interview\\
\bottomrule
\end{tabular}}
\caption{Instances of metapaths generated by random walk. $E$ represents eventuality while $C$ represents concept. The concepts in the instances are marked as $\textit{italic}$ texts. For example, ``I go to bed $\rightarrow$ I go to sleep $\rightarrow$ I sleep'' is an instance of the meta-path ``$E_1$ $\xrightarrow{\texttt{Conjunction}}$ $E_2$ $\xrightarrow{\texttt{Contrast}}$ $E_3$.''}
\label{tab:metapath}
\end{table}
\section{From Selectional Preference to Commonsense Knowledge}\label{sec:transferability}
In this section, we investigate the connection between ASER and existing commonsense knowledge bases. Specifically, we check the coverage and similarities between the selectional preference knowledge in ASER and the human-defined commonsense knowledge in ConceptNet~\cite{liu2004conceptnet} and ATOMIC~\cite{Maarten2019Atomic}.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{image/transomcs_match_stat.pdf}
\caption{
The matching statistics of ConceptNet assertions in ASER grouped by each relation. The coverage indicates the proportion of ConceptNet assertions where both heads and tails can be matched to an ASER unit, i.e., a discourse edge or an eventuality.
}
\label{fig:TransOMCS-stat}
\end{figure}
\begin{table}[t]
\begin{minipage}[t]{0.45\textwidth}
\centering
\footnotesize
\begin{tabular}{l|l}
\toprule
Relation & Dependency Pattern \\
\midrule
AtLocation & \verb|()->compound->()| \\
CapableOf & \verb|()<-nsubj<-()| \\
Causes & \verb|(-compound-)<-pobj<-of<-prep<-()| \\
CausesDesire & \verb|()<-pobj<-to<-prep<-()| \\
CreatedBy & \verb|()<-dobj<-make->nsubj->()| \\
DefinedAs & \verb|()<-nsubj<-be->attr->(-amod-)| \\
Desires & \verb|()<-nsubj<-()| \\
HasA & \verb|()<-nsubj<-have->dobj->()| \\
HasPrerequisite & \verb|()->dobj->()| \\
HasProperty & \verb|()<-nsubj<-be->acomp->()| \\
\bottomrule
\end{tabular}
\end{minipage}
\hspace{0.05\textwidth}
\begin{minipage}[t]{0.45\textwidth}
\centering
\footnotesize
\begin{tabular}{l|l}
\toprule
Relation & Dependency Pattern \\
\midrule
HasSubevent & \verb|(-dobj-)->neg->()| \\
HasFirstSubevent & \verb|(-prep-)<-Succession<-()| \\
HasLastSubevent & \verb|()<-acomp<-be<-Reason<-()| \\
InstanceOf & \verb|()<-acomp<-be->nsubj->()| \\
LocatedNear & \verb|()<-nsubj<-be->prep->on->pobj->()| \\
MadeOf & \verb|()<-compound<-()| \\
MotivatedByGoal & \verb|()<-xcomp<-()| \\
PartOf & \verb|()->compound->()| \\
ReceivesAction & \verb|()<-dobj<-()| \\
UsedFor & \verb|()<-pobj<-(-prep-)| \\
\bottomrule
\end{tabular}
\end{minipage}
\makeatletter\def\@captype{table}\makeatother\caption{Examples of extracted dependency patterns in TransOMCS. We select the pattern ranked as most plausible for each relation. \texttt{()} are placeholders for words, and attributes like \texttt{nsubj} are names of the dependency edges.} \label{tab:OMCS-pattern}
\end{table}
\begin{table}[t]
\begin{minipage}[t]{0.45\textwidth}
\centering
\footnotesize
\begin{tabular}{l|l|l}
\toprule
Head& Relation & Tail \\
\midrule
student & \texttt{AtLocation} & school \\
curator & \texttt{AtLocation} & museum \\
leader & \texttt{AtLocation} & group \\
glue & \texttt{CapableOf} & dry \\
anyone & \texttt{CapableOf} & think \\
door & \texttt{CapableOf} & open \\
love & \texttt{Causes} & be friendly \\
attract & \texttt{Causes} & be vulgar \\
want & \texttt{Causes} & be closer \\
music & \texttt{CausesDesire} & listen \\
friend & \texttt{CausesDesire} & talk \\
choice & \texttt{CausesDesire} & entitle \\
art & \texttt{CreatedBy} & artist \\
playoff & \texttt{CreatedBy} & team \\
money & \texttt{CreatedBy} & bank \\
earth & \texttt{DefinedAs} & world \\
god & \texttt{DefinedAs} & truth \\
door & \texttt{DefinedAs} & entrance \\
idea & \texttt{Desires} & come \\
word & \texttt{HasA} & meaning \\
house & \texttt{HasA} & wall \\
bathroom & \texttt{HasA} & sink \\
save & \texttt{HasPrerequisite} & do part \\
enter & \texttt{HasPrerequisite} & ask i \\
\bottomrule
\end{tabular}
\end{minipage}
\hspace{0.05\textwidth}
\begin{minipage}[t]{0.45\textwidth}
\centering
\footnotesize
\begin{tabular}{l|l|l}
\toprule
Head& Relation & Tail \\
\midrule
talk & \texttt{HasProperty} & cheap \\
future & \texttt{HasProperty} & uncertain \\
be sure & \texttt{HasSubevent} & ask \\
be hungry & \texttt{HasSubevent} & eat \\
intrude into & \texttt{HasFirstSubevent} & shoot \\
go at & \texttt{HasFirstSubevent} & work \\
closer & \texttt{HasLastSubevent} & go \\
world & \texttt{MadeOf} & country \\
whole & \texttt{MadeOf} & part \\
run & \texttt{MotivatedByGoal} & afraid \\
eat & \texttt{MotivatedByGoal} & hungry \\
sleep & \texttt{MotivatedByGoal} & tired \\
wall & \texttt{PartOf} & house \\
child & \texttt{PartOf} & family \\
bone & \texttt{PartOf} & fish \\
crime & \texttt{ReceivesAction} & commit \\
game & \texttt{ReceivesAction} & play \\
video & \texttt{ReceivesAction} & watch \\
table & \texttt{UsedFor} & sit at \\
radio & \texttt{UsedFor} & listen to \\
pool & \texttt{UsedFor} & swim in \\
nose & \texttt{LocatedNear} & eye \\
heat & \texttt{LocatedNear} & fire \\
beaver & \texttt{LocatedNear} & dam \\
\bottomrule
\end{tabular}
\end{minipage}
\makeatletter\def\@captype{table}\makeatother\caption{TransOMCS generated ConceptNet-like commonsense knowledge tuples. } \label{tab:OMCS-case}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{image/ASER-ATOMIC-update.pdf}
\caption{An illustration about exploring novel inferential commonsense knowledge about events. The center of the figure is a real subgraph of ASER. The grey ovals across ASER nodes are the relations that can be transferred to plausible \textit{if-then} commonsense relations. For example, the $\langle$``I eat'', \texttt{Succession}, ``I eat''$\rangle$ tuple in ASER can be intuitively written as the ATOMIC format, $\langle$``\textit{PersonX} eats'', \texttt{Effects on X}, ``be full''$\rangle$.}
\label{fig:aser-atomic-sketch}
\end{figure}
\begin{table}[t]
\centering
\small
\begin{tabular}{c|c|p{10cm}}
\toprule
\multicolumn{2}{c|}{} & Mapping rules \\
\midrule
\multicolumn{2}{c|}{Head} & Replace \textit{PersonX} and \textit{\textit{PersonY}} with concrete singular personal pronouns, i.e., I/he/she/man/women/person \\
\midrule
\multirow{4}{0.7cm}{Tail} &\tabincell{c}{\texttt{xWant}/\texttt{oWant}/\\\texttt{xIntent}/\texttt{xNeed}} & Add a personal pronoun in front of the tail and remove the initial ``to'' \\
&\texttt{xEffect}/\texttt{oEffect}&Add a personal pronoun in front of the tail \\
&\texttt{xReact}/\texttt{oReact}&Add a personal pronoun and ``be'' in front of the tail \\
&\texttt{xAttr}& Add a personal pronoun and ``be'' in front of the tail \\
\bottomrule
\end{tabular}
\caption{Mapping rules from ATOMIC to ASER.}
\label{table:atomic-preprocess}
\end{table}
\begin{table}[t]
\small
\renewcommand\arraystretch{1.0}
\centering
\begin{tabular}{l|c|c|c}
\toprule
Relation & Nodes & Edges & Avg. Shortest Path Length \\
\midrule
oEffect&31.1\% & 25.36\% & 2.41 \\
oReact&87.3\% & 51.53\% & 2.22 \\
oWant&61.6\% & 36.95\% & 2.47 \\
xAttr&95.8\% & 53.67\% & 2.38 \\
xEffect&33.1\% & 21.81\% & 2.51 \\
xIntent&33.8\% & 21.06\% & 2.56 \\
xNeed&52.9\% & 24.91\% & 2.67 \\
xReact&88.7\% & 52.66\% & 2.25 \\
xWant&58.8\% & 30.60\% & 2.59 \\
\midrule
Average & 62.9\% & 35.91\% & 2.44 \\
\bottomrule
\end{tabular}
\caption{Mapping statistics of ATOMIC nodes and edges in ASER. The \textit{Nodes} and \textit{Edges} columns denote the percentage of ATOMIC nodes or edges that can be found in ASER. The \textit{Avg. Shortest Path Length} column presents the average shortest path length of the matched ATOMIC edges in ASER.}\label{table:mapping_stat}
\end{table}
\begin{table}[t]
\renewcommand\arraystretch{1.0}
\centering
\small
\begin{tabular}{l|l|c|c}
\toprule
Head& Tail& ATOMIC-Rel& ASER-Rel\\
\midrule
\rowcolor{Gray}
\textit{PersonX} bites \textit{PersonX}'s tongue& \textit{PersonX} cries& \texttt{xWant}& \texttt{Precedence}\\
\rowcolor{Gray}
\textit{PersonX} feels hungry& \textit{PersonX} eats& \texttt{xWant}& \texttt{Conjunction}\\
\rowcolor{Gray}
\textit{PersonX} opens the envelope & \textit{PersonX} read the letter& \texttt{xWant}& \texttt{Co\_Occurance}\\
\rowcolor{Gray}
\textit{PersonX} pays \textit{PersonX}'s bill& \textit{PersonX} leaves the restaurant& \texttt{xWant}& \texttt{Co\_Occurance}\\
\textit{PersonX} bleeds profusely& \textit{PersonX} passes out& \texttt{xEffect}& \texttt{Co\_Occurance}\\
\textit{PersonX} goes to party& \textit{PersonX} gets drunk& \texttt{xEffect}& \texttt{Conjunction}\\
\textit{PersonX} plays well& \textit{PersonX} wins& \texttt{xEffect}& \texttt{Co\_Occurance}\\
\textit{PersonX} wins the lottery& \textit{PersonX} becomes rich& \texttt{xEffect}& \texttt{Co\_Occurance}\\
\rowcolor{Gray}
\textit{PersonX} would better go& \textit{PersonX} is busy& \texttt{xAttr}& \texttt{Condition}\\
\rowcolor{Gray}
\textit{PersonX} bites \textit{PersonX}'s nail& \textit{PersonX} is nervous& \texttt{xAttr}& \texttt{Synchronous}\\
\rowcolor{Gray}
\textit{PersonX} eats \textit{PersonX}'s breakfast& \textit{PersonX} is hungry& \texttt{xAttr}& \texttt{Condition}\\
\rowcolor{Gray}
\textit{PersonX} holds \textit{PersonX}'s tongue& \textit{PersonX} is quiet& \texttt{xAttr}& \texttt{Co\_Occurance}\\
\textit{PersonX} can not sleep& \textit{PersonX} is stressed & \texttt{xReact}& \texttt{Reason}, \texttt{Condition}\\
\textit{PersonX} is away from home & \textit{PersonX} is lonely& \texttt{xReact}& \texttt{Conjunction}\\
\textit{PersonX} is looking forward to it& \textit{PersonX} is excited& \texttt{xReact}& \texttt{Conjunction}\\
\textit{PersonX} tells \textit{PersonY} everything& \textit{PersonX} is trusted& \texttt{xReact}& \texttt{Conjunction}\\
\rowcolor{Gray}
\textit{PersonX} accepts the challenge& \textit{PersonX} wins & \texttt{xIntent}& \texttt{Co\_Occurance}\\
\rowcolor{Gray}
\textit{PersonX} bows \textit{PersonX}'s head& \textit{PersonX} prays & \texttt{xIntent}& \texttt{Co\_Occurance}\\
\rowcolor{Gray}
\textit{PersonX} removes \textit{PersonX}'s hat& \textit{PersonX} shows respect& \texttt{xIntent}& \texttt{Co\_Occurance}\\
\rowcolor{Gray}
\textit{PersonX} sits in car& \textit{PersonX} waits for \textit{PersonY} & \texttt{xIntent}& \texttt{Co\_Occurance}\\
\textit{PersonX} begins \textit{PersonX}'s work& \textit{PersonX} gets up& \texttt{xNeed}& \texttt{Contrast}, \texttt{Conjunction}\\
\textit{PersonX} closes the door& \textit{PersonX} has opened it & \texttt{xNeed}& \texttt{Synchronous}\\
\textit{PersonX} gets a divorce& \textit{PersonX} gets married& \texttt{xNeed}& \texttt{Reason}\\
\textit{PersonX} makes amends& \textit{PersonX} apologizes& \texttt{xNeed}& \texttt{Conjunction}\\
\rowcolor{Gray}
\textit{PersonX} calls \textit{PersonY}'s name& \textit{PersonY} turns around& \texttt{oEffect}& \texttt{Co\_Occurance}\\
\rowcolor{Gray}
\textit{PersonX} receives a text& \textit{PersonY} waits& \texttt{oEffect}& \texttt{Synchronous}\\
\rowcolor{Gray}
\textit{PersonX} takes \textit{PersonY} a picture& \textit{PersonY} smiles& \texttt{oEffect}& \texttt{Co\_Occurance}\\
\rowcolor{Gray}
\textit{PersonX} tries to tell \textit{PersonY}& \textit{PersonY} refuses to listen& \texttt{oEffect}& \texttt{Contrast}\\
\textit{PersonX} gets pregnant& \textit{PersonY} wants to marry \textit{PersonX}& \texttt{oWant}& \texttt{Precedence}\\
\textit{PersonX} has not seen \textit{PersonY} in years& \textit{PersonY} wants to see \textit{PersonX}& \texttt{oWant}& \texttt{Co\_Occurance}\\
\textit{PersonX} puts \textit{PersonX}'s arm around \textit{PersonY}& \textit{PersonY} pushes \textit{PersonX} away& \texttt{oWant}& \texttt{Conjunction}\\
\textit{PersonX} steals \textit{PersonY}'s wallet& \textit{PersonY} calls the police& \texttt{oWant}& \texttt{Conjunction}\\
\rowcolor{Gray}
\textit{PersonX} complains to the manager& \textit{PersonY} is sorry& \texttt{oReact}& \texttt{Co\_Occurance}\\
\rowcolor{Gray}
\textit{PersonX} did an excellent job& \textit{PersonY} is happy& \texttt{oReact}& \texttt{Conjunction}\\
\rowcolor{Gray}
\textit{PersonX} gives \textit{PersonY} money& \textit{PersonY} is grateful& \texttt{oReact}& \texttt{Conjunction}\\
\bottomrule
\end{tabular}
\caption{Overlaps of ASER and ATOMIC. }\label{table:ASER-ATOMIC-cases}
\end{table}
\subsection{Relationship with ConceptNet}
After around 20 years development, ConceptNet 5.0~\cite{speer2013conceptnet} now contains 21 million edges over 8 million nodes, built from the original ConceptNet~\cite{liu2004conceptnet}.
The core of ConceptNet, which is inherited from the Open Mind CommonSense (OMCS) project~\cite{liu2004conceptnet}, only contains 600K pieces of high-quality commonsense knowledge in the format of tuples, e.g., (`song', \textit{UsedFor}, `sing'). However, there is a huge gap between the small scale of existing commonsense knowledge resources and the broad demand of downstream applications, motivating us to acquire more commonsense knowledge cheaply and feasibly.
Based on the observation that selectional preference can naturally reflect commonsense knowledge about word choice in various contexts \cite{resnik1997selectional}, we proposed TransOMCS \cite{DBLP:conf/ijcai/ZhangKSR20} to transfer the selectional preference knowledge in ASER to ConceptNet-like commonsense tuples.
Specifically, we adopt the English subset of ConceptNet 5 \cite{speer2013conceptnet} as seed commonsense knowledge, and only relations covered by the original OMCS project~\cite{liu2004conceptnet} are selected.
Different from OpenIE~\cite{angeli-etal-2015-leveraging} and Hearst patterns~\cite{hearst-1992-automatic}, where human-defined patterns are leveraged to extract relations, we develop a pipeline to discover dependency patterns automatically.
As shown in Figure~\ref{fig:TransOMCS}, for each commonsense relation $r$ in ConceptNet, we first try to find patterns over dependency and discourse relations in ASER automatically with the overlap of ConceptNet assertions and ASER sub-graphs.
The percentage of ConceptNet knowledge that can be matched in ASER is presented in Figure~\ref{fig:TransOMCS-stat}.
After that, a pattern selection scoring function is designed to select highly plausible patterns. We present the most plausible dependency patterns for each relation as an illustration in Table~\ref{tab:OMCS-pattern}. Based on the patterns, we can traverse the whole ASER to acquire a large-scale commonsense knowledge graph in the format of ConceptNet.
A running example of TransOMCS is shown in Figure~\ref{fig:TransOMCS}.
For the OMCS-like assertion $\langle$``Good grades'', \texttt{Causes}, ``Graduate''$\rangle$, we can extract the corresponding dependency relation from the ASER edge $\langle$``he gets good grades'', \texttt{Result}, ``he graduates colledge''$\rangle$. Such dependency pattern is in turn used for other ASER edges to extract novel knowledge.
As a result, we successfully acquire 18 million ConceptNet-like commonsense assertions with high novelty and accuracy.
From the case study in Table~\ref{tab:OMCS-case} we can see that interesting commonsense knowledge is indeed contained in ASER.
For example, with the help of ASER, we can know that students are often at school, artists often create art, and the wall is part of the house.
\subsection{Relationship with ATOMIC}
Besides ConceptNet, another substantial commonsense knowledge base is ATOMIC \cite{Maarten2019Atomic}, a large-scale human-annotated commonsense knowledge graph that provides inferential knowledge about daily events. Like ASER, the ATOMIC nodes are events described in free-form text, while not parsed to be canonical. There are nine \textit{if-then} relationships defined across ATOMIC, measuring the daily causes and effects for certain base events. To tackle the limitations in terms of novelty and coverage of current \textit{if-then} commonsense acquisition methods, we proposed a novel framework DISCOS (from DIScourse to COmmonSense)~\cite{fang2020discos}, which transfers selectional preference knowledge in ASER to complex commonsense knowledge in ATOMIC.
As a result, we acquire 3.4 Million \textit{if-then} commonsense knowledge in the format of ATOMIC. An illustration of the process in DISCOS is presented in Figure~\ref{fig:aser-atomic-sketch}.
Specifically, we first conduct an alignment from ATOMIC to ASER. In ATOMIC, the personal pronouns are represented with wildcards like ``\textit{PersonX}'' and ``\textit{PersonY},'' and in ASER, the subjects of events are concrete personal pronouns like ``she'' and ``he.'' Moreover, as all of the tail events in ATOMIC are written by human annotators, the form of ATOMIC tails can be arbitrary and sometimes subjects are omitted. Based on those observations, we develop some string substitution rules to align the nodes in ATOMIC and ASER, as illustrated in Table~\ref{table:atomic-preprocess}. After conducting the string substitution operations, we use the parser in ASER to parse the acquired text into standard ASER format.
Table~\ref{table:mapping_stat} presents the coverage statistics between ATOMIC and ASER.
We first conduct the string match to check the coverage of ATOMIC nodes in ASER, and find that the average percentage of ATOMIC nodes found in ASER is 62.9\%. For edges, we present the percentage of ATOMIC edges whose head and tail are both covered by ASER, which is 35.91\% on average.
On top of the matched edges, we check the shortest path length between the matched head and tail in ASER and report the average among all edges in the \textit{Avg. Shortest Path Length} column.
The range of shortest path length starts from 1, where the shortest path length between two directly connected nodes is 1.
We can conclude that, within a few hops of reasoning in ASER, a decent percentage of ATOMIC relations can be inferred.
Some examples are presented in Table~\ref{table:ASER-ATOMIC-cases}. For instance, the knowledge that if \textit{\textit{PersonX} bites \textit{PersonX}'s tongue} then the person would want to \textit{cry}, can be entailed from the \texttt{Precedence} discourse relation in ASER.
As the heads and tails in ATOMIC are all arbitrary sentences, the aforementioned pattern mining approach used in TransOMCS is no longer suitable.
To effectively convert ASER knowledge into the ATOMIC format, we propose to use a neural network based classifier instead of hard patterns.
After we match ATOMIC and ASER, we will use the matched eventualities and associated sub-graph as the positive training examples.
For each matched eventuality, we consider its one-hop or two-hop neighbors in ASER to be the candidate eventualities for populating commonsense knowledge of the corresponding ATOMIC relation, whose examples are shown in Table~\ref{table:ASER-neighbor}.
With the help of a graph-based knowledge graph population model and the random negative example sampling, we successfully acquire large-scale commonsense knowledge in the format of ATOMIC.
As demonstrated in Figure~\ref{fig:aser-atomic-sketch} and Table~\ref{table:ASER-concept}, both the original extracted eventualities and edges and those after the conceptualization can help us find rich commonsense about daily events.
For example, before the conceptualization, we can find some knowledge like $\langle$``She takes antibiotic'', \texttt{Result}, ``She gets better''$\rangle$, which is rather specific. After the conceptualization, we can get a more abstract level commonsense that $\langle$``\textit{PersonX} takes medicine'', \texttt{Result}, ``\textit{PersonX} gets better''$\rangle$.
Further experiments in~\cite{fang2020discos} also show that compared with a pure supervised model, the knowledge populated with our approach is much more novel and diverse with the comparable high quality.
\begin{table}[t]
\centering
\footnotesize
\begin{tabular}{l|l|p{0.1\textwidth}<{\centering}|l}
\toprule
ATOMIC Head& ATOMIC Tail& ATOMIC-Rel& Add. Neigh. by ASER\\
\midrule
\rowcolor{Gray}
\textit{PersonX} bites \textit{PersonX}'s tongue& \textit{PersonX} cries& xWant& \textit{PersonY} strikes \textit{PersonX} carefully with back\\
\rowcolor{Gray}
\textit{PersonX} bows \textit{PersonX}'s head& \textit{PersonX} prays& xWant& \textit{PersonX} cover \textit{PersonX}'s face with hands\\
\rowcolor{Gray}
\textit{PersonX} catch \textit{PersonY}'s eye& \textit{PersonX} makes an impression & xWant& \textit{PersonY} is interested\\
\textit{PersonX} becomes angry& \textit{PersonX} yells& xEffect& \textit{PersonX} asks for explanation\\
\textit{PersonX} goes to the party& \textit{PersonX} gets drunk& xEffect& \textit{PersonX}'s stomach hurts\\
\textit{PersonX} wins the lottery& \textit{PersonX} becomes rich& xEffect& \textit{PersonX} would quit \textit{PersonX}'s job\\
\rowcolor{Gray}
\textit{PersonX} can not sleep& \textit{PersonX} is stressed& xReact& \textit{PersonX} had a bad day at work\\
\rowcolor{Gray}
\textit{PersonX} is away from home& \textit{PersonX} is lonely& xReact& \textit{PersonX} tries to talk to people\\
\rowcolor{Gray}
\textit{PersonX} is looking forward to it& \textit{PersonX} is excite& xReact& \textit{PersonX} is working hard to get there\\
\textit{PersonX} accepts the challenge& \textit{PersonX} win& xIntent & \textit{PersonY} plays\\
\textit{PersonX} bows \textit{PersonX}'s head& \textit{PersonX} prays& xIntent& \textit{PersonX} is silent\\
\textit{PersonX} sits in car& \textit{PersonX} waits for \textit{PersonY}& xIntent & the police gets \textit{PersonX} out\\
\rowcolor{Gray}
\textit{PersonX} gets a divorce& \textit{PersonX} gets married& xNeed& \textit{PersonX}'s spouse cheats on \textit{PersonX}\\
\rowcolor{Gray}
\textit{PersonX} makes amends& \textit{PersonX} apologizes& xNeed& \textit{PersonX} did wrong\\
\bottomrule
\end{tabular}
\caption{Additional commonsense neighbors that ASER can provide, which can be learned by a knowledge graph population model. }\label{table:ASER-neighbor}
\end{table}
\begin{table}[t]
\renewcommand\arraystretch{1.0}
\centering
\footnotesize
\begin{tabular}{l|l|l}
\toprule
&Head& Tail\\
\midrule
Extracted & she take antibiotic& she get better\\
Conceptualized & \textit{PersonX} take \textit{Medicine}& \textit{PersonX} get better\\
\midrule
Extracted & he pay he bill& money be not plentiful with he\\
Conceptualized & \textit{PersonX} pay \textit{Short-Dated-Asset}& money be not plentiful with \textit{PersonX}\\
\midrule
Extracted & i win the lottery& i become rich\\
Conceptualized & \textit{PersonX} win \textit{Form-of-Gambling}& \textit{PersonX} become rich\\
\midrule
Extracted & he spill coffe& i ask for refill\\
Conceptualized & \textit{PersonX} spill \textit{Beverage}& \textit{PersonY} ask for refill\\
\bottomrule
\end{tabular}
\caption{Examples of \textit{if-then} commonsense knowledge in ASER.The knowledge before and after with Conceptualization are indicated with ``Extracted'' and ``Conceptualized.''}\label{table:ASER-concept}
\end{table}
\section{Applications}\label{sec:extrinsic-evaluation}
In this section, we introduce the application of the selectional preference knowledge in ASER on downstream tasks.
First of all, as shown in~\cite{zhang2019sp-10k,DBLP:conf/www/ZhangLPSL20}, even with a trivial approach like string match, such knowledge can be used to answer a subset of the challenging commonsense reasoning benchmark (i.e., Winograd schema challenge~\cite{levesque2011winograd}) with high precision.
Besides that, the ASER knowledge can also help the machines to understand longer documents. For example, as shown in~\cite{DBLP:conf/ijcai/ZhangKSR20}, after converting the ASER knowledge into the ConceptNet format, we can apply that knowledge on the commonsense reading comprehension task~\cite{ostermann2018semeval} and achieve more significant improvement than the original ConceptNet~\cite{liu2004conceptnet}.
As the higher-order selectional preference reflects humans' understandings about daily life, we can apply them to help understand the daily dialogue and generate better responses.
For example, on the daily dialogue dataset~\cite{li2017dailydialog}, compared with adding the original conceptnet, adding ASER knowledge can double the improvement in terms of the BLEU score.
Last but not least, we also explored how to combine the higher-order SP in ASER with large-scale pre-trained language models~\cite{DBLP:journals/corr/abs-2012-15643}.
Experiments on multiple commonsense reasoning tasks including story completion~\cite{DBLP:conf/naacl/MostafazadehCHP16}, temporal relation prediction~\cite{DBLP:conf/acl/RothWN18}, and causal relation prediction~\cite{DBLP:conf/semeval/GordonKR12} demonstrate the ASER knowledge can be a good supplement of existing language models.
We leave more applications on other natural language processing and understanding tasks as future work.
\section{Conclusions}\label{sec:conclusion}
In this paper, we propose to represent commonsense knowledge with higher-order selectional preference over eventualities.
Motivated by this, we construct ASER, which is a large-scale eventuality knowledge graph that contains 438 million eventualities and 648 million edges.
Considering the large scale of commonsense, we propose an unsupervised pipeline to extract rich commonsense knowledge about events from the raw corpus instead of human annotation.
To effectively represent humans' preference about daily events, we design ASER to be weighed, and larger weight indicates that the eventuality or edge is more likely to happen.
Finally, to overcome the challenge that trivial commonsense might be often omitted, we propose to leverage the conceptualization process to generalize the knowledge about seen eventualities to unseen ones.
We conduct human evaluations, case studies, and extrinsic evaluations to show that ASER is a promising eventuality-centric commonsense knowledge graph with great potential in many downstream tasks.
All codes and data are published to encourage further research on commonsense and event understanding.
\section*{Acknowledgements}
This paper was supported by the Early Career Scheme (ECS, 26206717), General Research Fund (GRF, 16211520), Research Impact Fund (RIF, R6020-19 and R6021-20) from Research Grants Council in Hong Kong. We thank Dan Roth and Daniel Khashabi for their insightful comments on this work.
\section*{Contributions}
The contributions of all authors are as follows.
\begin{itemize}
\item \textbf{Hongming Zhang}: Proposing the idea of using higher-order selectional preference over eventualities to represent commonsense knowledge, designing the ASER structure, designing the eventuality and edge patterns, designing the eventuality extraction algorithm, Selecting data, conducting intrinsic evaluation, conducting extrinsic evaluations (except dialogue system), and writing the paper.
\item \textbf{Xin Liu}: Pre-processing raw data, constituency parsing and clause analyzing, extracting relations with discourse parsing systems, designing the ASER database schema and the construction pipeline, providing APIs for SQLite and MongoDB,
and drafting the major of Section~\ref{sec:aser-construction} and~\ref{sec:statistics}.
\item \textbf{Haojie Pan}: Exploring and implementing the conceptualization with Probase and drafting attached sections, conducting the extrinsic evaluation on the dialogue system, designing the client-server model for the distributed ASER system, and preparing the online demo.
\item \textbf{Haowen Ke}: Pre-processing raw data with CoreNLP to acquire lemazied tokens, pos-tags, name entities, dependency tree, and constituency tree. Analyzing the inference results in ASER and drafting Section~\ref{sec:inference}.
\item \textbf{Jiefu Ou}: Implementing the rule-based inference over ASER with the AMIE+ system. Assisting Xin Liu for discourse relation extraction and assisting Haowen Ke for analyzing the inference results in ASER.
\item \textbf{Tianqing Fang}: Analyzing the relation between ASER and other commonsense Knowledge bases, and drafting Section~\ref{sec:transferability}.
\item \textbf{Yangqiu Song}: Proposing the ideas of building an eventuality centric knowledge graph, using conceptualization for abstraction and instantiation, managing the ASER project, and revising the paper.
\end{itemize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,668 |
\section{Introduction}
Given two Riemannian manifolds $(M, g)$ and $(N,h)$, the solutions $\varphi: M \to N$ of the variational problem associated to the \textit{Dirichlet energy}
\begin{equation*}
\mathcal{E}(\varphi) =
\frac{1}{2} \int_M \abs{\mathrm{d} \varphi}^2 \nu_{g}.
\end{equation*}
are called \textit{harmonic maps}. Derrick's \textit{scaling argument} \cite{der} implies that, in any dimension other than $m = 2$, there are no non-constant finite-energy harmonic maps defined on $\RR^m$.
This feature is not entirely specific to the euclidean metric, as proved by Sealey \cite{sea}. If $\mathcal{E}(\varphi)$ is allowed to contain a potential term $V \circ \varphi$, then there are no non-trivial solutions if $m > 2$. A counterpart for compact domains is Xin's theorem \cite{xin} asserting that if $m > 2$, then there is no non-constant \textit{stable} harmonic map defined on the unit sphere $\mathbb{S}^m$. A mirror result by Leung \cite{leu} holds for mapping taking values into the sphere. Both facts are proved using the Lawson-Simons \textit{averaging argument} \cite{ls}.
Derrick and Xin-Leung theorems can be evaded by considering higher power energies, the price to be paid consisting in restrictions on the ellipticity of the corresponding Euler-Lagrange equations.
For instance, if we consider consider the $p$-\textit{energy} ($p>2$)
\begin{equation*}
\mathcal{E}_p(\varphi) =
\frac{1}{p} \int_M \abs{\mathrm{d} \varphi}^p \nu_g ,
\end{equation*}
both restrictions are relaxed to $m > p$. Moreover, the Hopf map from $\mathbb{S}^3$ to $\mathbb{S}^2$ minimizes the $p$-energy in its homotopy class for $p\geq 4$ \cite{riv}.
Another natural choice of high power functional was introduced in the seminal paper on harmonic maps \cite{eell} as
\begin{equation*}
\mathcal{E}_{\sigma_p}(\varphi)=
\frac{1}{2} \int_{M} \abs{\wedge^p \mathrm{d} \varphi}^2 \nu_g,
\end{equation*}
and was called $\sigma_p$-\textit{energy} since the integrand can be also seen as $\sigma_p(\varphi^*h)$, the $p^{th}$ elementary symmetric function of the eigenvalues of $\varphi^* h$ with respect to $g$. The fourth power case, $\mathcal{E}_{\sigma_2}(\varphi)$, was already known as the self-interaction term of Skyrme's sigma-model \cite{sky} in nuclear physics.
Motivated by the strong coupling limit of Faddeev-Niemi model \cite{fad}, Speight and Svensson \cite{sve, svee} studied the \textit{symplectic Dirichlet energy}:
\begin{equation*}
\mathcal{F}(\varphi) = \frac{1}{2}
\int_M \abs{\varphi^* \Omega}^2 \nu_g,
\end{equation*}
suited for maps taking values in a symplectic manifold $(N, \Omega)$.
While Derrick's result extends immediately to these alternative energy functionals asserting the non-existence of non-trivial finite energy solutions in dimensions above the highest degree of derivatives appearing in the integrand, Xin-Leung restriction needs a more elaborate case-by-case analysis.
This has already been done for the Yang-Mills energy (of instantons) \cite{bourg}, for the volume functional (of immersions) \cite{ls, sim}, for the $p$-energy \cite{leu, take} and for the $L^2$ norm of the pullback metric \cite{kaw}. In this short note we complete the picture for the fourth power energies by proving analogue results for $\mathcal{E}_{\sigma_2}$ and $\mathcal{F}$, and by pointing out their global counterpart. This allows us to derive stability properties also for the case when we couple each of these two functionals with the Dirichlet energy, as it is usually done in the original sigma-models.
\medskip
\begin{small}
Throughout the paper, manifolds, metrics, and maps are assumed to be smooth. On a connected Riemannian manifold $(M,g)$ with Levi-Civita connection $\nabla$, we use the following sign conventions for the curvature tensor field $R(X,Y)Z=\nabla_X\nabla_Y Z-\nabla_Y \nabla_X Z-\nabla_{[X,Y]}Z$, and $\Delta f = \tr \nabla \mathrm{d} f$ for the Laplacian on functions.
\end{small}
\section{Symplectic Dirichlet stability on spheres}
Let $(M,g)$ and $(N,J,h)$ be Riemannian manifolds, the second being endowed with an almost K\"ahler structure with the fundamental 2-form $\Omega(\cdot,\cdot)=h(\cdot,J\cdot)$. A map $\varphi:M \to N$ is $\mathcal{F}$-critical if the first variation of $\mathcal{F}$ at $\varphi$ vanishes, and this is proved (\cite{sve}) to be equivalent with the Euler-Lagrange equations
\begin{equation}\label{fh}
\mathrm{d} \varphi\left((\delta \varphi^* \Omega)^{\sharp}\right)=0.
\end{equation}
A vacuum solution (i.e. $\varphi^* \Omega=0$) is called \textit{isotropic}. A critical map is moreover a local minimizer (stable critical point) if the second variation of the energy (the Hessian) evaluated at this map is positive definite.
For any $v \in \Gamma(\varphi^{-1}TN)$, and any $\mathcal{F}$-critical map $\varphi$, the Hessian of $\mathcal{F}$ can be calculated as (\cite{sve})
\begin{equation}\label{fhess}
\mathrm{Hess}_{\varphi}^{\mathcal{F}}(v,v)= \int _M \{\abs{\mathrm{d} (\varphi^* \imath _v \Omega)}^{2} + \Omega(v, \nabla^{\varphi}_{Z_\varphi} v)\} \nu_g,
\end{equation}
where $Z_\varphi=(\delta \varphi^* \Omega)^{\sharp}$. In particular, if $\delta \varphi^* \Omega=0$, then we see that $\varphi$ is stable (it actually minimizes $\mathcal{F}$ in its homotopy class \cite{svee}).
\begin{lm}[\cite{slobo}]\label{magic}
Let $\varphi:(M,g) \to (N,h)$ be a mapping between Riemannian manifolds. Then for any $X,Y,Z \in \Gamma(TM)$ we have
\begin{equation*}
\left( \nabla_X \varphi^* h \right)(Y, Z)= h(\nabla \mathrm{d} \varphi(X, Y), \mathrm{d} \varphi(Z)) + h(\mathrm{d} \varphi(Y), \nabla \mathrm{d} \varphi(X,Z)).
\end{equation*}
\end{lm}
\begin{re}[Averaging argument]\label{av} The method introduced in \cite{ls} in order to find necessary conditions for stability on/into spheres consists in averaging the second variation of the respective energy functional on a particular family of gradient conformal vector fields.
Let $(a_\alpha)_{\alpha=1,...,m+1}$ be an orthonormal basis in $\RR^{m+1}$. Define $f_\alpha : \mathbb{S}^{m} \to \RR$, $f_\alpha(x)=\langle a_\alpha, x\rangle$ and take $\gr f_\alpha \in \Gamma(T\mathbb{S}^{m})$.
We have $(\gr f_\alpha)_ {x} = a_\alpha - f_\alpha(x) x$, $\abs{\gr f_\alpha}^2=1 - f_\alpha^2$ and
\begin{equation}\label{nabgr}
\nabla_X \gr f_\alpha=-f_\alpha X \qquad (X \in \Gamma(T\mathbb{S}^{m})),
\end{equation}
where $\nabla$ is the Levi-Civita connection of the canonical metric $g$ on $\mathbb{S}^{m}$. In particular, $f_\alpha$ are eigenfunctions of the Laplace operator
corresponding to the first non-zero eigenvalue: $\Delta f_\alpha=-m f_\alpha$. It is immediate to see that $\sum_\alpha f_\alpha^2=1$ (so $\sum_\alpha f_\alpha \gr f_\alpha=0$), and that, for any $X \in \Gamma(T\mathbb{S}^{m})$, $X=\sum_\alpha g(X, \gr f_\alpha)\gr f_\alpha$.
\end{re}
\begin{pr}
If $m > 4$ there is no non-isotropic stable $\mathcal{F}$-critical map from $\mathbb{S}^m$ to any almost K\"ahler manifold.
\end{pr}
\begin{proof}
Let $\varphi:\mathbb{S}^{m} \to (N, \Omega)$ be a smooth $\mathcal{F}$-critical map and $v_\alpha=\mathrm{d} \varphi(\gr f_\alpha) \in \Gamma(\varphi^{-1}TN)$, $\alpha=1,...,m+1$ be defined using Remark \ref{av}.
Observe that $\nabla^{\varphi}_{V} \mathrm{d} \varphi(X) = \mathrm{d} \varphi([V,X])$, for any $V \in \ker \mathrm{d} \varphi$ and any $X\in\Gamma(TM)$. Since $\varphi$ is $\mathcal{F}$-critical, $Z_\varphi \in \ker \mathrm{d} \varphi$ and we have
\begin{equation}\label{h1}
\begin{split}
\Omega(v_\alpha, \nabla^{\varphi}_{Z_{\varphi}} v_\alpha)
&= - \varphi^* \Omega([Z_{\varphi}, \gr f_\alpha], \gr f_\alpha)\\
&=\varphi^* \Omega \left(\nabla_{\gr f_\alpha}Z_{\varphi}, \gr f_\alpha\right)\\
&=-\left(\nabla_{\gr f_\alpha} \varphi^* \Omega\right) \left(Z_{\varphi}, \gr f_\alpha\right),
\end{split}
\end{equation}
so by summing over $\alpha$ we obtain
\begin{equation}\label{h1S}
\sum_\alpha
\Omega(v_\alpha, \nabla^{\varphi}_{Z_{\varphi}} v_\alpha)
= -\abs{\delta \varphi^* \Omega}^2.
\end{equation}
Using Lemma \ref{magic} and Remark \ref{av}, we obtain
\begin{equation}
\mathrm{d} (\varphi^* \imath _{v_\alpha} \Omega)(X, Y)=
\left(\nabla_{\gr f_\alpha} \varphi^* \Omega\right)(X,Y)
-2f_\alpha \, \varphi^* \Omega(X,Y),
\end{equation}
so by taking the norm and summing over $\alpha$,
\begin{equation}\label{h2S}
\sum_\alpha\abs{\mathrm{d} (\varphi^* \imath _{v_\alpha} \Omega)}^{2}=\abs{\nabla \varphi^* \Omega}^2 + 4 \abs{\varphi^* \Omega}^2.
\end{equation}
Combining \eqref{h1S} and \eqref{h2S} we see that calculating the trace of the Hessian requires the following Weitzenb\"ock formula for $p$-forms
(see \cite[(1.32)]{eel} and references therein)
\begin{equation*}
-\tfrac{1}{2}\Delta\abs{\sigma}^2
= \langle \Delta \sigma, \sigma\rangle
- \abs{\nabla \sigma}^2- \langle S(\sigma), \sigma \rangle ,
\end{equation*}
which by integration over a compact manifold without boundary gives:
\begin{equation}\label{weitz}
\int_M \abs{\mathrm{d} \sigma}^2 + \abs{\delta \sigma}^2 - \abs{\nabla \sigma}^2- \langle S(\sigma), \sigma \rangle =0.
\end{equation}
If $\sigma \in \Lambda^2 (M)$, then the curvature operator $S$ acts as follows
$$
S(\sigma)(X_1, X_2)=\sigma(\Ric X_1, X_2)+ \sigma(X_1, \Ric X_2)
+ \sum_s \sigma(e_s, R(X_1, X_2)e_s).
$$
In particular, for 2-forms on $\mathbb{S}^m$ we simply have
$$S(\sigma)(X_1, X_2)=(2m-4)\sigma(X_1, X_2).$$
Applying \eqref{weitz} for the closed 2-form $\varphi^* \Omega$ on $\mathbb{S}^m$ we obtain
\begin{equation*}
\begin{split}
\sum_\alpha \mathrm{Hess}_{\varphi}^{\mathcal{F}}(v_\alpha, v_\alpha)&=\int_{\mathbb{S}^m} \big\{-\abs{\delta \varphi^* \Omega}^2+\abs{\nabla \varphi^* \Omega}^2 + 4 \abs{\varphi^* \Omega}^2\big\}\nu_{can}\\
&=2(4-m) \int_{\mathbb{S}^m}\abs{\varphi^* \Omega}^2 \nu_{can}
\end{split}
\end{equation*}
and the conclusion follows.
\end{proof}
\noindent This generalizes \cite[Prop. 3.4]{slobo} in the case of Hopf maps. Recall that \cite{svee} the Hopf map $\varphi : \mathbb{S}^3 \to \mathbb{C} P^1$ minimizes $\mathcal{F}$ in its homotopy class.
\subsection{Full Faddeev-Niemi model} Let us now turn attention to the coupled energy
$$
\mathcal{E}(\varphi)+ \kappa \mathcal{F}(\varphi),
$$
where $\kappa$ is a positive coupling constant. A mapping $\varphi$ will be a critical point for this action if and only if:
\begin{equation}\label{fhfull}
\tau(\varphi)-\kappa J\mathrm{d} \varphi\left((\delta \varphi^* \Omega)^{\sharp}\right)=0,
\end{equation}
where $\tau(\varphi)=\tr \nabla \mathrm{d} \varphi$ is the \textit{tension field} of $\varphi$. Even if the the Hessian of a coupled energy is still a linear combination of the two individual Hessians, combining the averaging arguments requires caution, since in the computation of $\sum_\alpha \mathrm{Hess}_{\varphi}^{\mathcal{E}, \mathcal{F}}(v_\alpha, v_\alpha)$ we employed again the (individual) Euler-Lagrange equations. So by carefully redoing the same steps for the full energy and using this time \eqref{fhfull}, we obtain
\begin{equation*}
\begin{split}
\sum_\alpha \mathrm{Hess}_{\varphi}^{\mathcal{E}+\kappa\mathcal{F}}(v_\alpha, v_\alpha)
&=\int_{\mathbb{S}^m}\left\{(2-m)\abs{\mathrm{d} \varphi}^2 + 2\kappa(4-m)\abs{\varphi^* \Omega}^2 \right\}\nu_{can}.
\end{split}
\end{equation*}
In particular, if $m\geq 4$, then there is no non-constant stable $(\mathcal{E}+\kappa\mathcal{F})$-critical map from $\mathbb{S}^m$ to any almost K\"ahler manifold. If $m=3$, a necessary condition for a non-isotropic $(\mathcal{E}+\kappa\mathcal{F})$-critical map $\varphi:\mathbb{S}^3 \to N^2$ to be stable is
$$
\kappa\geq \frac{\int_{\mathbb{S}^3}\abs{\mathrm{d} \varphi}^2 \nu_{can}}{2\int_{\mathbb{S}^3}\abs{\varphi^* \Omega}^2\nu_{can}}
$$
For the Hopf map $\mathbb{S}^3 \to \mathbb{C} P^1 \cong \mathbb{S}^2(\tfrac{1}{2})$ this reads $\kappa \geq 1$ and it is also a sufficient condition, as proved in \cite{sve}.
\section{$\sigma_2$-Stability on spheres}
For any map $\varphi: (M^m,g) \to (N^n,h)$ between Riemannian manifolds of dimensions $m,n \geq 2$, we denote by $\sigma_2(\varphi^* h)=\sum_{i<j}\lambda_i^2\lambda_j^2$ and we call $\sigma_2$-\textit{energy} the action functional $\mathcal{E}_{\sigma_2}(\varphi)=\tfrac{1}{2}\int_M \sigma_2(\varphi^* h) \nu_g$, where $\lambda_i^2$ are the eigenvalues of $\varphi^* h$ with respect to $g$. The corresponding Euler-Lagrange equations are (\cite{cri}, cf. also \cite{slo})
$$
\tr \nabla(\abs{\mathrm{d} \varphi}^2 \mathrm{d} \varphi-\mathrm{d} \varphi \circ \mathfrak{C}_\varphi)=0,
$$
where $\mathfrak{C}_\varphi=\mathrm{d} \varphi^t \circ \mathrm{d} \varphi$, the (1,1)-"dual" of $\varphi^* h$, is called the \textit{Cauchy-Green tensor}.
The second variation formula is given below. Here we shall investigate its behaviour for mappings defined on spheres and we expect to recover the result of the previous section in this case (cf. \cite{sachs} for the identity map). Since $2\sigma_2(\varphi^* h)=\abs{\mathrm{d} \varphi}^4 - \abs{\varphi^* h}^2$ in order to prove this we would be tempted to simply combine the result in \cite{leu, take}
\begin{equation*}
\sum_\alpha \mathrm{Hess}_{\varphi}^{\mathcal{E}_4}(v_\alpha, v_\alpha)=(4-n) \abs{\mathrm{d} \varphi}^4
\end{equation*}
with the corresponding result in \cite{kaw} for $\mathcal{G}(\varphi)=\tfrac{1}{4}\int_M \abs{\varphi^* h}^2 \nu_g$
\begin{equation*}
\sum_\alpha \mathrm{Hess}_{\varphi}^{\mathcal{G}}(v_\alpha, v_\alpha)=(4-n) \abs{\varphi^* h}^2.
\end{equation*}
But, as already mentioned, in the derivation of these "trace" formulae the individual Euler-Lagrange equations have been employed again, so we need to identify the respective terms in order to see that this approach actually gives us the expected result. For the convenience of the reader we present the main lines of the complete proof.
\begin{lm}[The second $\sigma_2$-variation \cite{cri}]\label{sigma_2hess}
The second variation of the $\sigma_2$-energy along $v\in\Gamma(\varphi^{-1}TN)$ evaluated on a $\sigma_2$-critical map $\varphi$ is
\begin{equation}\label{sigma_2hes}
\begin{split}
\mathrm{Hess}_{\varphi}^{\mathcal{E}_{\sigma_2}}(v, v)&=\int_{M} \big\{ 2(\di^\varphi v)^2 + \abs{\mathrm{d} \varphi}^2\left(\abs{\nabla^\varphi v}^2 - \Ric^\varphi(v,v)\right)\big\}\nu_g\\
&-\int_{M} \big\{\tfrac{1}{2}\abs{H_v}^2 + \sum_i\lambda_i^2\left(\abs{\nabla_{e_i}^\varphi v}^2 - \langle R^N(v,\mathrm{d} \varphi(e_i))\mathrm{d} \varphi(e_i), v \rangle \right)\big\}\nu_g\\
\end{split}
\end{equation}
where $\{e_i\}_{i=1,...,m}$ is a (local) orthonormal frame of eigenvectors of $\varphi^* h$ on $M$, $H_v(X,Y)=h(\nabla_{X}^\varphi v, \mathrm{d} \varphi(Y))+h(\nabla_{Y}^\varphi v, \mathrm{d} \varphi(X))$, for any $X$ and $Y$ tangent vectors to $M$,
$\di^\varphi v=\tfrac{1}{2}\tr H_v$, and $\Ric^\varphi \left(v,w \right)=$ \\ $\sum_j h\left( R^N(v,\mathrm{d} \varphi(e_j))\mathrm{d} \varphi(e_j), w \right)$.
\end{lm}
\noindent As for the symplectic Dirichlet energy, we need a Weitzenb\"ock formula.
\begin{lm}[Weitzenb\"ock type formula \cite{naka}]\label{boc}
For any smooth map $\varphi:(M,g)\to(N,h)$ the following identy holds
\begin{equation*}
\begin{split}
\frac{1}{4}\Delta\abs{\varphi^* h}^2 =&\frac{1}{2}\abs{\nabla \varphi^* h}^2
+\sum_{i}\lambda_i^2 \abs{\nabla \mathrm{d} \varphi(e_i, \cdot)}^2 \\
&+\sum_{i}\lambda_i^2 \left[h\left(\mathrm{d} \varphi(\Ric^M e_i), \mathrm{d} \varphi(e_i)\right) - \Ric^\varphi \left(\mathrm{d} \varphi(e_i),\mathrm{d} \varphi(e_i)\right)\right]\\
&+\di(\mathfrak{C}_\varphi(\mathrm{d} \varphi^t(\tau(\varphi)))-h(\tau(\varphi), \tr \nabla (\mathrm{d} \varphi \circ \mathfrak{C}_\varphi)),
\end{split}
\end{equation*}
where $\{e_i\}_{i=1,...,m}$ is a (local) orthonormal frame of eigenvectors for $\varphi^* h$ with respect to $g$, corresponding to the eigenvalues $\{\lambda_i^2\}_{i=1,...,m}$.
\end{lm}
This identity is obtained by direct computation and not by deriving it from the general Weitzenb\"ock formula \cite[(1.34)]{eel} applied to $\mathrm{d} \varphi \circ \mathfrak{C}_\varphi$. By combining it with the Weitzenb\"ock formula used in the regularity theory of $p$-harmonic maps ($p=4$) \cite{xinp} we can obtain a Weitzenb\"ock formula suited for $\sigma_2$-critical maps.
We are now ready to prove the following stability property of $\sigma_2$-energy.
\begin{pr}
If $m > 4$, then there is no stable $\sigma_2$-critical map of rank $ \geq 2$ from $\mathbb{S}^m$ to any Riemannian manifold $N^n$ $(n\geq 2)$.
\end{pr}
\begin{proof}
Let $\varphi:\mathbb{S}^{m} \to N$ be a smooth $\sigma_2$-critical map and $v_\alpha=\mathrm{d} \varphi(\gr f_\alpha)$, $\alpha=1,...,m+1$ be defined in Remark \ref{av}. For the first two terms in \eqref{sigma_2hes}, a computation corresponding to the 4-harmonic case yields
\begin{equation*}
\begin{split}
&\sum_\alpha 2(\di^\varphi v_\alpha)^2 + \abs{\mathrm{d} \varphi}^2\left(\abs{\nabla^\varphi v_\alpha}^2 - \Ric^\varphi(v_\alpha,v_\alpha)\right)\\
&=(4-m)\abs{\mathrm{d} \varphi}^4
+h\left(\tau(\varphi), \abs{\mathrm{d} \varphi}^2\tau(\varphi)+\mathrm{d} \varphi(\gr \abs{\mathrm{d} \varphi}^2)\right)+\di(\dots).
\end{split}
\end{equation*}
For the third term in \eqref{sigma_2hes}, by using Lemma \ref{magic} we obtain
\begin{equation*}
\sum_\alpha \tfrac{1}{2}\abs{H_{v_\alpha}}^2=\tfrac{1}{2}\abs{\nabla \varphi^* h}^2 + 2\sum_i \lambda_i^4.
\end{equation*}
Finally, we directly check that
$$
\sum_{i}\lambda_i^2 \abs{\nabla \mathrm{d} \varphi(e_i, \cdot)}^2=
\sum_{i, \alpha}\lambda_i^2 \abs{\nabla_{e_i}^{\varphi} \mathrm{d} \varphi(\gr f_\alpha)}^2-
\sum_i \lambda_i^4,
$$
which, combined with the Weitzenb\"ock formula (Lemma \ref{boc}), yields
\begin{equation*}
\begin{split}
&\sum_{i,\alpha}\lambda_i^2\left[\abs{\nabla_{e_i}^\varphi v_\alpha}^2 - h\left( R^N(v_\alpha,\mathrm{d} \varphi(e_i))\mathrm{d} \varphi(e_i), v_\alpha \right) \right]\\
&= -\tfrac{1}{2}\abs{\nabla \varphi^* h}^2 +(2-m)\sum_i \lambda_i^4 + h\left(\tau(\varphi), \tr \nabla(\mathrm{d} \varphi \circ \mathfrak{C}_\varphi)\right)+\di(\dots).
\end{split}
\end{equation*}
Inserting all in the Hessian formula \eqref{sigma_2hes} and noticing that the $\sigma_2$-Euler-Lagrange equations satisfied by $\varphi$ assure the cancellation of $h(\tau(\varphi), \dots)$, we obtain
\begin{equation*}
\begin{split}
\sum_\alpha \mathrm{Hess}_{\varphi}^{\mathcal{E}_{\sigma_2}}(v_\alpha, v_\alpha)=2(4-m) \int_{\mathbb{S}^m}\sigma_2(\varphi^*h) \nu_{can}
\end{split}
\end{equation*}
and, since $\sigma_2(\varphi^*h)\geq 0$ with equality iff $\rank \mathrm{d} \varphi_x <2$ for all $x \in \mathbb{S}^m$, the conclusion follows.
\end{proof}
Starting from Equation \eqref{sigma_2hes} and applying the averaging argument (Remark \ref{av}) with $v_\alpha=(\gr f_\alpha)\circ \varphi$ yields
\begin{lm}[Stability inequality]For any stable $\sigma_2$-critical map $\varphi:M \to \mathbb{S}^n$ the folowing inequality holds
\begin{equation*}
\int_M \big\{(n-1)\left(\abs{\gr f}^2\abs{\mathrm{d} \varphi}^2 -\abs{\mathrm{d} \varphi (\gr f)}^2 \right) + 2(4-n)f^2 \sigma_2(\varphi^* h)\big\}\nu_g \geq 0,
\end{equation*}
where $f$ is a smooth function with compact support on $M$.
\end{lm}
Letting $f=1$ gives the non-existence result analogous to \cite{le, take, xinp},
\begin{pr}
If $n > 4$, then there is no stable $\sigma_2$-critical map of rank $ \geq 2$ from any compact Riemannian manifold $M^m$ $(m \geq 2)$ into $\mathbb{S}^n$.
\end{pr}
\subsection{Full Skyrme model} Let us consider the coupled energy
$$
\mathcal{E}(\varphi)+ \kappa \mathcal{E}_{\sigma_2}(\varphi),
$$
where $\kappa$ is a positive coupling constant. With the same argument as in the previous section, if $m\geq 4$, then there is no non-constant stable $(\mathcal{E}+\kappa\mathcal{E}_{\sigma_2})$-critical map from $\mathbb{S}^m$ to any Riemannian manifold. If $m=3$, a necessary condition for a non-constant $(\mathcal{E}+\kappa\mathcal{E}_{\sigma_2})$-critical map $\varphi:\mathbb{S}^3 \to N$ to be stable is
$$
\kappa\geq \frac{\int_{\mathbb{S}^3}\abs{\mathrm{d} \varphi}^2 \nu_{can}}{2\int_{\mathbb{S}^3}\sigma_2(\varphi^* h)\nu_{can}}
$$
For the identity map of $\mathbb{S}^3$ (of unit radius) this reads $\kappa \geq \tfrac{1}{2}$ and one knows that it is also a sufficient condition \cite{los, man}; see also \cite{slo}.
\section{Infima in homotopy classes}
In this section we point out a global analogue of the results in the previous sections.
\begin{lm} \label{ineq} $(i)$ \ Let $\varphi: (M, g) \to (N^n, \Omega, h)$ be a (smooth) map into an almost K\"ahler manifold with fundamental $2$-form $\Omega$. Then
$$
\abs{\varphi^* \Omega}^2 \leq \sigma_2(\varphi^*h) ,
$$
where the equality is reached if and only if $n=2$.
\noindent $(ii)$ \ Let $\varphi: (M, g) \to (N^n,h)$ be a (smooth) map between Riemannian manifolds. Then
$$
\sigma_2(\varphi^*h) \leq \tfrac{n-1}{2n}\abs{\mathrm{d} \varphi}^4,
$$
where the equality is reached if and only if $\varphi$ is semi-conformal (i.e., the eigenvalues of $\varphi^*h$ are all equal).
\end{lm}
\begin{proof}
$(i)$ Let $\{e_i\}_{i=1,...,m}$ be a (local) orthonormal frame of eigenvectors of $\varphi^*h$. Applying Cauchy inequality $\abs{\varphi^* \Omega}^2 =\sum_{i<j}h(\mathrm{d} \varphi(e_i), J\mathrm{d} \varphi(e_j))^2$ $\leq
\sum_{i<j}\abs{\mathrm{d} \varphi(e_i)}^2\abs{\mathrm{d} \varphi(e_j)}^2=\sum_{i<j}\lambda_i^2 \lambda_j^2
= \sigma_2(\varphi^*h)$, gives us the result.
$(ii)$ This is one of the Newton's inequalities.
\end{proof}
Since, by \cite{wei, whi}, the infimum of the 4-energy in each homotopy class of mappings from (or into) a sphere of dimension greater than 4 is zero, Lemma \ref{ineq} implies the following
\begin{pr}
If $m > 4$, then the infimum of the symplectic Dirichlet energy and of $\sigma_2$-energy in any homotopy class of maps $\mathbb{S}^m \to N$ or $M \to \mathbb{S}^m$ ($M,N$ compact) is zero.
\end{pr}
We include here an elementary proof for the first part of the result, analogous to the Dirichlet energy case \cite{eel, eell}. If a homotopy class of mappings $M \to \mathbb{S}^m$ contains a Riemannian submersion, the proof of the second statement is similar.
\begin{proof}
Let $m \geq 5$, $c>0$ and $\phi_c$ be defined (by suspension) between charts of $\mathbb{S}^m$ as
$$
(\cos s, \sin s \cdot z) \mapsto (\cos \alpha(s), \sin \alpha(s) \cdot z); \quad \alpha(s)=2\arctan(c \tan(\tfrac{s}{2})),
$$
where $0\leq s < \pi$ and $z \in \mathbb{S}^{m-1}$. Notice that this defines indeed a smooth map $\phi_c:\mathbb{S}^m \to \mathbb{S}^m$ (regular at the poles), which has topological degree 1, and is conformal of dilation
$$
\lambda^2=\frac{c(1+\tan^2(\tfrac{s}{2}))}{1+c^2\tan^2(\tfrac{s}{2})},
$$
where we considered $\mathbb{S}^m$ endowed with the canonical metric (we can show as in \cite{eel} that the statement we wish to prove is independent of the choices of metrics on the domain or codomain). Since $m \geq 5$,
\begin{equation*}
\begin{split}
\mathcal{E}_4(\phi_c) &= \frac{\vol(\mathbb{S}^{m-1})}{4}\int_{0}^{\pi} \lambda^4 \sin^{m-1}s \, \mathrm{d} s\\
&=
\frac{\vol(\mathbb{S}^{m-1})}{4}\int_{0}^{\pi}\left(\frac{2c\tan(\tfrac{s}{2})}{1+c^2\tan^2(\tfrac{s}{2})}\right)^4\sin^{m-5}s \, \mathrm{d} s\\
&\leq
\frac{\vol(\mathbb{S}^{m-1})}{4}\int_{0}^{\pi}\left(\frac{2c\tan(\tfrac{s}{2})}{1+c^2\tan^2(\tfrac{s}{2})}\right)^4 \mathrm{d} s=
\frac{\vol(\mathbb{S}^{m-1})\pi c(c^2+4c+1)}{4(c+1)^4},
\end{split}
\end{equation*}
so $\lim_{c\to 0}\mathcal{E}_4(\phi_c)=0$.
Now let $N$ be a compact manifold and $\varphi:\mathbb{S}^m \to N$. Then $\varphi_c = \varphi \circ \phi_c$ is homotopic with $\varphi$. By an elementary (algebraic) property of Hilbert-Schmidt norm,
$$
\abs{\mathrm{d} \varphi_c}^4 \leq \abs{\mathrm{d} \varphi}^4\abs{\mathrm{d} \phi_c}^4,
$$
so we can conclude that $\lim_{c\to 0}\mathcal{E}_4(\varphi_c) = 0$. Combining with Lemma \ref{ineq} allows us to conclude that the infimum of the symplectic Dirichlet energy and of $\sigma_2$-energy in the homotopy class of $\mathbb{S}^m \to N$ is zero.
\end{proof}
\section{Final remarks}
In this note we restricted to the $\sigma_2$ and symplectic Dirichlet energies since they correspond to Lagrangians which are at most
quadratic in first time derivatives (a requirement for any field theory with standard Hamiltonian). Nevertheless the results here should have straightforward extensions to other higher power functionals as $\sigma_p$.
Also we discussed only the sphere case, but we had in mind that the same phenomena should occur not only on product of spheres but also on other symmetric spaces as it was proved for ($p$-)harmonic maps (\cite{ohn, mont, wei}). Most notably it would be interesting to find the stability properties of the symplectic Dirichlet energy for maps defined on a complex projective space. Direct application of the averaging argument suited to $\mathbb{C} P^m$ (\cite{oh}) has failed to provide us with an effective criterion of stability. In this case the use of a different basis of vectors for the averaged Hessian seems to impose (most probably symplectic vectors).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,384 |
{"url":"https:\/\/homework.cpm.org\/category\/CCI_CT\/textbook\/pc3\/chapter\/12\/lesson\/12.3.1\/problem\/12-136","text":"### Home > PC3 > Chapter 12 > Lesson 12.3.1 > Problem12-136\n\n12-136.\n\nThis problem is a checkpoint for working with parametrically-defined functions.\n\n1. Graph $x(t)=t^2+3,$\u00a0$y(t)=2t\u22121\\text{ for }$$\u22123\\le t\\le3$.\n\n2. Rewrite the parametrically-defined function in part (a) in rectangular form.","date":"2021-10-23 19:15:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 3, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5747931599617004, \"perplexity\": 12639.259407395382}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585737.45\/warc\/CC-MAIN-20211023162040-20211023192040-00377.warc.gz\"}"} | null | null |
Isthmohyla calypsa is een kikker uit de familie boomkikkers (Hylidae).
De soort werd voor het eerst wetenschappelijk beschreven door Karen R. Lips in 1996. Oorspronkelijk werd de wetenschappelijke naam Hyla calypsa gebruikt. De soort komt voor in Midden-Amerika en wordt met uitsterven bedreigd, onder meer door chytridiomycose. De boomkikker is momenteel door de IUCN als kritiek geclassificeerd.
Uiterlijke kenmerken
De mannetjes worden ongeveer 2,6 centimeter lang, vrouwtjes worden groter tot 4,1 centimeter. De lichaamskleur is groen met lichtere bruine vlekken. De onderzijde is lichter tot geel. De kikker heeft een karakteristiek uiterlijk door de opvallende stekelige bulten op de bovenzijde van de kop, het lichaam en de ledematen. Deze doen de kikker lijken op een wandelende pluk mos en de uitsteeksels dienen waarschijnlijk als camouflage. De kop valt op door de grote ogen en elliptische pupillen.
De larven worden vrij groot tot 5,4 centimeter, ze hebben een afgeplat lichaam. De staartvin is doorzichtig en het uiteinde is afgerond.
Verspreiding en habitat
Het verspreidingsgebied van deze boomkikker omvat bosgebieden tussen de 1810 en 1920 meter in de zuidelijke delen van de Cordillera de Talamanca. In Costa Rica is de soort bekend van de Pacifische zijde van de bergketen, terwijl Isthmohyla calypsa in Panama aan beide zijden voorkomt. De aantallen zijn sterk afgenomen en in Costa Rica is de soort inmiddels mogelijk uitgestorven. Tot begin jaren negentig was Isthmohyla calypsa algemeen in Las Tablas bij Internationaal park La Amistad met een transectiestudie gemiddeld 12,5 individuen per vierhonderd meter. In 2001 werden gedurende zeven dagen veldonderzoek nog slechts vijf bergboomkikkers waargenomen. In Panama leeft de kikker mogelijk nog rondom Volcán Barú.
Bronvermelding
Hylinae
Dier uit het Neotropisch gebied
IUCN-status kritiek | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,034 |
Q: How to extract date, hours and minutes from datetime I have a pandas dataframe and I want extract date, hours and minutes to a new column in the following way. I also want to add integer variable at the end of the extracted date/hours/minte:
2017-10-25 10:11:12.002000+00:00
-> int_variable = 7
-> 201710251011 + int_variable
-> 2017102510117
What's the best way doing it?
A: Use Series.dt.strftime with join int_variable by convert to str:
df['date'].dt.strftime('%Y%m%d%H%M') + str(int_variable)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,693 |
#### THE LADYBIRD BOOKS FOR GROWN–UPS SERIES
### BALLS
#### _by_
## J.A. Hazeley, N.S.F.W. and J.P. Morris, O.M.G.
#### (Authors of 'Windmill Warriors: Heroes of Crazy Golf')
Publishers: Ladybird Books Ltd., Loughborough
_Printed in England. If wet, Italy._
## Contents
Balls
#### THE ARTISTS
Martin Aitchison
Robert Ayton
Frank Hampson
Frank Humphris
John Theodore Earley Kenney
B. Knight
Jack Matthew
Bernard Robinson
Harry Wingfield
Eric Winter
Gerald Witcomb
Harry Woolley
THE AUTHORS would like to record their gratitude and offer their apologies to the many Ladybird artists whose luminous work formed the glorious wallpaper of countless childhoods. Revisiting it for this book as grown-ups has been a privilege.
_This delightful book is the latest in the series of Ladybird books which have been specially planned to help grown-ups with the world about them._
_As in the other books in this series, the large clear script, the careful choice of words, the frequent repetition and the thoughtful matching of text with pictures all enable grown-ups to think they have taught themselves to cope. The subject of the book will greatly appeal to grown-ups._
Series 999
This is a ball.
A ball can bounce anywhere.
Nobody knows where a ball might go.
But the manager thinks he knows, and the fans think they know and the well–paid pundits think they know, so they speculate about it for most of their waking lives and everybody seems fine with that.
If Ben were only happy when his favourite team did well, he would not be happy very often.
Instead Ben is happy whenever a team he hates does badly.
There are lots and lots of those.
Some sports are more than a game. They have become part of our national identity.
The crack of leather on willow. The gentle clatter of applause as a cherry–red Dukes arcs over the measured sward. Drunken Smurfs messily assembling a forty–foot beer snake.
This is who we are.
"Right, that's it, I'm off," says Carl after the opposing team is awarded a throw–in.
There are still 88 minutes of play left, but Carl knows how these things go.
Mark's wife bought him some tickets to the big match as a treat for working so hard since the twins were born.
Mark met his friends in the pub before–hand.
He cannot remember who won. He cannot remember the score. He cannot remember what game they were playing.
Mark loves sport.
It is the football World Cup.
The famous players from the national team do impossible, beautiful things with the ball in slow–motion in all the advertisements.
Sadly, the team have to play the actual matches at full speed, so they will be going home after the group stage as usual.
A person is considered to be starting an argument about the off–side rule if, when someone asks "what is the off–side rule?" they are nearer to the television than both the person asking "what is the off–side rule?" and the second–last person explaining the off–side rule using empty beer cans and Pringles tubes, unless that person is not actively involved in the argument.
Terry is nearing the end of his prison sentence, so has been allowed out on day release. His children are waiting for him on the platform.
"How do you feel about staying with Aunt Pat today, kids?" says Terry.
"Only Barnsworth are playing a friendly at home," he explains. "And I've not seen them in years."
BBC Radio 5 Live has cut away from coverage of the worst mainland terrorist atrocity in decades because there is an important update from the curling.
Otherwise, regular listeners will complain.
Sometimes when we are angry, we want to shout terrible, spiteful things at strangers in the street, but there are laws. We cannot do what we want, even if it makes us feel better.
In a stadium it is different.
You can even set your terrible, spiteful things to music.
Sam has been a county–level fast–bowler since she was ten. She has a season ticket for her local non–league football club and is putting together a Euro Champions League sweepstake in the office.
"You like the men in shorts, eh?" says absolutely everyone to her, all the time. "Those lovely legs!"
Sam could brain them with a cricket ball so quickly they wouldn't know it was her.
Some animals like sport as much as people do.
And when they retire, they can play sport for fun.
These retired race–horses are enjoying a relaxing round of golf.
Every year, Geoff buys his team's replica kit, their replica away kit, a premium TV satellite sports package and a season ticket that costs more than a second–hand car. They usually play so badly that he does not stay for the end.
"Football is the working man's game," says Geoff. "It's not stuck up. It's not like opera."
It is possible to see a world–class opera for around £15.
It is the quarter finals of the Women's 400kg Welterwidth Berrington, a newly introduced Olympic event in which Team GB have a good chance of a surprise medal placing.
Very few people had heard of the sport before Tuesday, but everybody in the country is suddenly an expert on it now.
Kiran's dad is looking after him for the weekend.
"Don't just sit and watch sport and ignore him," says Kiran's mum to Kiran's dad. "He's got homework to do."
"It'll be fine," says Kiran's dad.
Now he is older, Bob is realistic. He no longer thinks he could, at a push, come on as a maverick substitution from the stands in the 85th minute and put his team level in a vital cup tie.
He is balding and pudgy now and has to stop for breath climbing from the burger stand to his seat.
His fantasies have now moved into management.
For his birthday, Michael always asks for the new Wisden.
He takes the book to his shed at the bottom of the garden and spends his big day looking for mistakes. Then he e–mails Wisden with corrections.
Happy birthday, Michael.
The umpire or referee must know all the laws of the game.
Otherwise players may try and take advantage to help themselves or distract their opponents.
This team has found loopholes in the umpire's knowledge of recent rulings on both bat size and Slade impressions.
Vince has been a fan of Framley North End since he was six. He watches them play every week, through flashes of triumph and troughs of defeat. He is a loyal supporter, and will be until the day he dies and is buried in a Framley North End coffin.
Vince cannot imagine what his life would have been like if he had supported another team.
Probably identical, apart from the colour of the coffin.
Ali's team hate their local rivals. It is something to do with a refereeing decision in 1938.
When the plumber comes round to fix her sink, Ali swaps her team badge tooth–mug for an unmarked glass in case the plumber is a rival fan.
She does not want him sticking her toothbrush anywhere unsavoury.
Many players perform special celebrations when they do well.
This fly–half may have forgotten his kit and been made to do the Six Nations in his vest and pants, but he still celebrates a try by jumping right over the posts.
Brocklyn Fudginelli of the San Domingo Thunderjacks is one of the most famous athletes in the United States of America.
In other countries, sports stars can earn even more by taking up lucrative offers to play in overseas leagues.
Unfortunately for Mr Fudginelli, no–one else on the planet cares about whatever this sport might be that he is doing here.
During the First World War, soldiers stopped fighting to play football in No–Man's–Land.
Hostilities then resumed until the Allies defeated Germany in 1918.
Historians agree that had it not ended so decisively, Germany would almost certainly have won the war on penalties.
"We have walked many miles," pants Darabont.
"Do you have Sky Sports?"
##### MICHAEL JOSEPH
UK | USA | Canada | Ireland | Australia
India | New Zealand | South Africa
Michael Joseph is part of the Penguin Random House group of companies whose addresses can be found at global.penguinrandomhouse.com
First published 2017
Copyright © Jason Hazeley and Joel Morris, 2017
All images copyright © Ladybird Books Ltd, 2017
The moral right of the authors has been asserted
ISBN: 978-1-405-93406-0
# Contents
1. Cover
2. Title Page
3. The Artists
4. Balls
5. Copyright Page
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
1. Cover
2. Table of Contents
3. Begin Reading
| {
"redpajama_set_name": "RedPajamaBook"
} | 1,618 |
Published by Yvette Ulloa on February 22, 2016 February 22, 2016
Yvette Ulloa, the fiery Latina was born in NY and currently lives in California. Yvette is passionate about health/fitness, travelling the world full time through Worldventures Dreamtrips, and teaching others to achieve success!
Yvette Ulloa and her husband Dave Ulloa started their journey with WorldVentures 9 years ago and have lived a life of total fun, freedom and fulfillment. They were a police officer and government employee turned self-made millionaires, lovingly called The Blue Collar Millionaires.
Yvette Ulloa has been through a journey of challenges, however they have turned out to be some of the biggest blessings in her life. She has been happily married for 14 years to the love of her life Dave Ulloa, a former professional basketball player in Australia and self-made entrepreneur. Yvette's philosophies are all about contribution, sharing lessons from her journey with others (see her healing blog), and she is lucky to travel around the world inspiring people with her story. She has spoken to audiences as big as 12,000 in Vegas, South Africa, Hungary, Greece and many other places, where she is highly acclaimed speaker and coach. Yvette has a beautiful love for people which she is able to express daily through her job of empowering people to believe in themselves.
In 2010, Yvette Ulloa and her husband Dave Ulloa achieved one of their life-long dreams to travel the world with Anthony Robbins and become Platinum Partners. They got to experience a whole year of immersion in Scotland, Egypt, Israel, Fiji and get personally mentored by Tony Robbins.
One of Yvette Ulloa's biggest passions is helping women around the world to believe in themselves and have a voice. She will reach out her hand to you to help you climb up to the top of the mountain! And nothing like enjoying the journey and making many pit stops to smell the roses, but also embracing challenges and chaos that come your way. Yvette Ulloa and Bethany Webster led the first ever spa dreamtrip and an empowering weekend for women from all over the country.
A couple of years ago, Dave's dad tragically passed away in a truck accident. In his honor, Dave and Yvette Ulloa built an orphanage in Cuenca Ecuador, La Esperanza which at its peak times housed 65 children in 3 homes. This was one of the first Worldventures Foundation voluntour trips. The Huffington Post recently published an article about the Worldventures Foundation bottle schools we are building in Guatemala, read the article and join us on a future dream voluntour vacation! It's a new way of traveling the world, vacation on purpose!!
This is how Yvette Ulloa became a world traveler!! You can find her sharing her world photos on Instagram and she's always twitter.com/yvetteulloa">tweeting out great tips on traveling! Yvette does not hang much on Pinterest but every once in a while she will pin some cool things!
Yvette Ulloa also has a beautiful healing blog. Half way through her journey with WorldVentures, Yvette was diagnosed with a deadly tumor in her jaw. They were going to do surgery and after three surgeries, $150,000 cost and wiring her jaw shut for nine months, perhaps there would be hope to recovery. Yvette Ulloa chose the alternative route and years later, the tumor has been completely contained and she didn't have to do the surgery. Follow Yvette's Healing Blog for inspiration and her journey. Her healing blog has helped thousands cope with difficult situations in their life through positivity, prayer and alternative routes. See her youtube account for the latest update on her health and her healing! Drop her a note on her Facebook any time, she would love to hear from you and teach you how you too can travel the world!!!
Categories: EcotourismFamily & Kids
Tags: Dave UlloaworldventuresYvette UlloaYvette Ulloa spoiled traveler
Magical Moments in Costa Rica: A Monkey Thief
So we were excited about visiting Costa Rica for the first time! Had heard about the amazing rain forests, aqua blue water and amazing culture. Little did we know that we would encounter a thief - not just a regular thief, but one dressed up as a monkey! Really? Not really, but it was an actual little monkey stealing our food!
Ecotourism in Puerto Vallarta
Ecotourism in Puerto Vallarta Explore the lush rainforests and vast oceans of Puerto Vallarta. Puerto Vallarta is one of the few destinations in the world that offers so many untouched ecological resources. The rainforests that Read more…
Jellyfish in Puerto Vallarta
Snorkeling at Los Arcos in Mismaloya is an absolute must for anyone who enjoys sealife. Algae gardens and coral are surrounded by vibrantly colored schools of fish including Angel fish, Parrot Fish (yes, they look Read more… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,388 |
Manitowish Waters Airport is a town owned public use airport located one mile (1.6 km) south of the central business district of Manitowish Waters, a town in Vilas County, Wisconsin, United States. It is included in the National Plan of Integrated Airport Systems for 2021–2025, which categorized it as a local general aviation facility.
Although most U.S. airports use the same three-letter location identifier for the FAA and IATA, this airport is assigned D25 by the FAA but has no designation from the IATA.
Facilities and aircraft
Manitowish Waters Airport covers an area of 439 acres (177 ha) at an elevation of 1,610 feet (491 m) above mean sea level. It has two runways: 14/32 is 3,498 by 60 feet (1,066 x 18 m) with an asphalt surface and 4/22 is 3,094 by 120 feet (943 x 36 m) with a turf surface.
For the 12-month period ending August 12, 2021, the airport had 6,200 aircraft operations, an average of 17 per day: 97% general aviation and 3% air taxi.
In February 2023, there were 12 aircraft based at this airport: all 12 single-engine.
See also
List of airports in Wisconsin
References
External links
Airport page at Manitowish Waters website
Airports in Wisconsin
Buildings and structures in Vilas County, Wisconsin | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,337 |
Nida Allam (born December 15, 1993) is an American politician, political activist, and data analyst. She currently serves on the Durham County Board of Commissioners, to which she was elected in 2020, making her the first Muslim woman to serve in public office in North Carolina. Allam is one of five women to serve on the Durham County Board of Commissioners; this is the first time the board has consisted entirely of women in its 139-year history.
Since 2018, Allam has served as the Chair of the Durham Mayor's Council for Women. She was elected as the Third Vice-Chair of the North Carolina Democratic Party and served from 2017 to 2021, becoming the first Muslim to serve on the party's executive board. On November 8, 2021, Allam announced that she would be seeking the Democratic Party's nomination for Congress in North Carolina's newly redrawn 4th Congressional District.
Early life and education
Allam was born on December 15, 1993, in Ottawa, Canada. She is the daughter of immigrants, her father is from India and her mother is from Pakistan. Allam has two older sisters. When she was five years old her family moved to Brier Creek, a suburb between Raleigh and Durham in North Carolina, after her father took a job with IBM at Research Triangle Park. When she was six years old the family moved to the nearby town of Cary. She became a naturalized United States citizen as a teenager. Her mother, Iffat Allam, served as the Chair of the Women's Committee at their mosque. Allam and her mother volunteered at local food banks and helped set up homes for single mothers and refugees in the Research Triangle. A devout Muslim, she did not begin wearing the hijab full-time until she was in eighth grade.
Allam graduated from Needham B. Broughton High School, a magnet school in downtown Raleigh, where she was a member of the varsity lacrosse team. As a high school student, she chaired the Triangle Health Fair, a Muslim student-led campaign to partner with local doctors, chiropractors, and dentists to provide free health care to low-income community members.
She graduated from North Carolina State University with a degree in sustainable materials and technology. While at university, she founded the NC State For Bernie Club and became Co-Chair of the Triangle For Bernie Club.
Political career
Allam was inspired to become politically involved after her best friend, Yusor Mohammad Abu-Salha, was one of the three people killed in the 2015 Chapel Hill shooting. She had been a bridesmaid at Abu-Salha's wedding that December. The shooting targeted members of North Carolina's Muslim community. Allam became involved in the grassroots movement and worked as a political director for U.S. Senator Bernie Sanders' 2016 presidential campaign, as well as an organizing director for Justice Cheri Beasley's campaign for the North Carolina Supreme Court. She was the 2016 Political Director in North Carolina, South Carolina, New Jersey and New York for Sanders' presidential campaign.
She is a 2019 alumna of Durham's chapter of the New Leaders Council.
Initial runs for office
Allam decided to run for public office after having worked behind the scenes in the progressive movement and with voter mobilization efforts because she believed there needed to be more progressive candidates representing the diversity of the American people. She was elected as the Third Vice Chair of the North Carolina Democratic Party in January 2017, becoming the first Muslim American to serve on the party's executive council, and was appointed as the Chair of the Durham Mayor's Council for Women in 2018.
As a member of the Mayor's Council for Women, she advised Mayor Steve Schewel on issues pertaining to the rights of women and LGBTQIA community members, especially non-binary and transgender people. As Third Vice Chair of the Democratic Party in North Carolina, she served alongside Second Chair Matt Hughes, former First Chair Aisha Dew, Party Secretary Melvin Williams, and former State Party Chairman Wayne Goodwin. She also served as a delegate at the 2016 Democratic National Convention and the 2020 Democratic National Convention.
Durham County Board of Commissioners
Allam was elected to the Durham County Board of Commissioners in 2020 with endorsements from the Durham Association of Educators, Equality North Carolina, and the People's Alliance PAC. When Allam announced her candidacy for Durham County Commissioner, her family members received Islamophobic hate mail via social media platforms. She was elected to serve alongside Nimasheena Burns, Wendy Jacobs, Heidi Carter, and Brenda Howerton. This was the first time that Durham County has had an all-woman board of commissioners in its 139-year history. Upon her election, she became the first Muslim woman to hold an elected office in North Carolina. She received 39,523 votes in the primary election and 122,947 votes in the general election, finishing ahead of all other candidates. Her election was celebrated by the Council on American–Islamic Relations and Muslim Advocates.
Congressional candidacy
On November 8, 2021, Allam announced that she would be seeking the Democratic Party's nomination for Congress in North Carolina's newly redrawn 4th Congressional District. If she were elected, she would be the third Muslim woman to serve in Congress, after Ilhan Omar and Rashida Tlaib, both of whom endorsed Allam's candidacy. Allam lost the primary to her more moderate opponent by 9 points.
Political views
Allam ran for Durham County Commissioner on a platform centered on addressing economic inequality. Campaign priorities included a $15 minimum wage for county workers, boosting mental health services in schools and investing in businesses run by women and people of color.
She believes that charter schools have increased racial segregation in Durham schools. As a county commissioner, Allam stated she plans to increase the minimum wage of Durham Public Schools classified staff to U.S. $15 an hour and enact property tax assistance programs. She has stated that evictions and lack of affordable housing opportunities are also a crisis in the county, and referenced the issue of gentrification misplacing Black families from their homes in Durham's historical African-American neighborhoods. Allam has also called for more funding and community investment into Durham Public Schools and Durham Technical Community College, saying that education is tied to economic and racial justice issues. Allam supports organized labor unions. She blames the North Carolina General Assembly for inadequate funding for public schools.
Allam has been criticized for her past statements and tweets in regards to Israel that some have seen as Anti-Zionist or anti-Semitic. In 2021, Allam apologized to the Jewish community for her past statements and committed to "a movement for justice and peace, in which anti-Semitism must have no home." While Allam received endorsements in her race from figures like Ilhan Omar, who have been in the past accused of antisemitism, Valerie Foushee, her primary opponent, began to receive funds and assistance from pro-Israel groups such as AIPAC and Sam Bankman-Fried's Protect Our Future PAC, prompting allegations that Foushee's campaign had succeeded primarily due to support from dark money as the race became "the most expensive Democratic congressional primary in North Carolina history".
Personal life
She lives in Durham with her husband, Towqir Aziz, and two dogs named Otis and Nala. She and Aziz met in a Muslim Sunday school. Allam is a member of the Women's Islamic Initiative in Spirituality & Equality. She is a citizen of the United States and Canada, and also holds Pakistani citizenship by descent.
In April 2022, Allam announced that she was pregnant. Allam had been pregnant in 2021, but had an abortion due to medical issues.
Electoral History
References
1993 births
21st-century American politicians
21st-century American women scientists
Bernie Sanders 2016 presidential campaign
American left-wing activists
American people of Indian descent
American people of Pakistani descent
American Muslim activists
Canadian emigrants to the United States
Canadian people of Indian descent
Canadian people of Pakistani descent
Candidates in the 2022 United States House of Representatives elections
County commissioners in North Carolina
Indian Muslim activists
Living people
Needham B. Broughton High School alumni
North Carolina Democrats
North Carolina State University alumni
Pakistani Muslim activists
People from Durham County, North Carolina
Politicians from Ottawa
Scientists from North Carolina
Women data scientists
Women in North Carolina politics
People from Cary, North Carolina | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,020 |
New Text Confucianism () is a school of thought in Confucianism that was based on Confucian classics recompiled in the early Han dynasty by Confucians who survived the burning of books and burying of scholars during the Qin dynasty. The survivors wrote the classics in the contemporary characters of their time, and these texts were later dubbed as "New Text". New Text school attained prominence in the Western Han dynasty and became the official interpretation for Confucianism, which was adopted as the official ideology by Emperor Wu of Han.
Represented by Confucians such as Dong Zhongshu, this school advocated a holistic interpretation of Confucian classics and viewed Confucius as a charismatic, visionary prophet, a sage who deserved the Mandate of Heaven but did not attain kingship due to circumstances. The school competed with Old Text Confucianism in the later Han dynasty and its dominance waned as the latter became the new orthodoxy. The school fell into obscurity during the chaotic period after the fall of the Han dynasty and remained so until late Ming dynasty in the 17th century.
The school was reinvigorated by a group of scholars who were dissatisfied with the popular Neo-Confucianism at the time in the late Ming dynasty. The movement gained momentum in eighteenth century with the rise of the Changzhou School of Thought. It became a major intellectual trend in Chinese philology and political ideology. As formulated by B. Elman, it was intended to offer a solution to the crisis of confidence between the Chinese state and its gentry constituency in the transition from the Qianlong era to the Jiaqing era in the Qing dynasty.
Ideological features
The New Text school focuses on the philosophical and metaphysical meaning of the Confucian texts, using an apocryphal and prognostic approach to the reading. This was criticised by the rival Old Text school as superstitious as the Old Text School favoured reading them from a historical perspective. The New Text school viewed Confucius as an uncrowned king (su wang) while the Old Text school viewed him as a teacher of ancient knowledge.
Qing dynasty
The scholar Zhuang Cunyu (1718-1788), a secretary to the Qianlong emperor, was the pioneer of the Changzhou New Text school revival. Dissatisfied with his apolitical colleagues in the Han learning movement, Zhuang published studies based on the New Texts aiming to interpret the writings of Confucius' as prescriptions on government, especially with regard to the corruption and lawlessness of his contemporaries. Using the evidential research methods of Han learning, this school of thought sought to interpret moral and political lessons from the Confucian classics, so as to create a legitimate framework to combat the political corruption of the time.
Downplaying the role of Mencius, as a sign of opposition to Confucianism during the Song dynasty.
Much attention paid to the Gongyang Zhuan as the text revealing the true wisdom of Confucius. The reading of Chunqiu therein had prophetic overtones, which the opponents of the school condemned as superstition.
High esteem of the work of He Xiu (何休), the Han dynasty author of the commentary to the Gongyang Zhuan (春秋公羊解詁).
Attacks on the Zuo Zhuan as a purported Liu Xin's fabrication intended to overturn the Gongyang Zhuan.
By far the most important feature of the New Text Confucianism as political movement was advocating reform (Zhuang Cunyu, Liu Fenglu), drawing from the Gongyang legalist-style ideology of "weighing the circumstances". The reforms were seen necessary since the Heshen-related crisis of power. According to Wei Yuan 魏源 (1794-1857),
"The ancients had what pertained to the ancients. To force the ancients upon the moderns is to misrepresent the moderns. To use the moderns as the standard for the ancients is to misrepresent the ancients. If one misrepresents the present, then there can be no way to order [the contemporary world]... If one read the [medical] works of the Yellow Emperor and Shennong, and used them to kill people, one would be labelled a mediocre doctor. If one read the works of the Duke of Zhou and Confucius and used them to harm the empire, would one not be labelled a mediocre Confucian? Not only would such [incompetence] bring no benefit to any particular age, but in addition it would cause people no longer to believe in the Way of the sages."
Scholarly genealogies
Zhuang Cunyu (1719—1788), grandfather of
Liu Fenglu, mentor of
Wei Yuan (1794—1857) and Gong Zizhen (1792—1841);
Kang Youwei
See also
Gongyang Zhuan
Old Texts
Literature
Elman, Benjamin A. Classicism, politics, and kingship: the Chang-chou school of New Text Confucianism in late imperial China. Berkeley: University of California Press, 1990.
References
Chinese Classical Studies
Confucian schools of thought | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,657 |
Iunia Torquata (* vor 10 v. Chr.; † 55 n. Chr.) war eine Vestalin aus der Familie der Junier. Sie war die Tochter eines Gaius Iunius Silanus (Torquatus?) und Schwester des Konsuls des Jahres 10 n. Chr., Gaius Iunius Silanus, der 22 n. Chr. als Prokonsul von Asia wegen laesa maiestas angeklagt wurde. Torquata setzte sich dafür ein, dass er um des Ansehens der Familie willen nicht auf die unwirtliche Kykladeninsel Gyaros, sondern auf die benachbarte, größere Insel Kythnos verbannt wurde. Von den Bewohnern einer weiteren Kykladeninsel, Tenos, erhielt Iunia Torquata eine nur unvollständig erhaltene Ehreninschrift, aus der der Grund für diese Ehrung nicht zu erschließen ist. Zwei Inschriften bezeugen, dass Iunia Torquata noch im Alter von 64 Jahren Vestalin war und später zur Vestalis maxima aufstieg.
Quellen
PIR2 I 866
Literatur
Einzelnachweise
Torquata, Iunia
Vestalin
Geboren im 1. Jahrhundert v. Chr.
Gestorben 55
Frau | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,671 |
(Hyogo, 25. lipnja 1940. – 2. lipnja 1978.) japanski je nogometaš.
Klupska karijera
Igrao je za Mitsubishi Motors.
Reprezentativna karijera
Za japansku reprezentaciju igrao je od 1961. do 1965. godine. Odigrao je 12 utakmica postigavši 4 pogotka.
S japanskom reprezentacijom je igrao na Olimpijskim igrama 1964.
Statistika
Vanjske poveznice
National Football Teams
Japan National Football Team Database
Japanski nogometaši | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,898 |
<?php
class Application_Service_Translator_Category {
private $errors = array();
// {{{ translate(Application_Model_Category $category, array $post): public void
public function translate(Application_Model_Category &$category, array $post) {
$category->name = $post['name'];
return true;
}
// }}}
// {{{ getErrors()
public function getErrors() {
return $this->errors;
}
// }}}
}
?>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,777 |
Q: How to assign controller to iFrame in Ionic App? How would i achieve controller assigning to iframe places in an ionic app?
should i use ng-controller attribute on a div in iframe or will contoller of parent pafe, i.e. page that contains iframe be used from within iframe?
Edit:
To set a context,
Case1: I want to bind a property within iFrame whose src is a local html page, which is fetching some data and I want to set dynamic URL or set some property (dynamically) which is bound to controller.
Case2: Suppose iFrame is rendering some local html page which has tag, now I want to call controller function on click of that anchor, how do I call that?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,392 |
Saying g'day, kia ora to bubble boost
Charlotte Heck-Parsch & Felicity Emmett
Economist & Senior Economist, ANZ
News of a trans-Tasman travel bubble commencing on April 19 has sparked hope for Australians and New Zealanders eager to travel abroad and should lift confidence in the tourism outlook.
From a solely economic point of view, the opening might have a small negative impact on Australian trade, but the confidence boost of a return to some semblance of travel normality should provide some offset.
"The immediate uptake by Australians and New Zealanders is difficult to gauge, given pent-up demand, but also some reluctance to fly given ongoing health and cancellation worries."
A travel capacity between New Zealand and Australia of 80 per cent compared to pre-pandemic levels would imply a net negative impact of $A1 billion for Australia, equalling about 0.05 per cent of annual gross domestic product (GDP), although there is a great deal of uncertainty around this.
Given each other's position as the only travel destination and some pent-up demand, there could well be higher demand for holidays in New Zealand (and vice versa) such that airlines might upwardly adjust their capacities over time.
Since Australians spend more money overseas than New Zealanders, such a scenario could magnify the gap between travel imports and exports and lead to a larger negative impact for Australia's GDP.
Nevertheless, this impact would still be comparatively small on an aggregate level.
The news is particularly welcome across Australia given the apparent delays in the vaccination rollout, and has sparked hopes of more jobs for Australia's airline industry, where the trans-Tasman route is one of the busiest in terms of travel volume.
The immediate uptake by Australians and New Zealanders is difficult to gauge, given pent-up demand, but also some reluctance to fly given ongoing health and cancellation worries.
Already, airlines have responded with increased services. Qantas announced an initial lift in capacity from 3 per cent pre-COVID-19 capacity to 83 per cent, with Qantas and Jetstar together offering up to 122 return flights per week to New Zealand.
Air New Zealand will run about 210 flights a week, andwill bring 330 staff back from furlough to prepare for the restart of trans-Tasman travel.
There are also suggestions other countries could soon open up to Australian travellers. Australia's Federal Tourism Minister, Dan Tehan, has suggested Singapore could be next, with Japan and South Korea other potential destinations in the future.
There are around 600,000 New Zealanders living in Australia and around 75,000 Australians living in New Zealand. Many of these people will be keen to visit families and friends across the ditch despite the uncertainties.
Still, the economic contribution of those visiting family and friends is smaller compared to the average tourist. A New Zealander who travels to Australia to visit relatives or friends spends on average two thirds of what a tourist on holiday would spend.
In pre-pandemic times, New Zealand was Australia's second most popular international holiday destination, after Indonesia.
Australians spent more money on New Zealand holidays than New Zealanders spent in Australia. In 2019, New Zealanders spent $A2.5 billion on holiday in Australia, and Australians spent $A3.8 billion in New Zealand, which resulted in a negative net effect trade impact for Australia of $A1.3 billion.
Now, with both countries being each other's only quarantine-free destination, things might look even more rosy for the two neighbouring countries and the overall spending for New Zealanders as well as Australians in their neighbouring countries could easily surpass 2019 levels.
In 2018, Australians spent $A45.4 billion in overall overseas travel, whereas New Zealanders spent $A6.1 billion. There is a much bigger potential for Australian holiday spending in New Zealand than vice versa. This would mean a more negative net trade effect than in 2019.
In terms of the broader impact, there will be a lift in airline jobs, but there could also be some substitution of domestic spending for NZ holidays. Australians could well swap holidays on Fraser Island for a ski trip to Queenstown.
Spending may drop for both domestic hospitality (cafes and restaurants) and goods consumption (which retail data show has been a big beneficiary of closed international borders). However, this substitution impact could be quite small at least initially.
There are still deterrents to travel as new outbreaks may see borders shut any moment. As such, some level of hesitance will likely remain, particularly for those not travelling to see friends and family.
The possibility of having to quarantine – and possibly finding out mid-flight – might make some more comfortable holidaying in their home country. Moreover, Australians burned by state border closures and the resulting holiday cancellations may be wary of booking their NZ holiday only to have it cancelled because of another outbreak.
However, with the vaccine roll-out – although delayed – on its way, outbreaks and lockdowns will become somewhat less frequent and encourage the uptake of travel opportunities. Moreover, some travellers in both countries have strong incentives to use this long-awaited travel opportunity.
Ultimately, the trans-Tasman bubble is a win-win for travel-loving Australians and New Zealanders.
Charlotte Heck-Parsch is an Economist and Felicity Emmett is a Senior Economist at ANZ
This story is an edited version of an ANZ Research note and was originally published on ANZ's Institutional website. You can read the original note HERE.
Business Finance COVID-19 Economy
Kiwi data show a tentative return to a - post-COVID - normal
Sharon Zollner | Chief Economist New Zealand, ANZ
New Zealanders may be avoiding parks still but life is beginning to return to some sort of normal.
Coronavirus grounds airport spend
Travel bans and coronavirus-conscious travellers are hitting tourism in Australia, according to airport spend data. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,055 |
\section{Appendix}\label{appendix}
\section{Introduction}
Existing vehicle coordination methods, which are designed primarily for human drivers, tend to fall short in leveraging the increased sensitivity and precision of autonomous vehicles (AVs). With the progress of self-driving technologies, the bottleneck of roadway efficiency will no longer be attributed to drivers but rather to the automation scheme underpinning the coordination of AVs' actions. A crucial challenge for these schemes lies in detecting and reasoning about uncertainties in the operating environment. In urban scenarios, uncertainty arises predominantly\footnote{Other sources of uncertainty could stem from vehicle perception and trajectory tracking error due to road and weather conditions.} from human-driven vehicles (HVs) as their intention could exhibit a stochastic and oftentimes risky behavior. Unlike streets/highways with well-delimited lanes, intersections typically lack clear marking, thereby creating ``conflict zones'' with elevated potential for crashes. In fact, as reported in~\cite{grembek2018introducing}, 40\% of all crashes and 20\% of fatalities happen to occur at intersections.
\iffalse
\begin{figure}[!t]
\centering \vspace{2mm}
\includegraphics[width=0.8\linewidth]{Figures/intersection_colide_points1.pdf}
\caption{The set of interaction points (collision points) between agents at an intersection.}
\label{fig:colide-points}\vspace{-10pt}
\end{figure}
\fi
\begin{figure*}[t!]
\begin{subfigure}{.254\textwidth}
\centering
\includegraphics[trim={2.3cm 0 2.3cm 0},clip, width=\textwidth]{Figures/intersection_colide_points1.pdf}
\caption{}
\label{fig:colide-points}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/PFT-1.pdf
\caption{}\label{fig:test2}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/PFT-2.pdf
\caption{}\label{fig:pft}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/PFT-3.pdf}
\caption{}\label{fig:intent}
\end{subfigure}
\caption{(a) The set of interaction points (collision points) between agents at an intersection.
(b) Multiple trajectory generation for the same maneuver. (c) PFT representation of a maneuver with tracking error. (d) Intent recognition output for an HV.}
\end{figure*}
These uncertainties can be mitigated by a \textit{risk-aware Intelligent Intersection} system in which AVs' actions are judiciously planned and coordinated in a centralized or decentralized fashion. While coordination can be achieved either way, the former approach is less prone to communication overheads and packet loss, does not suffer from synchronization issues and offers enhanced system-wide controllability, hence the focus of the present work on the centralized scheme. In line with this rationale, Dresner et al. \cite{v2iintersect} developed a protocol for multi-vehicle coordination via a centralized controller relying on {\em Vehicle-to-Intersection} (V2I) communication network~\cite{v2i}. The protocol employs a reservation mechanism wherein AVs declare their destinations to the controller and wait until permission is granted. The controller provides each vehicle with trajectory details, modeled as a path on an abstracted grid map, as well as destination speed. The work is extended in~\cite{au2010motion} with a refined trajectory representation considering vehicle kinematics. To avoid collisions, a grid-based collision map is constructed with inflated grid cells as a buffer. However, tuning cell size heuristically yields more conservative behavior when the cell is large and deteriorated performance when small, with no clear relationship to the actual probability of collision. Furthermore, the coordination mechanisms in~\cite{au2010motion,v2iintersect} rest on the {\em First-come-first-serve} (FCFS) principle, which \textit{lacks guarantees} on the global optimality of the attained objective value (e.g., maximum throughput).
Different from existing centralized planners for optimizing intersection management (e.g., \cite{au2010motion,v2iintersect,zhao2021bilevel, ahn2016semi}), we develop a \textit{chance-constrained model} which alongside optimal control \textit{ensures} that collision probability remains within prescribed limits despite the imposed uncertainty. In planning under uncertainty, the Multi-agent Markov Decision Process (MMDP), which extends the classical MDP \cite{howard1960dynamic} to multiple agents, is a well-established mathematical formalism~\cite{boutilier1999sequential}. A special class of MDPs with non-negative utilities is known as the Stochastic Shortest Path (SSP) \cite{bertsekas1991analysis} (a.k.a. Stochastic Longest Path). Notoriously, MMDPs suffer from an exponential joint action space and a state space that is typically exponential in the number of agents. Notable research efforts have been directed towards exploiting agents' interactions to facilitate the problem's tractability (see, e.g., \cite{becker2004decentralized,spaan2008local}). Of particular interest, we consider MMDPs with {\em local interactions} \cite{melo2011decentralized,scharpff2016solving}, where agents have to coordinate their actions \textit{only at certain} interaction zones. We explore such an interaction scheme within an SSP framework with additional constraints that capture the probability of failure (e.g., collision) under several risk criteria. In particular, as exemplified in Fig.~\ref{fig:colide-points}, the studied intersection is modelled as a collection of finite ``interaction'' points where vehicles (i.e., agents) are likely to collide. However, contrary to prior methods which encode failure as a negative penalty \cite{scharpff2016solving,melo2011decentralized}, we bound the probability of collision by a preset threshold. This allows for a more versatile representation of safety requirements and has been adopted recently in the literature for the single AV scenario \cite{huang2018hybrid,huang2019online}. We remark that the solution techniques for MMDPs provided in \cite{melo2011decentralized,scharpff2016solving} are not amenable to these newly introduced constraints.
To further the design of safe and efficient traffic management programs, the present study proposes a risk-aware Intelligent Intersection system for AVs and HVs that, in spite of possible uncertainties, maximizes intersection's throughput without infringing the desired risk tolerance level. More concretely, the contributions and roadmap of this paper can be summarized as follows:
\begin{enumerate}
\item In Sec.~\ref{sec:system}, we lay out the architecture of the proposed Intelligent Intersection framework where AVs are proctored by a centralized controller whereas HVs follow traffic light signals. As illustrated in Figs.~\ref{fig:test2} and~\ref{fig:intent}, the system is augmented with a \textit{probabilistic intent detector} as well as \textit{robust motion model generator} for HVs, allowing to cater for real-world nuances (e.g., imperfect trajectory estimation, ambiguity in drivers' decisions).
\item In Sec.~\ref{sec:ccssp}, we formulate a novel Multi-agent Chance-Constrained Stochastic Shortest Path (MCC-SSP) model with local interactions and devise \textit{an exact solution method} based on Integer Linear Programming (ILP). Through rigorous analytical scrutiny, the probabilistic constraints in MCC-SSP are proved reducible to equivalent linear ones in ILP (Theorem~\ref{lem1}). More importantly, the ILP formulation features \textit{polynomial number} of variables and constraints when the number of agents per interaction is small, thereby \textit{improving upon the state-of-the-art} designed for the single agent case~\cite{alyassi2021dual, sungkweon2021ccssp}.
\item Drawing on MCC-SSP formalism, Sec.~\ref{sec:model} develops a conditional planner that enables AVs to react to potential contingencies (e.g., an unexpected maneuver from an HV). The planner supplies AVs with {\em safe} (w.r.t. permissible risk limit) contingency plans -- a maneuver for every possible scenario. A hybrid risk calculation method is employed, permitting the computations to be carried out offline in most cases. Subsequently, Sec.~\ref{sec:traj_optimization} models the reference trajectories via multi-variate Gaussian processes known as Probabilistic Flow Tubes (PFT)~\cite{pft} (see Fig.~\ref{fig:pft}) and presents a computationally efficient means of estimating their collision probabilities.
\item Lastly, Sec.~\ref{sec:experiment} validates the effectiveness and practicality of the proposed risk-aware intelligent intersection system through a series of simulations. Specifically, taking the classical grid problem as a case study we first demonstrate the \textit{scalability} of the introduced ILP formulation against the number of agents and horizons as well as its \textit{invariance} to the number of states. Next, we investigate the proposed planner's computational feasibility and contrast its performance with that of common approaches: FCFS and standard signalized scheme. As simulations indicate, the featured planner outperforms the two benchmarks by up to a factor of 2 in maximizing the intersection's throughput and supports rapid planning for multiple horizons ($> 1Hz$).
\end{enumerate}
\section{Related Work}\label{sec:related}
Parallel to the aforementioned centralized planners, a separate line of research has been devoted to developing decentralized intersection management schemes resting on {\em Vehicle-to-Vehicle} (V2V) communication. For instance, the works in~\cite{carlino2013auction, bashiri2017platoon} present game-theoretic decentralized approaches in the context of platooning so as to optimize traffic flow under several scenarios, including intersections. Chandra et.al.~\cite{chandra2022gameplan} propose a game-theoretic approach for unsignaled intersections, where priory is defined based on the driver's behavior (aggressive having higher priority).
Zhang et.al.~\cite{zhang2021priority} present a decentralized priority-based intersection system for cooperative AVs. The system incorporates three levels of planning; the first level is based on FCFS for vehicles at the intersection zone, the second considers AVs followed by emergency AVs arriving at the intersection (far zone) by scheduling their arrival time accordingly, and the third level is for the lowest priorities. However, unlike their centralized counterparts, decentralized schemes could inflict diminished system-wide controllability and efficiency, let alone communication overheads. In this work, we focus on the centralized case, with V2I communication, allowing the system to achieve better overall global optimality.
An essential component of autonomous vehicle planning is uncertainty estimation in the environment. In particular, we are interested in tactical-decision making in the presence of uncertainty in HVs' intentions \cite{bandyopadhyay2013intention} and tracking error, where AVs may not be able to follow precisely a reference trajectory, mainly due to control uncertainty. Also, AVs from different vendors may run different controllers, hence have trajectory variations from the same reference trajectory. To cope with uncertainty in AVs' tactical decision-making, several works in the literature model the problem as Markov decision process variants. Brechtel et al. \cite{brechtel2014probabilistic} model the decision-making problem as a continuous state partially observable MDP (POMDP), where observation stochasticity resembles perception noise (e.g., LiDAR point cloud noise). Hong et al. \cite{sungkweon2021ccssp} model intention recognition of human-driven vehicles as {\em hidden} state variables in POMDP.
Besides optimizing an objective function in POMDPs, \cite{huang2018hybrid,huang2019online} consider an additional chance constraint that bounds the collision probability below a safety threshold.
Arguably, with a properly modeled state space, as we illustrate in this work, the driving problem can be modeled as a fully observable chance-constrained MDP, which is more scalable than the POMDP counterpart.
MDP \cite{howard1960dynamic} is a widely used model for planning under uncertainty.
Moreover, the problem has a dual linear programming (LP) formulation \cite{d1963probabilistic} which solves the problem as a minimum cost flow problem also known as the Stochastic Shortest Path (SSP) problem.
A special class of MDP optimizes an objective function while also bounding the probability of constraint violations is often called {\em chance-constrained} MDP (CC-MDP) \cite{dolgov2003approximating} which is an NP-Hard problem. To reduce the problem, an approximation of the chance constraint using Markov's inequality was proposed by \cite{dolgov2003approximating}, effectively converting the problem into an MDP with a secondary {\em cost function}, called constrained MDP (C-MDP). Another approach \cite{de2017bounding} applies Hoeffding's inequality to improve the approximation. Both methods provide conservative policies with respect to safety thresholds. Exact methods for solving CC-MDP rely on those used for the partially observable MDPs (CC-POMDP) \cite{santana2016rao,khonji2019approximability}. However, even for a single agent, such methods suffer from scalability. They require full history enumeration in the worst case, which makes the solution space exponentially large with respect to the planning horizon \cite{sungkweon2021ccssp}. To the best of our knowledge, our technique, even for the single-agent case, is the first {\em exact} method for solving CC-MDPs that does not require history enumeration. Also, our method extends the approximate method \cite{de2017bounding}, which considers independent agents with a shared risk budget, to situations where agents can interact at specific locations.
\section{Intelligent Intersection System} \label{sec:system}
The intersection system unders tudy considers a risk-aware architecture \cite{khonji2020risk} for both AVs and HVs, with the former following a centralized controller and HVs adhering to traffic signals. We adopt the protocol specified in \cite{v2iintersect} for AVs, where each vehicle declares a target destination to the coordinator via V2I communication. The coordinator is a computing unit installed on a road side unit equipped with communication and sensing modalities. Given vehicles' target destinations, the coordinator optimizes the intersection's overall performance and transmits a trajectory specification to each vehicle. The coordinator also predicts the intention of HVs and incorporates the uncertainty in the AV plans. Overall, the proposed system comprises six modules, as depicted in Fig. \ref{fig:high-level}, which are elaborated in the paragraphs to follow.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{Figures/arch2.pdf}
\caption{High-level architecture of the featured Intelligent Intersection system.} \label{fig:high-level}
\end{figure}
\vspace{2pt}
\noindent{\textit{\bfseries Probabilistic Perception}:}
Perception uncertainty can be specified as a distribution in which the object is located. One way to obtain a distribution over the object's shape and location is through point clouds. The points are segmented according to the face-plane of the obstacle they belong to; points belonging to the same face can be used to perform a Bayesian linear regression on the plane's parameters. Given a Gaussian prior on the face-plane parameters and under reasonable assumptions about the underlying noise-generation process, the posterior on the face-plane parameters will also be Gaussian \cite{axelrod2018provably}.
A recent work in \cite{senanayake2018automorphing} presents a fast algorithm that generates a probabilistic occupancy model for dynamic obstacles in the scene with few sparse LIDAR measurements. Typically, the occupancy states exhibit highly nonlinear patterns that cannot be captured with a simple linear classification model. Therefore, deep learning models and kernel-based models can be considered as potential candidates.
This module obtains a two-dimensional top-down representation of all vehicles in the scene along with uncertainty in their locations. For AVs, accurate location and pose (along with Gaussian uncertainty in location, which could be estimated via Kalman filter) can be transmitted through V2I~\cite{hartenstein2009vanet}. Moreover, infrastructure sensors, such as cameras, can obtain HVs poses and validate AV pose estimations via segmentation methods \cite{khan2013image}.
\vspace{2pt}
\noindent{\textit{\bfseries Motion Model Generator}:} The system continuously elicits trajectory traversal data of both AVs and HVs to learn probabilistic motion models in an online (real-time) fashion. These models are helpful in two ways: (i) for AVs, the learned motion models capture the uncertainty in the control system (ii) for HVs, the learned motion models represent how humans drive according to different driving patterns associated with tracking uncertainties.
To encode uncertainty in trajectory traversal, we leverage a probabilistic representation, called Probabilistic Flow Tube (PFT) \cite{pft}, that encodes a nominal trajectory and uncertainties for a given driving pattern.
The PFT is learned from a set of demonstrating trajectories, where each trajectory is composed of a sequence of positions. The output is a Gaussian process, a sequence of means and covariances that represents a nominal trajectory and uncertainties in traversals. We refer to \cite{pft} for further details on PFTs. The generation of the initial trajectory for AVs is described in Sec.~\ref{sec:traj_optimization}.
\iffalse
\setlength{\textfloatsep}{0pt}
\begin{algorithm}[t!]
\SetAlgoLined
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{Demonstrating trajectories $\{D_i\}_{i=1}^N$}
\Output{Learned motion model $M$}
\For{$t=1$ to $T$}{
$\mathbf{x}_t \gets \{(x_i^t, y_i^t)\}_{i=1}^N$,
$\mu_t = \textrm{Mean}(\mathbf{x}_t)$,
$\Sigma_t = \textrm{Cov}(\mathbf{x}_t)$
}
$M = \{(\mu_t, \Sigma_t)\}_{t=1}^T$\\
\textbf{return} $M$
\caption{Generate PFT}
\label{alg:pft}
\end{algorithm}
\fi
Existing works \cite{pft,huang2019online} assume the demonstrating trajectories are pre-labeled with maneuver types and learn a PFT for each maneuver type, which entails extra labeling effort and confines to a fixed taxonomy of maneuvers. Instead, we leverage an unsupervised clustering algorithm to create distinct clusters \cite{sung2012trajectory} given demonstrating trajectories representing different driving patterns and learn a distinct PFT for each cluster. As we continuously acquire more trajectories from the perception system, we update the clusters when the variance of a PFT is excessively high. We visualize an example of PFT and its demonstrating trajectories in Figs.~\ref{fig:pft} and \ref{fig:test2}, respectively. Since the demonstrating trajectories are collected with different sizes, we use dynamic time warping (DTW) \cite{myers1980performance} to ensure equally sized trajectories prior to applying the PFT generator.
Then, PFTs are computed by sampling multiple trajectories, each obtained by following a nominal trajectory using a PID controller with slightly perturbed parameters to emulate manufacturer-specific controller deviations.
\iffalse
\begin{figure}[!t]
\centering
\includegraphics[width=.85\linewidth]{Figures/pft.png}
\caption{PFTs generated using PID sampled trajectories.}
\label{fig:pft_pid}
\end{figure}
\fi
\vspace{2pt}
\noindent{\textit{\bfseries HV Management}:}
Considering that the intersection system lacks communication means with HVs, we assume HVs follow traffic signals and the employed intent recognition module predicts the trajectory of HVs to generate safe plans for AVs. For instance, if the traffic light is green on a given road, only HVs from that road may enter the intersection immediately. In contrast, AVs from any road may enter the intersection upon the system's command.
\noindent{\textit{\bfseries Intent Recognizer}:} This module predicts human driver's intention as a distribution over a discrete set of candidate PFTs, thereby allowing AVs to maintain contingency plans for all potential scenarios. This is a departure from current approaches in the literature, where an AV is often provided with a single trajectory, whereas we provide a response trajectory \textit{for each potential scenario}. Contingency planning enables fast response to risky scenarios. The set of candidate PFTs are elected based on road policies that determine the set of legal maneuvers, each with its corresponding PFT.
Intent recognition, in its own field, has been extensively studied and there are more sophisticated learning-based methods, such as \cite{huang2019uncertainty,ma2019trafficpredict}, that can produce accurate motion predictions conditioned on more detailed priors.
Given the tracked trajectory points of HVs, the module predicts the set of PFTs that the human driver is likely to follow through Bayesian filtering.
First, we supplement the tracked vehicle trajectory with more data points based on polynomial fit,
then extend it into future horizons based on the polynomial coefficients to obtain an augmented trajectory. The augmentation allows us to provide sufficient data in case of short observation tracks.
Second, we compute the observation probability of the augmented trajectory conditioned on each candidate PFT. The observation probability is multiplied by a prior distribution over PFT probabilities to obtain a posterior distribution of a candidate PFT the driver intends to follow.
The result is a distribution over a set of candidate PFTs, which allows us to leverage a risk-aware planner for each contingency, as explained in the following sections.
Given a prior of all possible PFTs, the module computes the likelihood of the tracked trajectory against each PFT and updates the posterior distribution over discrete PFT choices. The future trajectory points are then predicted given the nominal trajectory and uncertainties associated with each PFT.
\vspace{2pt}
\noindent{\textit{\bfseries Risk-Aware Planner}:}
The planner takes as input the perception data, including observed AV states, PFT motion models learned for AVs, intentions of HVs, and outputs contingency plans for each AV in the scene. In Sec.~\ref{sec:model}, we formulate the intersection problem as an MCC-SSP and devise \textit{an exact} ILP-based solution approach.
\vspace{2pt}
\noindent{\textit{\bfseries Risk Estimator}:}
The online nature of the system and the high number of queries from the planner call for a computationally lightweight risk calculation method. Since PFTs are multi-dimensional Gaussian processes, collision at a given time amounts to solving multi-variate integrations.
A straightforward alternative to the exact risk computation method is a high-resolution Monte Carlo sampling. Moreover, in Sec.~\ref{subsec:risk_precomp} we show how to compute the risk offline for most cases.
\iffalse
\color{blue}
\noindent{\bf Pedestrian Management.}
The hybrid intersection of AVs and HVs presents a unique problem for pedestrians crossing the intersection, which requires the system to accurately predict the future motions of the pedestrians. The pedestrian prediction problem has been widely studied in the recent years using learning-based methods~\cite{alahi2016social,mohamed2020social}.
In this work, we divide the problem into three cases: 1) a pedestrian crosses an entering lane while the vehicle light signal is green, 2) a pedestrian crosses an entering lane while the vehicle light signal is red, and 3) a pedestrian crosses an exiting lane.
In the first case, HVs are exiting the lanes, so the pedestrians are not allowed to cross (red pedestrian light). In the second case, only AVs are exiting the lanes, so a prediction system is used to predict the movement of pedestrians, and the road side unit will plan for the AVs accordingly by adding a collision point for pedestrians (flashing amber pedestrian light). In the third case, a prediction system is needless due to the delay between vehicles entering the intersection and reaching the exit lane. Therefore, pedestrians are allowed to cross when HVs signal is red, and AVs are also prevented from the given exit lane (green pedestrian light). To improve the efficiency, the pedestrian signal at exiting lanes is activated only between HVs signal light change, when pedestrians are available.
\color{black}
\fi
\section{Multi-Agent Chance-Constrained Stochastic Shortest Path} \label{sec:ccssp}
In this section, we formally define the fixed-horizon Multi-agent Chance-Constrained Stochastic Shortest Path (MCC-SSP) model.\footnote{Although the paper's results extend to multi-agent CC-MDP, we emphasize the stochastic shortest path variant as we believe that non-negative utility values fully capture relevant applications. We also want to discourage using negative values in the objective to penalize risk since chance constraints provide a more natural approach to risk representation.}
\subsection{Problem Definition}
\subsubsection{Agent Model}
We are given a set of agents $\mathcal X$ (e.g., vehicles), each following a Markov decision process model
$M^v=\langle \mathcal S^v, \mathcal A^v, T^v,U^v, s_0^v\rangle, v \in \mathcal X,$ where
\noindent
\begin{itemize}
\item $\mathcal S^v$ and $\mathcal A^v$ are finite sets of {states} and {actions} for agent $v$, respectively.
\item $T^v: \mathcal S^v \times\mathcal A^v\times \mathcal S^v \rightarrow [0,1]$ is a probabilistic {transition function} between states,
$T^v(s^v,a^v,\bar s^{v}) = \Pr(\bar s^{v} \mid a^v,s^v)$, where $s^v,\bar s^{v} \in \mathcal S^v$ and $a^v\in \mathcal A^v$.
\item $U^v: \mathcal S^v \times \mathcal A^v\rightarrow \mathbb R_+$ is a non-negative {utility function}.
\item $s^v_0$ is an initial state of agent $v\in \mathcal X$.
\end{itemize}
\subsubsection{MCC-SSP Model}
The examined model considers situations where agents interact only at certain {\em interaction points} (see Fig.~\ref{fig:colide-points} for an illustration), indexed by a set $\mathcal N$. Each point $i\in \mathcal N$ is in charge of coordination among multiple agents denoted by subset $\mathcal X^i \subseteq \mathcal X$. We assume that every agent is assigned to at least one coordination point (i.e., $\mathcal X = \cup_{i\in \mathcal N} \mathcal X^i$).
Such interactions among agents entail risk of failure (e.g., collision). We consider multiple risk criteria, indexed by a set $\mathcal J$.
Formally, we define the
MCC-SSP problem as a tuple
$M\triangleq\langle \mathcal X, \mathcal N, \mathcal J, (\mathcal S^i, \mathcal A^i, T^i,U^i, s_0^i)_{i\in \mathcal N}, h, r^j(v,v')_{v,v' \in \mathcal X}, (\Delta^j)_{j\in \mathcal J} \rangle$ where,
\begin{itemize}
\item $\mathcal X$ is the set of agents, $\mathcal N$ is the set of interaction points, and $\mathcal J$ is the set of risk criteria.
$\mathcal S^i\triangleq \times_{v\in \mathcal X^i} \mathcal S^v$ is a factored set of {\em interaction states} of the coordinated agents, and $\mathcal A^i\triangleq\times_{v\in \mathcal X^i} \mathcal A^v$ is a factored set of joint {actions} for interaction $i\in\mathcal N$, respectively. For $a^i\in\mathcal A^i$, we write $a^i_v$ to denote the action of vehicle $v\in \mathcal X^i$.
\item
$T^i: \mathcal S^i \times\mathcal A^i\times \mathcal S^i \rightarrow [0,1]$ is the joint transition function such that $T(s^i,a,\bar s^{i}) \triangleq \prod_{v\in\mathcal X^i} T^v(s^v, a^v,{\bar s^v})$, where $\bar s$ is the next state.
\item
$U^i: \mathcal S^i \times \mathcal A^i\rightarrow \mathbb R$ is the total utility such that $U^i(s^i, a^i)\triangleq \sum_{v \in \mathcal X^i} U^v(s^v, a^v).$ When an agent belongs to multiple interaction points, then we count utility of that agent at one interaction only. In other words, if $v \in \bigcap_i \mathcal X^i$ then there is only one interaction $i'$ such that $U^{i'}(s^{i'},a^{i'})\triangleq \sum_{v' \in \mathcal X^{i'}} U^{v'}(s^{v'}, a^{v'})$ and the rest will have
$U^i(s^i,a^i)\triangleq \sum_{v' \in \mathcal X^i\backslash \{v\}} U^{v'}(s^{v'}, a^{v'}).$
\item
$s^i_0\triangleq (s^v_0)_{v \in \mathcal X^i} $ is the joint initial state.
\item $h$ is the planning horizon.
\item $r^j(v,v'): \mathcal S^v\times \mathcal S^{v'} \rightarrow [0,1]$ provides the probability of failure (e.g., collision) due to interaction between agent $v$ and $v'$ at their respective states according to risk criterion $j$; and $\Delta^j$ is the corresponding risk budget, a threshold on the probability of failure over the planning horizon.
\end{itemize}
We represent the joint actions of all interaction points by $A \triangleq \times_{v \in \mathcal X} \mathcal A^v$, and joint states by $\mathcal S \triangleq \times_v S^v.$ For convenience, we write $U(s, a)\triangleq \sum_{v \in \mathcal X} U^v(s^v, a^v)$ to denote to the total utility of state $s\in \mathcal S$ and action $a \in \mathcal A$.
A {\em deterministic} and {\em non-stationary} policy $\pi(\cdot,\cdot)$ is a function that maps a state and time step into an action, $\pi:\mathcal S\times \{0,1,...,h-1\} \rightarrow \mathcal A$. A {\em stochastic} policy $\pi: \mathcal S \times \{0,1,...,h-1\}\times \mathcal A \rightarrow [0,1]$ is a probability distribution over actions from a given state and time.\footnote{To avoid clutter, we write $\pi(s_k, a)$ instead of $\pi(s_k, k,a)$ for state $s_k$.}
We write $\pi^i: \mathcal S^i \rightarrow \mathcal A^i$ to encode the joint action of agents $\mathcal X^i$ at interaction $i\in \mathcal N$, as per a feasible policy $\pi$.
A {\em run} is a sequence of random joint states $S_0, S_1,\ldots, S_{h-1}, S_h$ resulting from policy execution, where $S_0 = s_0\triangleq(s^v_0)_{v\in\mathcal X}$ is known. We use superscript $i$ to denote the corresponding run with respect to interaction $i\in \mathcal N$ (similarly we use superscript $v$ to that of agent $v$).
Let $R^{j}(s^v, s^{v'})$ be a Bernoulli random variable for failure between $s^v\in \mathcal S^v$ and $s^{v'} \in \mathcal S^{v'}$ with respect to criterion $j\in \mathcal J$. As per MCC-SSP model $M$, $\Pr(R^{j}(s^v, s^{v'}) = 1) = r^j(s^v, s^{v'})$.
The objective of {\sc MCC-SSP} is to compute a policy (or a conditional plan) $\pi$ that maximizes the cumulative expected utility (or minimizes the cumulative expected cost) while maintaining risks below the given thresholds $\Delta^j$. Formally,
{\small{\begin{align}
\small{\textsc{(MCC-SSP)}} & \nonumber\\
\max_{\pi}& ~\mathbb{E} \Big[ \sum_{\mathclap{t=0}}^{h-1} U(S_{t}, \pi(S_{t}) ) \Big] \nonumber \\
\text{s.t.}&~ \Pr\Big(\bigvee_{t=0}^{h} \bigvee_{v, v' \in \mathcal X}R^{j}(S^v_t, S^{v'}_t) \mid \pi \Big) \le \Delta^j, j \in \mathcal J. \label{con1}
\end{align}}}
According to the definition above, if $\mathcal X= \mathcal N$, i.e., single-agent interaction points, then agents are independent, except for sharing the risk budget. Such variant has been studied extensively in the literature for constrained MDPs (see \cite{de2021constrained} for a comprehensive survey). As reported below, we provide an exact computation method, which improves upon the approximate approach provided in \cite{de2017bounding}.\footnote{We note that the chance constraint in \cite{de2017bounding} is slightly more general than ours, as it bounds the total probability of a sum of costs exceeding a certain threshold. The current approach can capture such constraints by augmenting the state space to include all possible distinct values of the sum. Arguably, in some applications, the number of distinct values could be exceedingly large. Therefore, we can discretize the set of values such that, in the worst case, we violate the constraint by at most a factor of $(1+\epsilon),$ often referred as resource augmentation model.}
\subsection{Execution Risk}
Define the {\em execution risk} of a run at joint state $s_k$ as
$\textsc{Er}^j(s_k) \triangleq \Pr\Big(\bigvee_{t=k}^h \bigvee_{v, v' \in \mathcal X}R^{j}(S^v_t, S^{v'}_t) \mid S_k = s_k\Big).$ By definition, Cons.~\raf{con1} is equivalent to $\textsc{Er}^j(s_0) \le \Delta^j$. Here, it is assumed that any pair of agents may fail in at most one interaction point. This assumption will be important to obtain a {\em linear} constraint and holds for the studied intersection application (see Fig.~\ref{fig:colide-points}) where each collision point is between a unique set of agents. For applications where such assumption may not hold, Eqn.~\raf{eq:bool} below establishes an upper bound on the execution risk based on the Union bound. That being so, the proposed approach in Sec.~\ref{sec:ilp} can still generate a feasible yet possibly suboptimal solution.
The execution risk can be written as
\begin{flalign}
\textsc{Er}^j(s_k)=&\Pr\Big(\bigvee_{t=k}^h \bigvee_{i\in \mathcal N} \bigvee_{v, v' \in \mathcal X^i}R^{j}(S^v_t, S^{v'}_t) \mid S_k = s_k\Big)&& \notag\\%
=&\sum_{i \in \mathcal N} \Pr\Big(\bigvee_{t=k}^h \bigvee_{v, v' \in \mathcal X^i}R^{j}(S^v_t, S^{v'}_t) \mid S^i_k = s^i_k\Big),\label{eq:bool}&& \raisetag{25pt}
\end{flalign}\noindent where the last equation holds by the assumption on mutual exclusivity of events and by conditional independence.
We write execution risk at interaction point $i$ as $\textsc{Er}^j(s^i_k) \triangleq \Pr\Big(\bigvee_{t=k}^h \bigvee_{v, v' \in \mathcal X^i}R^{j}(S^v_t, S^{v'}_t) \mid S^i_k = s^i_k\Big)$. Thus, $\textsc{Er}^j(s_k) = \sum_{i \in \mathcal N} \textsc{Er}^j(s^i_k)$.
\begin{lemma}\label{lem:er}
The execution risk at interaction point $i\in \mathcal N$ can be written recursively as
{\footnotesize{\begin{align}
\textsc{Er}^j(s^i_k)= \sum_{s^i_{k+1}\in \mathcal S^i}\sum_{a^i \in \mathcal A^i} \textsc{Er}^j(s^i_{k+1}) \pi(s^i_k, a^i) \widetilde T^j(s^i_k, a^i, s^i_{k+1})&
+ \widetilde r^j(s_k),& \notag
\end{align}}}\noindent where $\widetilde r^j(s^i_k)\triangleq 1 - \prod_{v,v' \in \mathcal X^i}(1-r^{j}(s^v_k, s^{v'}_k))$ is the probability of failure at $s^i_k$, and $\widetilde T^{i,j}(s^i_k, \pi(s^i_k), s^i_{k+1})\triangleq T^i(s^i_k, \pi(s^i_k), s^i_{k+1})\prod_{v,v' \in \mathcal X^i}(1-r^{j}(s^v_k, s^{v'}_k))$.
\end{lemma}
\begin{proof}
See Sec.~\ref{lemproof} in the Appendix.
\end{proof}
\subsection{Integer Linear Programming Formulation}\label{sec:ilp}
Define a variable $x_{s,k,a}^{i,j} \in [0,1]$ for each state $s^i_k$ at time $k$, action $a^i\in \mathcal A^i$, agent $i\in\mathcal N$, and constraint $j \in \mathcal J$ such that
\begin{align}
&\sum_{a^i \in \mathcal A^i} x_{s,k,a}^{i,j} = \sum_{s^i_{k-1} \in \mathcal S^i} \sum_{a^i \in \mathcal A^i} x^{i,j}_{s,k-1,a} \widetilde T^{i,j}(s^i_{k-1}, a^i, s^i_{k}), \notag \\
& \hspace{10mm} k = 1,...,h-1, s^i_k \in \mathcal S^i, i \in \mathcal N, j \in \mathcal J\cup \{0\}, \label{conf1}\\
&\sum_{a^i \in \mathcal A^i} x^{i,j}_{s,0,a} = 1, \quad i\in\mathcal N, \label{conf2}
\end{align}
where $\widetilde T^{i,0}(\cdot,\cdot, \cdot) \triangleq T^i(\cdot,\cdot, \cdot)$.
As such, the above {\em flow} equations for $j=0$ represent the standard dual-space constraints for SSP~\cite{10.5555/2074158.2074203}. In the context of SSP, $x^{i,0}_{s,k,a}$ stand for the probability of agent $i$ taking action $a^i$ from state $s^i_k$.
By recursively expanding the execution risk at $s_0$ using Lemma~\ref{lem:er} and Eqn.~\raf{eq:bool} we arrive at the following result.
\begin{theorem}\label{lem1}
Given a conditional plan $\mathbf{x}$ that satisfies Eqn.~\raf{conf1}-\raf{conf2}, the execution risk can be written as a linear function of $\mathbf{x}$,
{\scriptsize\begin{align}
\textsc{Er}^j(s_0) = \sum_{k = 1}^{h}\sum_{i\in\mathcal N~~} \sum_{\mathclap{\substack{s^i_{k-1}\in \mathcal S^i \\a^i \in \mathcal A^i, s^i_k\in \mathcal S^i }}} \widetilde r^{j}(s^i_{k}) x^{i,j}_{s,k-1,a} \widetilde T^{i,j}(s^i_{k-1}, a^i, s^i_{k})
+\sum_{i \in \mathcal N} \widetilde r^j(s^i_0).\nonumber
\end{align}}
\end{theorem}
\begin{proof}
See Sec.~\ref{thmproof} in the Appendix.
\end{proof}
Consequently, we can formulate MCC-SSP as an ILP that has a polynomial number of variables and constraints in terms of $h,|\mathcal N|, |\mathcal X|, |\mathcal A^v|$ when the number of agents per interaction is at most a constant $c$, i.e., $|\mathcal X^i|\le c$. This would only require enumerating agent actions within each interaction point which is significantly lower than the complete enumeration.
{\footnotesize\begin{align}
\max_{x, z}~~& \sum_{k=0}^{h-1}\sum_{i \in \mathcal N}\sum_{s^i_k \in \mathcal S^i, a^i \in \mathcal A^i} x^{i,0}_{s,k,a} U(s^i_k, a^i) \qquad \qquad \textsf{(MCC-SSP-ILP)}\notag\\
\text{s.t.}~~& \text{Cons.~\raf{conf1}-\raf{conf2}}, \notag\\
&\sum_{k = 0}^{h-1}\sum_{i\in \mathcal N} \sum_{\mathclap{\substack{~~~s^i_{k}\in \mathcal S^i\\~~~ a^i \in \mathcal A^i, s^i_{k+1}\in \mathcal S^i }}} \widetilde r^{j}(s^i_{k+1}) x^{i,j}_{s,k,a} \widetilde T^{i,j}(s^i_k, a^i, s^i_{k+1})\le \widetilde\Delta^{j}, j\in \mathcal J \label{riskcons} \\
&\sum_{a^i\in \mathcal A} z^i_{s,k,a}\le 1, \quad k=0,...,h-1, s^i_k \in \mathcal S^i, i \in \mathcal N\label{conzone}\\
& x^{i,j}_{s,k,a} \le z^i_{s,k,a}, \quad i\in \mathcal N, j \in \mathcal J, k=0,...,h-1, s^i_k \in \mathcal S^i\label{conzbind}\\
&\sum\limits_{\mathclap{a^i \in \mathcal A^i \mid a^i_v = \overline a^v}} z^{i}_{s,k,a} = \sum_{\mathclap{~~~~{a^i}' \in \mathcal A^{i'}\mid a^{i'}_v = \overline a^v}} z^{i'}_{s,k,a}, \ v\in \mathcal X, \overline a^v \in \mathcal A^v, i, i' \in \mathcal N \mid v \in \mathcal X^i \cap \mathcal X^{i'} \label{con:bind}\\
&z^i_{s,k,a}\in \{0,1\}, \quad x^{i,j}_{s,k,a} \in [0,1],~~\quad i \in \mathcal N, j \in \mathcal J, \notag\\
& s^i_k \in \mathcal S^i, k=0,...,h-1.
\end{align}}
In {\sc MCC-SSP-ILP}, Cons.~\raf{riskcons} follows directly from Theorem~\ref{lem1}, where $\widetilde\Delta^{j}\triangleq \Delta^j - \sum_{i\in \mathcal N} \widetilde r^j(s^i_0).$
The variable $z_{s,k,a}^i$ is used to bind the actions of $i$ across all flows. In other words, if action $a^i$ is selected with respect to risk criterion $j$, and $a'$ for criterion $j'$, then $a^i=a'$. Thus, Cons.~\raf{conzbind} ensures that the same action is selected. Since, $z_{s,k,a}\in \{0,1\}$, Cons.~\raf{conzone} guarantees at most one action is chosen at each node. Lastly, Cons.~\raf{con:bind} maintains the consistency of the selected action for each agent across all interaction points.
To quantify the planner's complexity, we next appraise the worst-case running time of MCC-SSP analytically by examining the maximum number of nodes in the solution space. As an upper bound, consider a tree-based structure that expands all the nodes for a defined horizon of $h$ without combining similar states as in the graph structure that the MCC-SSP follows. Given a set of $|\mathcal N|$ interaction points each containing $|\mathcal X^i|$ agents, and $|\mathcal A^v|$ number of actions per agent $v$, we have $\prod_{v\in \mathcal X^i}|\mathcal A^v|$ possible actions per interaction point $i \in \mathcal N$. Thus, the last level of the solution tree for interaction point $i$ contains $(\prod_{v\in \mathcal X^i}|\mathcal A^v|)^h$ nodes. Hence, The total number of nodes for the planning problem is $O(\prod_{i\in \mathcal N}((\prod_{v\in \mathcal X^i}|\mathcal A^v|)^h))$, which can be also written as $O(|\mathcal N|((|\mathcal X^{i^m}|\cdot|\mathcal A^{v^m}|)^h))$ where $i^m$ is the point with maximum agents, and $v^m$ is the agent with maximum actions.
\iffalse
\subsection{Complexity Analysis}
To evaluate the planning task's complexity, we will focus on the maximum number of nodes in the solution space. We consider the tree-based structure, which expands all the nodes for a defined horizon of $h$ without combining similar states as in the graph structure that MCC-SSP follows. Hence, we are analyzing the upper bound for the MCC-SSP. The intersection consists of $|\mathcal N|$ interaction points that contain up to $|\mathcal M^i|$ maneuvers at a time. Maneuvers are defined as the path without considering the speed. Given $|\mathcal A|$ number of actions per agent, we have (${|\mathcal A|}^{|\mathcal M^i|}$) possible actions per interaction point $i$. Thus, we get $({|\mathcal A|}^{|\mathcal M^i|})^h$ nodes for the last depth of the solution tree, and the total number of nodes is $O(|\mathcal N|\cdot({|\mathcal A|}^{|\mathcal M^i|})^h)$. Considering that an interaction point with more than two maneuvers ($|\mathcal M^i|>2$) can be divided into multiple interaction points to maintain a maximum of two maneuvers per interaction point ($|\mathcal M^i|=2$), we get $O(|\mathcal N|\cdot{|\mathcal A|}^{2h})$.
\fi
\iffalse
FOR FUTURE USE
TODO: update the notation based on the about
\textcolor{red}{\textbf{Assessment of the number of nodes:}
To assess the possible number of nodes, we consider the tree based structure where we expand the all the nodes for defined horizon without combining similar states such as the graph structure of the MCC-SSP, making the analysis an upper bound for the MCC-SSP. The intersection consists of 36 interaction points, 28 of them contain two maneuvers($t_2$), while the 8 others contain three maneuvers($t_3$).
For the case of two actions per vehicle (1-speed), we have four ($a_{t_2}=4$) possible actions (two actions to the power of two) for the two maneuver interaction point, and eight actions ($a_{t_3}=8$) (two actions to the power of three) for the three maneuver interaction point. For a given depth the number of nodes will be the number of actions to the power of the depth. Thus the total number of nodes is $\sum_{i=0}^h {a_{t}}^i$ for each maneuver interaction point. Using the geometric series closed-form sum, we get $({a_{t_2}}^h +2)/3$ for the two maneuver interaction point and $({a_{t_3}}^h +6)/7$ for the three maneuver interaction point. Thus the total number of nodes for the case of two actions per vehicle (1-speed) MCC-SSP is $28 \cdot ({a_{t_2}}^h +2)/3 + 8 \cdot ({a_{t_3}}^h +6)/7$, equivalent to $O({a_{t_3}}^h)$.
For the case of three actions per vehicle (2-speed), we have nine ($a_{t_2}=9$) possible actions (three actions to the power of two) for the two maneuver interaction point, and twenty seven actions ($a_{t_3}=27$) (three actions to the power of three) for the tree maneuver interaction point. For a given depth the number of nodes will be the number of actions to the power of the depth. Thus the total number of nodes is $\sum_{i=0}^h {a_{t}}^i$ for each maneuver interaction point . Using the geometric series closed-form sum, we get $({a_{t_2}}^h +7)/8$ for the two maneuver interaction point and $({a_{t_3}}^h +25)/26$ for the three maneuver interaction point. Thus the total number of nodes for the case of three actions per vehicle (2-speed) MCC-SSP is $28 \cdot ({a_{t_2}}^h +7)/8 + 8 \cdot ({a_{t_3}}^h +25)/26$, equivalent to $O({a_{t_3}}^h)$.
}
\fi
\section{Risk-Aware Planning Under MCC-SSP}\label{sec:model}
\subsection{Agents and Interaction Points}
Recall that in the MCC-SSP formulation vehicles are indexed by the set $\mathcal X$, wherein HVs constitute a subset $\mathcal Y\subset\mathcal X$ and have a single action (hence are uncontrollable). The set of interaction points $\mathcal N$ represents all possible collision points between agents in the intersection, including those right before the intersection, as pictured in Fig.~\ref{fig:colide-points}.
Each interaction point $i \in \mathcal N$ is associated with a set of {\em reference} maneuvers $\mathcal M^i$ (green dashed lines in Fig.~\ref{fig:colide-points}). A reference maneuver $m_v\in\mathcal M^i$ is defined as a path that an agent $v$ can follow regardless of the speed, while actions are defined as trajectories.
\subsection{Action Model}
Action $a^i\triangleq(a^i_v)_{v\in\mathcal X^i}$ at the interaction point $i \in \mathcal N$ resembles one possible combination of variants of reference maneuvers that pass through that point, with $a^i_v\in \mathcal A^v$ representing a variation of a corresponding reference maneuver $m_v$ with a specific speed.
For an agent, a typical scenario would constitute a two-action model: 1) perform the maneuver or 2) wait. We \textit{expand} the scope by introducing \textit{speed-sensitive maneuvers}, such as {\tt turn\_left\_slow, turn\_left\_fast, wait}.
\subsection{State Representation}\label{sec:staterpr}
An interaction point $i \in \mathcal N$ at time $k$ is associated with a state $s^i_k\triangleq (k,p^v)_{v \in \mathcal X^i}$, where $p^v$ is the PFT of the current executing maneuver of vehicle $v$ or the {\tt wait} command. Upon arriving at the intersection, the vehicle is added to the state with the {\tt wait} maneuver.
The maneuver state changes (under a transition function) when an action is applied.
The vehicle's position and velocity are computed based on the progression of the maneuver PFT. We define the risk $r^j(s^i)$ of state $s^i$ as the probability of collision between the agents in the state (formalised in Sec.~\ref{sec:riskc}).
\subsection{Transition Function}
The transition function $T^v(s^v,a^v,\overline s^v)$ is computed by utilizing the intent recognition subsystem. The corresponding PFT resulting from action $a^v$ can be computed based on prior data collected by the Motion Model generator, as well as other external factors such as road conditions~\cite{huang2018hybrid,huang2019online, pft}. For HVs, the Intent Recognizer can capture human uncertainty. We refer the reader to \cite{gindele2013learning} for more details. \textcolor{black}{Moreover, any significant deviation of an AV or HV from its trajectory (mainly taking an illegal turn) due to some fault or miscommunication is handled by setting the system to a halt state. A halt state causes all vehicles and light signals to stop until the vehicle clears the intersection.}
\vspace{-5pt}
\subsection{Operating Horizon}
We adopt a receding horizon approach for online execution. For each time step $t$, we solve an MCC-SSP model for a horizon of length $h$. Vehicles that leave the intersection are removed from $\mathcal X$, and the model is subsequently solved on a rolling basis for the remaining vehicles still in the intersection.
Agents that are still executing their maneuvers from the previous state are considered obstacles and thus, there are no actions that apply to them.
The horizon duration $\Delta t$ is chosen to be less than the action termination time $\tau$ in order to provide a smoother transition. However, it's noteworthy that a smaller horizon window will result in a shorter total planning time ($h \cdot \Delta t $).
\subsection{Objective Function}\label{sec:objjj}
The proposed Intelligent Intersection system seeks to optimize the following \textit{multi-criteria} objective function: 1) Maximize the rate of flow (vehicles per unit time); 2) Minimize the maximum waiting time (duration from the moment the vehicle reaches the intersection to the moment it enters the intersection) to ensure fair distribution of traffic; 3) Facilitate a priority-based objective for emergency vehicles.
Recall that the optimal policy $\pi^* = \arg\max_{\pi} \mathbb{E}\Big[\sum_{t=0}^{h-1} U(s_{t}, a_t) \mid \pi \Big]$ requires a definition of a utility function. For a state $s=(k,p_v)_{v\in \mathcal X}$, and action $a=(a^v)_{v\in\mathcal X}$, we define $U(s,a) \triangleq \sum_{v \in \mathcal X} (U^v(a^v) \mid a^v \neq \texttt{wait}), $ such that $U^v(a^v) \triangleq \lambda_0 {\texttt{vel}(a^v)} +\lambda_1 P^v + \lambda_2 \sqrt{w^v}+ \lambda_3\frac{\sum_{v' \in \texttt{lane}(v)} P^{v'}}{|\texttt{lane}(v)|}$,
where $\lambda_0 \text{ to } \lambda_3 \in \mathbb R$ are used to weigh each term. Here, $\texttt{vel}(\cdot)$ provides the velocity of a given reference trajectory (faster maneuvers provide higher rewards), $P^v$ represents a priority score range (e.g., from 1 to 10) and the last term is the average priority of the vehicle's lane, with $\texttt{lane}(v)$ standing for the set of vehicles that are assigned to vehicle $v$'s source lane. In the definition of $U^v(a^v)$, $w^v$ captures the waiting time associated with the state. This requires augmenting the state space to include total waiting time for each vehicle $v$, which increments whenever state maneuver $s^v = \texttt{wait}$. For exposition clarity, we omitted the waiting time from the state representation in Sec.~\ref{sec:staterpr}.
Though sufficient for current purposes, the presented objective function can be further extended to incorporate progress (or speed) and comfort requirements for each agent~\cite{wei2014behavioral}.
\subsection{Risk Computation }
\label{sec:riskc}
Given that maneuvers are represented as PFTs, i.e. multivariate Gaussian processes, the risk can be calculated via Monte Carlo sampling. Particularly, to compute the risk between two maneuver trajectories $p^v, p^{v'}$, we extrapolate over time (over a duration of $\tau$ ), but at higher resolution (i.e., PFT resolution). Define $\mathtt{Col^{vv'}}(t)$ to be a Bernoulli random variable such that $\mathtt{Col}^{vv'}(t)=1$ if and only if collision occurs at time $t\in \{0, \Delta \tau, 2 \Delta \tau,..., \tau \}$ following their corresponding PFTs, and $\mathtt{Col}^{vv'}(t)=0$ otherwise. Then, at a given time $t$, the Monte Carlo sampling approach can be invoked to determine $\mathtt{Col}^{vv'}(t)$. Given $\mathtt{Col}^{vv'}(t)$ for $\forall t$, the risk of collision throughout the time horizon $\tau$ between agents $v$ and $v'$ is defined as $r^j(v,v')\triangleq\Pr\Big(\bigvee_{t=1}^\tau \mathtt{Col}^{vv'}(t) \Big) = 1 - \Pr\Big(\bigwedge_{t=0}^\tau \neg \mathtt{Col}^{vv'}(t) \Big) = 1 - \prod_{t=0}^\tau \Pr(\neg \mathtt{Col}^{vv'}(t)) =1 - \prod_{t=0}^\tau (1- \Pr( \mathtt{Col}^{vv'}(t)))$.
\iffalse
\begin{align*}
r^j(v,v')& \triangleq \Pr\Big(\bigvee_{t=1}^\tau \mathtt{Col}^{vv'}(t) \Big) \notag\\
&= 1 - \Pr\Big(\bigwedge_{t=1}^\tau \neg \mathtt{Col}^{vv'}(t) \Big) = 1 - \prod_{t=1}^\tau \Pr(\neg \mathtt{Col}^{vv'}(t)) \notag\\
& =1 - \prod_{t=1}^\tau (1- \Pr( \mathtt{Col}^{vv'}(t))) \notag
\end{align*}
\fi
\subsection{Offline Risk Computation}\label{subsec:risk_precomp}
To streamline the planning process, we precompute the risk of all the possible states of every interaction point. For an interaction point $i$ with $|\mathcal M^i|$ maneuvers, we allocate an $|\mathcal M^i|-$dimensional lookup table of size $O(\tau'^{|\mathcal M^i|})$, where $\tau' \geq \tau$ is the maximum number of potential progressions of a vehicle through the reference maneuver ($ \tau(m_v)=|m_v| \quad \forall \; m_v\in \mathcal M^i$), which is dependent on the defined PFT time-step of the trajectory. After precomputing the risk for all collision points, we can efficiently obtain the risk of any state by retrieving the risk of every combination of vehicles in the state. We precompute the risk for multiple standard vehicle dimensions to generalize for any vehicle model.
\iffalse
To improve the planning time, we precompute the risk of all the possible states of an interaction point. For an interaction point $i$ with $|\mathcal M^i|$ maneuvers (maneuvers represent the path while action represents the trajectory), we use a $|\mathcal M^i|-$dimensional lookup table of size $O(\tau'^{|\mathcal M^i|})$, where $\tau' \geq \tau$ is the maximum number of potential progression of a vehicle through the reference maneuver ($ \tau(m_v)=|m_v| \quad \forall \; m_v\in \mathcal M^i$), which is dependent on the defined PFT time-step of the trajectory.
After saving the precomputed risk for collision points, we can efficiently compute the risk of any state by retrieving the risk of every combination of vehicles in the state.
To generalize for any vehicle model, we precompute the risk for multiple standard vehicle dimensions. To guarantee risk constraints, we only use a larger or equal predefined vehicle dimension to represent a vehicle.
\fi
\section{Trajectory Optimization for AVs} \label{sec:traj_optimization}
AVs in the intersection problem require a reference trajectory (action) to follow. Such trajectory should be optimal with respect to certain objectives (e.g., comfort) while respecting the vehicles' control limits. Moreover, the trajectory should be safe concerning static obstacles as well as other vehicles in the intersection (dynamic obstacles).
To generate a set of reference trajectories, we employ a technique which finds the optimal path given the system and collision avoidance constraints.
We model the trajectory optimization task as a finite-horizon discrete-time shooting problem \cite{mayne1966second}, and tackle it with the Control-Limited Differential Dynamic Programming (DDP) algorithm \cite{tassa2014control}. In the subsections to follow, we address each of the above aspects.
\subsection{Dynamics Model}
The adopted vehicle dynamics model is based on the kinematic bicycle model. The bicycle model provides an approximate yet efficient representation (by averaging the speed of both wheels on a given axle) of the vehicle dynamics. The dynamics model with the center of mass as a reference point is defined as:
\begin{gather}
\Dot{x}^c = v^c \cdot \cos(\theta+\beta) ~,~ \Dot{y}^c = v^c \cdot \sin(\theta+\beta) \nonumber\\
\Dot{\theta} = \omega = v^c \cdot \frac{\tan(\zeta) \cos(\beta)}{L} ~,~ \beta = \tan^{-1}(l_r \frac{\tan(\zeta)}{L}),\nonumber
\end{gather}where $(x^c, y^c, \theta)$ is the position and heading of the vehicle, $v^c$ is the vehicle's velocity, $\zeta$ is the steering angle, $L$ is the length of the vehicle, and $l_r$ is the distance from the back axle to the center of mass. The dot notation (e.g., $\Dot{\theta}$) represents the derivative of the variable.
The trajectory state is defined as $\tilde{x} = [x^c, y^c, \theta, \zeta, v^c]$ and the control is defined as $\tilde{u} = [a, \Dot{\zeta}]$ where $a$ is the vehicle's acceleration (i.e., $\Dot{v^c}$).
We convert the continuous dynamics model into a discrete one by updating the state every $\Delta t$ using an explicit Euler integration scheme. The subsequent state $\tilde{x}_{k+1}$ is defined as:
\begin{gather}
x_{k+1}^c = x_k^c + \Dot{x}^c \cdot \Delta t ~,~
y_{k+1}^c = y_k^c + \Dot{y}^c \cdot \Delta t \nonumber\\
\theta_{k+1} = \theta_k + \Dot{\theta} \cdot \Delta t ~,~
\zeta_{k+1} = \zeta_k + \Dot{\zeta} \cdot \Delta t, v^c_{k+1} = v^c_k + a \cdot \Delta t.\nonumber
\end{gather}
\subsection{Cost Function}
The purpose of the cost function is to yield a smooth trajectory, while also imposing safety requirements to bypass static and dynamic obstacles. The integral cost function is defined as $[\Dot{\zeta}, a_c, c_s, c_d]$ where $a_c$ is the rotational acceleration ($a_c = (v^c)^2 \cdot \frac{\tan(\zeta)\cos(\beta)}{L}$), and $c_s$, $c_d$ are the static and dynamic obstacle costs as defined in Secs.~\ref{stt} and \ref{dnn}, respectively. A quadratic barrier function is used to enforce the limits on steering angle $\zeta$ (the rate of change of the steering angle is handled by the solver as an input constraint). The boundary cost function is $[x^c-G_x, y^c-G_y, \theta-G_\theta, v^c-G_v]$ where $G$ is the goal state.
\subsection{Static Obstacles}\label{stt}
In the studied intersection, curbs or unpaved areas are treated as static obstacles. Common obstacle avoidance methods either suffer from local minima (e.g., Artificial Potential Fields \cite{khatib1986real}) or tend to direct the state away from obstacles as much as possible (e.g., Harmonic Potential Fields \cite{kim1992real}). This served as a motivation to resort to the Signed Distance Field (SDF) \cite{finean2021predicted} approach with the hinge loss function \cite{mukadam2018continuous}. In implementing SDF, we first generate a binary array of the obstacles (Fig.~\ref{fig:sdf2}) based on the intersection map (Fig.~\ref{fig:sdf1}). Next, we apply the signed distance field function, which returns the signed distance $D_s$ from the zero contour in an array (Fig.~\ref{fig:sdf3}). Finally, we apply the hinge loss function (Fig.~\ref{fig:sdf4}), which returns a zero value if the state is not near an obstacle.
The hinge loss function is defined as
$$
h(D_s)=
\begin{cases}
-D_s+\epsilon & \text{if } d \leq \epsilon\\
\quad\;\; 0 & \text{if } d > \epsilon
\end{cases},\notag
$$
where $\epsilon$ is a safety distance from the boundary of the obstacles. We set the static obstacle cost as $c_s = e^{-\frac{1}{2}h(D_s)^2}$.
\subsection{Dynamic Obstacles}\label{dnn}
We render other non-colliding trajectories
that do not share a collision point but might collide if they were generated close to each other as dynamic obstacles. Two factors are accounted for in the dynamic obstacle avoidance, namely the vehicle's geometric shape and the expected controller uncertainty when following a trajectory.
As illustrated in Fig.~\ref{fig:vehicle_circles}, vehicle's shape can be captured by three adjacently placed overlapping circles. To determine whether two vehicles would collide, one can compare the Euclidean distance between center-points and sum of radius of any pair of circles among the two vehicles.
\begin{figure}[!b]
\begin{subfigure}{.5\linewidth}
\centering
\includegraphics[width=\linewidth]{Figures/reg_model_30_ref_paths.png}
\caption{}
\label{fig:reg_model_paths}
\end{subfigure}
\hspace{.03\linewidth}
\begin{subfigure}{.43\linewidth}
\centering \vspace{2.5mm}
\includegraphics[width=\linewidth]{Figures/vehicle_circles1.pdf}\vspace{1mm}
\caption{}
\label{fig:vehicle_circles}
\end{subfigure}
\caption{(a) A set of 30 uniformly distributed trajectories, and (b) the distance between two vehicles each represented via three circles. }
\end{figure}
\begin{figure*}[t!]
\begin{subfigure}{.23\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/intersection1-SDF1.jpg}
\caption{}
\label{fig:sdf1}
\end{subfigure}
\begin{subfigure}{.23\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/intersection1-SDF2.jpg}
\caption{}
\label{fig:sdf2}
\end{subfigure}
\begin{subfigure}{.26\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/intersection1-SDF3.png}
\caption{}\label{fig:sdf3}
\end{subfigure}
\begin{subfigure}{.26\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/intersection1-SDF4.png}
\caption{}\label{fig:sdf4}
\end{subfigure}
\caption{Static obstacle representation: (a) The original intersection. (b) Static obstacles are highlighted in white. (c) Signed-distance field. (d) Application of the Hinge Loss function.}
\end{figure*}
To incorporate the resultant expected uncertainty from AVs' trajectory following, we construct a regression model of controller uncertainty. The model takes as input the vehicle state and control, namely $[\zeta, v^c, a, \Dot{\zeta}]$, and returns the expected error defined as a radius with a predefined confidence interval. To this end, we first generate a set of reference trajectories (with uniform goal points as visualized in Fig.~\ref{fig:reg_model_paths}) using the cost function defined earlier which incorporates all the potential trajectories at an intersection. Next, we execute each trajectory using the controller
and collect the data.
The data is then normalized by averaging over each subset of input trajectories and the expected error is calculated using the Quantile function.
The linear regression model is then trained using the normalized input and the expected error. As empirically observed, the model attained 98.96\% accuracy.
Given two sets of vehicle trajectories $S_x^1$ and $S_x^2$, we let the dynamic distance $D_d$ between the two be the sum of the geometric distance (positive value implies overlapping) and the controllers' expected uncertainty. Mathematically,
{\footnotesize\begin{align*}
D_d = \max\Big\{ \Upsilon (o_1, o_2) +\tt{Reg}(s_1)+\tt{Reg}(s_2) ~ \forall~ s_1\in S_x^1, s_2\in S_x^2, ~ 0 \Big\} \,,
\end{align*}}where $o_1,o_2$ are the circles that represent vehicle geometry per state, $\Upsilon (o_1, o_2) \triangleq \min\limits_{o_1\in s_1, o_2\in s_2} r_{o_1} + r_{o_2} - d(o_1,o_2)$ with $r_{o_1}, r_{o_2}$ denoting the radii of $o_1, o_2$ while $d(\cdot)$ capturing their Euclidean distance, and $\tt{Reg}(\cdot)$ encodes the output of regression models. Observe that $D_d = 0$ indicates no collision. The cost of dynamic obstacles is set to $c_d = e^{-\frac{1}{2}(D_d)^2}$.
\subsection{Trajectory Optimization}
We cast the trajectory optimization problem as a Shooting Method \cite{mayne1966second} and solve it using the Control-Limited Differential Dynamic Programming (DDP) algorithm \cite{tassa2014control}, which is an indirect method that admits quadratic convergence for any system with smooth dynamics \cite{jacobson1970differential} while bounding the control inputs. Hence, we are able to bound the acceleration based on the vehicle's specifications or the required action speed. The trajectory optimization algorithm, explained in Alg.~\ref{alg:traj}, consists of two loops. The first generates an initial trajectory with static obstacles, while the second one incorporates dynamic obstacles.
In Alg.~\ref{alg:traj}, $S_x$ is the set of trajectories' states, $S_u$ is the set of trajectories' controls, $\mathcal A$ is the set of all actions defined by a starting and a goal point. The expression $c_d(S_u^i, S_u^j | (i,j) \nsubseteq \mathcal N )$ represents the dynamic obstacle cost between $i$ and all actions $j$ that don't share a collision point from the set of collision points $\mathcal N$. The boolean variable \textit{Converge} represents the stopping criterion, which can be conditioned on a minimum threshold of difference between the previous trajectories and the new ones, or alternatively fixed to a certain number of iterations.
\iffalse
\begin{algorithm}[!ht]
\SetAlgoLined
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{Set of all Actions $\mathcal A$}
\Output{Set of Trajectories $S_x$, $S_u$}
\For{$\mathcal A^i \in \mathcal A$}{
$S_x^i, S_u^i = DDP(\mathcal A^i)$
}
\While{Not Converge}{
\For{$\mathcal A^i \in \mathcal A$}{
$S_x^i, S_u^i = DDP(\mathcal A^i, S_u^i, c_d(S_u^i, S_u^j | (i,j) \nsubseteq \mathcal N ))$ \\
}}
\textbf{return} $S_x, S_u$
\caption{Trajectory Optimization}
\label{alg:traj}
\end{algorithm}
\fi
\begin{algorithm}[!h]
\SetAlgoLined
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{Set of all actions $\mathcal A$}
\Output{$S_x$, $S_u$}
\For{$\mathcal A^i \in \mathcal A$}{
$S_x^i, S_u^i = DDP(\mathcal A^i)$
}
\While{Not Converge}{
\For{$\mathcal A^i \in \mathcal A$}{
$S_x^i, S_u^i = DDP(\mathcal A^i, S_u^i, c_d(S_u^i, S_u^j | (i,j) \nsubseteq \mathcal N ))$ \\
}}
\textbf{return} $S_x, S_u$
\caption{Trajectory Optimization}
\label{alg:traj}
\end{algorithm}
\iffalse
\subsection{Test Cases}
To visualize the intersection trajectory optimization formulation, we ran our model on two intersections, one in Russia (Fig.~\ref{fig:intersection1}) with an asymmetrical number of the lanes and another intersection in the UAE (Fig.~\ref{fig:intersection2}) with multiple static obstacles in the center of the intersection.
We utilized \textit{Crocoddyl} \cite{mastalli2020crocoddyl}, an open-source trajectory optimization software that implements the DDP algorithm to run our experiment.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{Figures/intersection1_traj.pdf}
\caption{Trajectories for an intersection at Tverskaya Zastava Square in Moscow, Russia.}
\label{fig:intersection1}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=.9\linewidth]{Figures/intersection2_traj.pdf}
\caption{Trajectories for an intersection near Khalifa University in Abu Dhabi, UAE.}
\label{fig:intersection2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=.7\linewidth]{Figures/reg_model_30_ref_paths.png}
\caption{A set of 30 evenly distributed paths used as reference path for the PID controller, used to generate the regression model. }
\label{fig:reg_model_paths}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=.5\linewidth]{Figures/vehicle_circles.pdf}
\caption{Vehicle representation using three circles.}
\label{fig:vehicle_circles}
\end{figure}
\fi
\section{Performance Evaluation} \label{sec:experiment}
To proceed with the evaluation, we first test the introduced ILP formulation's scalability on the multi-agent version of the well-known grid problem with independent agents (i.e., $\mathcal X = \mathcal N$) and shared risk constraint. Next, we employ the CARLA simulator \cite{dosovitskiy2017carla} as a test-bed to simulate the proposed risk-aware intelligent intersection system, verify its effectiveness and practicality as well as collect ground truth data on vehicle driving. In this section, we report the empirical findings on the planning time complexity of MCC-SSP (Table \ref{tab:planning_time}), the throughput of a fully-AV intersection (Table \ref{tab:t1}), the impact of HVs on the throughput (Fig. \ref{fig:HV-throughput}), and experimentations on variants of the objective function (Fig.~\ref{fig:obj_exp}). Lastly, we present two case studies of the trajectory optimization workflow.
\iffalse
We consider first a multi-agent version of the well known grid problem \cite{russell2002artificial} to showcase the MCC-SSP .
Moreover, in order to evaluate the performance of our proposed intersection system, vehicle driving and obtain ground truth data.
The experiment presents the planning time (Table \ref{tab:planning_time}), the throughput of a fully AV intersection (Table \ref{tab:t1}), and the impact of HVs on the throughput (Fig. \ref{fig:HV-throughput}) of the intersection problem, and an experiment on variants of the objective function (Fig.~\ref{fig:obj_exp}). Finally, we present two case studies for the intersection trajectory optimization formulation.
\fi
\subsection{Scalability Analysis}
As one demonstration, we apply the proposed MCC-SSP model to the multi-agent grid problem wherein robots can move in four directions inside a bounded discretized area. The movement, however, is uncertain with an 80\% success probability represented in the transition function, 5\% of the states are randomly defined as risky, and 10\% of the states are randomly set with a cost of $1$ while the rest have a cost of $2$. The grid size is set to (10000x10000) to assess the planner's performance at scale. The initial state is random for each agent. Fig. \ref{fig:multiagent-grid} plots the running time and the average objective value. As demonstrated by the figure, the formulation scales well with the horizon size and number of agents. In general, the running time is expected to increase with the number of agents and the length of planning horizon as more variables and constraints would be involved.
\begin{figure}[ht]
\begin{subfigure}{.495\linewidth}
\centering
\includegraphics[width=\linewidth]{Figures/multiagent_time_3d.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{.495\linewidth}
\centering
\includegraphics[width=\linewidth]{Figures/multiagent_obj_3d.pdf}
\caption{}
\end{subfigure}
\caption{Performance of the proposed MCC-SSP model on the multi-agent grid problem against the planning horizon and number of agents: (a) {\sf MCC-SSP-ILP} solving time, and (b) the objective value.
\label{fig:multiagent-grid}
\end{figure}
\subsection{Simulated Risk-aware Intelligent Intersection System}
\subsubsection{Intersection Throughput and Planning time}\hfill\\
\noindent
\textit{\bfseries Setup}:
CARLA simulator was used to generate PFTs for both AVs and HVs. For AVs, we executed a PID controller over nominal trajectories multiple times, each with slightly perturbed coefficients (uniformly chosen $P\in[0.4,1.2], I=0$, and $D\in[0.2,0.8]$)\footnote{Samples with tracking error higher than 1 meter were excluded.}. The motivation behind this setup is that AVs from different vendors may have different controller setups. There are other factors, such as vehicle drift, weather, and road conditions, that could affect performance in real life.
Similarly, we generate PFTs for HVs, with a 50\% chance of taking either action (e.g., go left or straight) when the traffic signal for HVs is green for the corresponding side (rotating every minute). As an HV is accessing the intersection, the probability gradually approaches 100\% for the respective action. For comparison, we employ the FCFS augmented with our risk detection approach as a benchmark planner. The simulations were repeated over 300-fold to reduce the uncertainties in the results.
The simulation setup consists of a two-lane four-sided (eight AVs) intersection (depicted in Figure \ref{fig:colide-points}), where the horizon duration is one second and a receding horizon is used for continuous planning. The horizon duration is the time between each planning horizon.
The trajectories are defined based on a PFT with 6Hz time-step.
\iffalse
\begin{table}[ht]
\footnotesize
\centerin
\begin{tabular}{|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{} & \multicolumn{2}{c|}{Planning time (sec)} \\ \hline
Actions & Horizon & Preprocessing & Solving \\ \hline
\multirow{6}{*}{2 actions} & $h=1$ & 0.00361 & 0.0143 \\ \cline{2-4}
& $h=2$ & 0.01379 & 0.0610 \\ \cline{2-4}
& $h=3$ & 0.02700 & 0.2344 \\ \cline{2-4}
& $h=4$ & 0.06879 & 0.6550 \\ \cline{2-4}
& $h=5$ & 0.17493 & 2.1739 \\ \cline{2-4}
& $h=6$ & 0.43763 & 6.6600 \\ \hline
\multirow{2}{*}{3 actions} & $h=1$ & 0.01426 & 0.0421 \\ \cline{2-4}
& $h=2$ & 0.06661 & 1.0946 \\ \hline
\end{tabular}%
\caption{The planning time for 16 random AVs evenly distributed over a two-lane four-sided intersection ($\Delta=5\%$).
\label{tab:planning_time}
\end{table}
\fi
\noindent
\textit{\bfseries Results}:
The first set of simulations, summarized in Table~\ref{tab:t1}, contrasts the performance (in terms of throughput) of FCFS and MCC-SSP under different risk thresholds and number of actions per agent. While the risk budget in MCC-SSP bounds the expected risk of the policy, the FCFS planner parses it as a bound for each action taken per agent and thus the expected risk in FCFS may exceed the bound for an MCC-SSP's single horizon.
As evident from Table~\ref{tab:t1}, MCC-SSP outperforms FCFS for any risk bound, and increasing the risk bound ($\Delta$) improves the performance since the system is taking more riskier actions. On the other hand, increasing the horizon doesn't necessarily improve the throughput. Even though a longer planning horizon provides a more optimal solution, the same risk bound gets distributed over the planning horizon. Thus, effectively, the single horizon case, for instance, has a higher risk threshold within, say, two receding horizons when compared to $h=2$.
The observed planning time and scalability of the ILP formulation are reported in Table~\ref{tab:planning_time} as a function of the planning horizon and number of actions per agent. We presume that a planning time less than or equal to a single horizon duration (the horizon duration is one second) is reasonable for an intersection system; thus, for two actions we are able to use a horizon of four, and with three actions we are able to use a horizon of two.
We also present the negative impact of HVs on the throughput in Fig.~\ref{fig:HV-throughput}. As anticipated, with more human-driven vehicles, the throughput decreases.
\subsubsection{Intersection Objective Function}\hfill\\
\noindent
\textit{\bfseries Setup}:
To investigate the effectiveness of the objective function put forth in Sec.\ref{sec:objjj}, we performed a test case (portrayed in Fig.~\ref{fig:obj_exp1}) where an infinite number of AVs are arriving from the north and south. All AVs are traveling forward, similar to a highway scenario. On the other hand, the ego vehicle (green circle) is attempting to turn left, starting from the west and heading north.
We test two variants of the objective function, the first without the waiting time parameter ($\lambda_2=0$) and the second with the waiting time parameter ($\lambda_2=4$).
\vspace{4pt}
\noindent
\textit{\bfseries Results}:
In the case of the first objective, we observe that the ego vehicle never enters the intersection and waits indefinitely (depicted in Fig.~\ref{fig:obj_exp2}), which is undesirable. On the other hand, with the second objective, the ego vehicle enters the intersection (depicted in Fig.~\ref{fig:obj_exp3}) after the 10th horizon ($4\sqrt{10} > 15$ where 15 is the number of AVs waiting at the intersection).
\begin{figure}[ht]
\begin{subfigure}{.327\linewidth}
\centering
\includegraphics[trim={15cm 25cm 20cm 25cm}, clip, width=\linewidth]{Figures/obj_experiment1.png}
\caption{}
\label{fig:obj_exp1}
\end{subfigure}
\begin{subfigure}{.327\linewidth}
\centering
\includegraphics[trim={15cm 25cm 20cm 25cm},clip, width=\linewidth]{Figures/obj_experiment2.png}
\caption{}
\label{fig:obj_exp2}
\end{subfigure}
\begin{subfigure}{.327\linewidth}
\centering
\includegraphics[trim={15cm 25cm 20cm 25cm},clip, width=\linewidth]{Figures/obj_experiment3.png}
\caption{}
\label{fig:obj_exp3}
\end{subfigure}
\caption{Experimentation with the objective function: (a) The initial state and intention of the ego vehicle. (b) Running the planner with an objective function without waiting time. (c) Enacting the waiting time parameter in the objective function.}
\label{fig:obj_exp}
\end{figure}
\vspace{-5pt}
\subsection{Trajectory Optimization}
To demonstrate the adopted trajectory optimization workflow, we ran our model on two intersections, one in Russia (Fig.~\ref{fig:intersection1}) with an asymmetrical number of the lanes and another intersection in the UAE (Fig.~\ref{fig:intersection2}) with multiple static obstacles in the center of the intersection. In the simulations, we utilized \textit{Crocoddyl} \cite{mastalli2020crocoddyl}, which is an open-source trajectory optimization software that implements the DDP algorithm.
The initial trajectories that were generated in the first loop of Alg.~\ref{alg:traj} were very close to each over and overlapped in some cases. However, after running the trajectory optimization with the dynamic obstacle cost (second loop of Alg.~\ref{alg:traj}), the trajectories diverged and no overlapping was observed.
\begin{figure}[!h]
\begin{subfigure}{.515\columnwidth}
\centering
\includegraphics[trim={1.5cm 0cm 0cm 0cm},clip,width=\textwidth]{Figures/intersection1_traj.pdf}
\caption{}
\label{fig:intersection1}
\end{subfigure}
\begin{subfigure}{.475\columnwidth}
\centering
\includegraphics[width=\textwidth]{Figures/intersection2_traj.pdf}
\caption{}
\label{fig:intersection2}
\end{subfigure}
\caption{Output trajectories of the employed method for different intersections: (a) Tverskaya Zastava Square in Moscow, Russia. (b) Intersection near Khalifa University, Abu Dhabi, UAE.}
\end{figure}
\section{Conclusion} \label{sec:conclusion}
This work proposes a risk-aware Intelligent Intersection system modeled as a novel class of MCC-SSP, wherein agents interact at certain localized zones (collision points).
The system admits an adjustable risk tolerance parameter that allows to enforce the desired guarantee level on the probability of collisions despite the presence of perception and planning uncertainties. We introduce an exact integer linear programming formulation of the problem, featuring polynomial number variables and constraints when the number of agents per localized zone is small. The system is demonstrated in a realistic driving simulator that involves both AVs and HVs. As validated through simulations, the proposed system provides optimal plans that translate to higher throughput than the existing approaches. In future work, we target to design an approximation algorithm for the problem that runs in polynomial time and provides certifiable worst-case performance guarantees (i.e., approximation ratio).
\iffalse
\color{blue}
\section{Prospective Extensions}
\noindent{\bf Pedestrian Management.}
The hybrid intersection of AVs and HVs presents a unique problem for pedestrians crossing the intersection, which requires the system to accurately predict the future motions of the pedestrians. The pedestrian prediction problem has been widely studied in the recent years using learning-based methods~\cite{alahi2016social,mohamed2020social}.
We propose dividing the problem into three cases: 1) a pedestrian crosses an entering lane while the vehicle light signal is green, 2) a pedestrian crosses an entering lane while the vehicle light signal is red, and 3) a pedestrian crosses an exiting lane.
In the first case, HVs are exiting the lanes, so the pedestrians are not allowed to cross (red pedestrian light). In the second case, only AVs are exiting the lanes, so a prediction system is used to predict the movement of pedestrians, and the road side unit will plan for the AVs accordingly by adding a collision point for pedestrians (flashing amber pedestrian light). In the third case, a prediction system is needless due to the delay between vehicles entering the intersection and reaching the exit lane. Therefore, pedestrians are allowed to cross when HVs signal is red, and AVs are also prevented from the given exit lane (green pedestrian light). To improve the efficiency, the pedestrian signal at exiting lanes is activated only between HVs signal light change, when pedestrians are available.
\color{black}
\fi
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,973 |
Q: jQuery FormData and files I'm attempting to do ajax file upload. When i check the variable form_data it is blank?
<form action='ajax_image_upload/process.php' method='post' enctype='multipart/form-data' class='upload_form'>
<textarea name='description' placeholder='Type Default Video Description'></textarea>
<span>
Channel Image: <input type='file' name='image' />
</span>
<input name='__submit__' type='submit' value='Upload'/>
</form>
var container = $(".upload_form");
var form_data = new FormData();
form_data.append('image', container.find("form > span").children("input[name='image']"));
var post_url = container.children("form").attr("action"); //get action URL of form
//jQuery Ajax to Post form data
$.ajax({
url : post_url,
type: "POST",
data: form_data,
contentType: false,
cache: false,
processData: false,
mimeType: "multipart/form-data"
}).done(function(res){
});
How can I solve this?
A: The issue is because you need to append the binary data from the file input to the FormData, not the jQuery object. Try this:
var fileData = container.find("form > span").children("input[name='image']")[0].files[0];
form_data.append('image', fileData);
Obviously, if there are multiple inputs, or multiple files within the input, you'll need to loop through them and append them individually.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,944 |
Q: per-user resolv.conf My linux server has differnt users [u01 - u04], can I have each user use a specific resolv.conf?
for example:
u01: nameserver=10.14.15.123
u02: nameserver=10.14.15.124
u03: nameserver=10.14.16.125
A: Short answer is no, see resolv.conf(5). However if you really want this feature, you can reimplement the functions in resolver(3) in a shared library, and use ld-magic to override the originals, at least in dynamically linked applications.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,175 |
ICON makes two senior appointments
Published on: 21/05/12
ICON has appointed Dr Susan Anton as director of epidemiology and risk management practice, and late phase and outcomes research.
It has also appointed Dr Pui Leung as its new senior clinical research physician, in ICON's development solutions division.
Dr Anton comes to ICON with over 25 years of experience international epidemiologic, health economic, and outcomes research, and other market access initiatives.
She joins ICON from Boehringer Ingelheim, where she was director of US medical outcomes research and registries.
In this role, Dr Anton provided strategic approach, design and analytic guidance for comparative effectiveness research, epidemiologic studies and health technology assessment within their medical outcomes research and registries division.
Prior to Boehringer Dr Anton directed health economic, epidemiologic and outcomes research efforts across therapeutic areas at Bristol-Myers Squibb.
Dr Leung joins ICON with over 23 years of hospital and pharma industry experience in a variety of medical specialties including diabetes, endocrinology and general medicine.
Prior to ICON, Dr Leung gained 10 years of experience as a senior Phase I clinical research physician in two UK CROs, and has been principal investigator in many trials with a variety of different study designs including First in Human studies.
Before joining the pharma industry in 2002, Dr Leung spent 13 years as a hospital physician in the NHS.
ICON acquires market access consultancy
PRA bolsters late phase services
Contract research news in brief
Former FDA and MHRA regulators join Parexel Consulting
Cedra appoints Anthony Busa | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,065 |
The "We Are a New World" festival is coming back to Madrid on the 1st and 2nd of February! A window given to those who wish to come as they are, offering a place with no prejudice, gender, race or preference, where you are free to express yourself however you wish, surrounded by great music, and an all-inclusive group of people, for 2 fantastic nights of freedom and fun!
Absolut Manifesto paves the way of what our future world can and should become!
With a focus on creativity and art, this festival will be home to painters, designers, musicians and more, WANR will brings artists, local and international! From London Rappers, to Spanish DJs, you're bound to dance and have a great time!
For more info regarding the tickets, check out the webpage!
Previous articleCocktail & Pizza: The Combo You Never Knew You Needed! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,348 |
This reflection comes courtesy of Theologika trustee Terry Hershey, quoting theologian Paul Tillich.
"Grace strikes us when we are in great pain and restlessness.
It strikes us when we walk through the dark valley of a meaningless and empty life.
Sometime at that moment a wave of light breaks into our darkness, and it is as though a voice were saying, "You are accepted. You are accepted, accepted by that which is greater than you, and the name of which you do not know.
Do not seek for anything, do not perform anything, do not intend anything.
Simply accept the fact that you are accepted.
May grace reach into your life and surpise you today and always. Amen.
I love this quote. Thank you! | {
"redpajama_set_name": "RedPajamaC4"
} | 543 |
Q: reloadData for a Table View in a View Controller Hello I have a view controller that loads from an xib i created. It has two toolbars and a table view in that.
I add this too the header file in the ViewController
@interface FilterViewController : UIViewController <UITableViewDelegate, UITableViewDataSource> {
When I do
[self.tableView reloadData]
It does throws up an error and does not build.
A: Just making your UIViewController conform to UITableViewDataSource and UITableViewDelegate does not automatically give you a tableView reference.
You need to create a tableView IBOutlet and connect it in Interface Builder.
Also, why not just inherit UITableViewController instead of a UIViewController?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,012 |
Legrad är en ort i Kroatien. Den ligger i länet Koprivnica-Križevcis län, i den nordöstra delen av landet, km nordost om huvudstaden Zagreb. Legrad ligger meter över havet och antalet invånare är .
Terrängen runt Legrad är platt. Den högsta punkten i närheten är meter över havet, km öster om Legrad. Runt Legrad är det ganska tätbefolkat, med invånare per kvadratkilometer. Närmaste större samhälle är Koprivnica, km söder om Legrad. Omgivningarna runt Legrad är en mosaik av jordbruksmark och naturlig växtlighet.
Inlandsklimat råder i trakten. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är juli, då medeltemperaturen är °C, och den kallaste är december, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är september, med i genomsnitt mm nederbörd, och den torraste är mars, med mm nederbörd.
Kommentarer
Källor
Externa länkar
Orter i Koprivnica-Križevcis län | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,308 |
Lingua-Video.com (Lingua-Video.com Medien GmbH) ist ein Filmverlag für Bildungsfilme mit Sitz in Bonn. Ein zweites Büro befindet sich in Hamburg. Das Filmprogramm des Verlags richtet sich speziell an Schulen, Bibliotheken, Universitäten und Medienzentren. Geschäftsführende Gesellschafterinnen sind Cäcilia Simon und Jeannine Simon.
Geschichte
Lingua-Video.com wurde 1989 von Cäcilia Simon in Königswinter gegründet. Die Grundidee bestand darin, originalsprachige Filme für die Fremdsprachenausbildung an Schulen und Bildungseinrichtungen zur Verfügung zu stellen. Neben diesem Kerngeschäft wurde das Filmprogramm im Laufe der Unternehmensgeschichte um gesellschaftspolitisch relevante Themen in deutscher Sprache erweitert. 2006 verlegte Lingua-Video.com seinen Sitz nach Bonn. 2008 wurde neben dem Bonner Hauptsitz ein weiteres Büro in Hamburg eröffnet. Lingua-Video.com ist jedes Jahr auf der Bildungsfachmesse didacta vertreten.
Programm
Lingua-Video.com bietet Filme in englischer, französischer, spanischer und deutscher Sprache an. Das Programm umfasst klassische und moderne Literaturverfilmungen, Theaterinszenierungen, landeskundliche, geschichtliche und sozialkritische Dokumentationen, Spielfilme und Künstlerbiographien.
Der Verlag versorgt Schulen, Medienzentren und Bibliotheken im Fach Deutsch und in den Fremdsprachen mit Filmen, die in den Lehrplänen vorgesehen und auf Schullektüren abgestimmt sind. Im Fremdsprachenunterricht weit verbreitete Filme von Lingua-Video.com sind z. B. Quiero ser (spanisch)
oder Pas d'histoires! 12 regards sur le racisme au quotidien (französisch).
Die Filme von Lingua-Video.com werden als DVD, DVD-ROM und über die Online-Distribution auf geschlossenen Bildungsservern angeboten.
Unter der Produktlinie Lingua-Video didactics veröffentlicht der Verlag didaktische Materialien, die im Verbund mit den dazugehörigen Filmen vertrieben werden (DVD-ROM).
Bekannte Filme
Beispiele für Filme von Lingua-Video.com sind Michael Kohlhaas – der Rebell von Regisseur Volker Schlöndorff, Rolltreppe abwärts, die Verfilmung des gleichnamigen Bestsellers von Hans-Georg Noack, und Poem – Ich setzte den Fuß in die Luft und sie trug von Ralf Schmerberg.
Außerdem veröffentlichte Lingua-Video.com die Poetry Clips von Bas Böttcher.
Preise und Auszeichnungen
Im April 2011 wurde der von Lingua-Video.com initiierte Film Sprich mit! mit dem Hauptstadtpreis für Integration und Toleranz des Vereins Initiative Hauptstadt Berlin ausgezeichnet.
Quellen
Weblinks
Internetpräsenz von Lingua-Video.com
Verlag (Bonn)
Lehrmittelverlag | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,856 |
Professor Peter Virdee unveiled as new Sporting Equals Corporate Ambassador
Sporting Equals is honoured to unveil its new Corporate Ambassador Mr Prof. Peter Virdee, as part of its Ambassador programme, supporting the future champions programme. This programme aims to help talented and dedicated young black and minority ethnic (BME) individuals to succeed at elite level sport, and serve as valuable role models to encourage other young people to get involved in sport.
Sporting Equals is the national organisation promoting greater participation by the black and minority ethnic population in sport and a healthy lifestyle. The not-for profit organisation, is at the heart of BME communities through its network of 5,000 community organisations. As a national partner of Sport England and strategic advisor to DCMS they have partnership agreements and affiliations with the leading sports governing bodies.
The objective of Future Champions is to recognise, celebrate and inspire. By shining a light on the stars of tomorrow, the programme not only nurtures talent but also encourages participation from other young people from similar backgrounds, motivated to get involved by the success of our Future Champions.
Prof. Peter Virdee said "even if I can inspire one child to take up sport, through my Corporate Ambassador role with Sporting Equals, I will be happy".
Born and bred in the inner cities of Birmingham, earning his first pounds as a car minder, Prof. Peter is now a highly established business and property entrepreneur, one of the most iconic Sikh's across the world. Peter's journey can serve as an inspiration for other young people including those living in inner city areas. With the right mind set, drive and motivation, Prof. Peter experience provides evidence that nothing is impossible.
Prof. Peter himself is an avid sportsman, playing badminton, polo and basketball as a way to both get fit and get into a mind frame of self-discipline that has helped him in his journey to business success. Prof. Peter believes that sport can provide many benefits and is a key tool to develop youngsters, helping them achieve their potential.
Prof. Peter said "There is so much sporting talent in the Asian community, not just in the UK but worldwide. The Asian diaspora is one of the largest globally. Statistically we should be bringing through top talent in the professional sporting arena. However in reality this is not the case. I believe this is largely centred around the lack of encouragement for youngsters to take up sports at a grassroots level and role models to inspire them. We need to re-educate our community and try and change these mind-sets, especially those of the third generation parents. Investment should be targeted, developing facilities in inner city areas with high levels of social deprivation including cities with high concentration of Asian communities to further encourage them to get involved in sport".
Arun Kang CEO at Sporting Equals said, "On behalf of the board and the team we are delighted to welcome Prof. Peter as a Corporate Ambassador. Prof. Peter brings a wealth of experience, drive, vision and strong networks all will considerably enhance our offer. As a philanthropist he has supported numerous good causes and charities both at home and abroad and has witnessed how sport can be utilised to bring communities together".
Prof. Peter said, "As my offices are based in Central London, I witnessed firsthand the energy around London during the summer it was truly incredible. Bridging gaps in society as well as highlighting the many sports that are available for youngsters to participate in. The time is now to build on the positives that the London 2012 Olympic and Paralympic games have left, carrying forward the legacy as we should be encouraging more young people into sport".
"Through my role as an Ambassador of Sporting Equals, I will work with the organisation to make this happen. I believe in giving back to the community, and via Sporting Equals I will further help build a better future for kids in our community. The key is to polish these 'un-cut' diamonds from the Asian community and develop them into the bright sports persons that we all know they can be".
Commenting on the Future Champions programme Prof. Peter said, "I was ecstatic watching the Bhangra dancers at the opening ceremony. This showcased the Asian community's strong presence in the UK. However what was disappointing was the lack of Asians in the top tier, participating and winning medals".
Arun said "London 2012 showed the very low number of British Asian's participating in the Olympic and Paralympic games, and Prof. Peter will help to increase interest within Asian families, businesses, communities and faith centres to ensure British Asians are not missing at Olympic and Paralympic games in the future".
Prof. Peter added "Hopefully this will kick start the Asian community to get into sports, however, this will take time and the right encouragement for youngsters at a grassroots level to help us find more elite talent through this programme. I am confident that we will be watching British Asians winning the gold's, silver's and bronze medals in years to come".
Notes for Editors:
Sporting Equals exists to actively promote greater participation by the black and minority ethnic population in sport and a healthy lifestyle. It is a not-for profit organisation. We have a strong national network of delivery partners on the ground, at the heart of BME communities. We also have partnership agreements and affiliations with the leading sports governing bodies. Our job is to bring these two networks together in order to:
increase participation
open up a vast new pool of talent for sport in the UK
offer opportunities for talented sportsmen and women
change outdated attitudes and assumptions
enable BME communities to access sports and fitness facilities
share the social, personal and health benefits of an active life more widely
For more information, please contact David Mbaziira, Head of Marketing & Communications on 0121 777 1375.
For further information about Sporting Equals, please visit the website at www.sportingequals.org.uk.
© Copyright Peter Virdee. Design by Lisa Tse Ltd
Don't Worry ! You will not be spammed | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,647 |
{"url":"https:\/\/nigerianscholars.com\/past-questions\/government\/question\/276301\/","text":"Home \u00bb \u00bb Which of the following is a means of limiting the rule of law?\n\n# Which of the following is a means of limiting the rule of law?\n\n### Question\n\nWhich of the following is a means of limiting the rule of law?\n\n### Options\n\nA) Supremacy of the law\n\nB) Equality before the law\n\nC) Immunity granted to diplomats\n\nD) Strike action by workers","date":"2022-05-23 02:39:49","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8549224138259888, \"perplexity\": 1872.8024650596683}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662552994.41\/warc\/CC-MAIN-20220523011006-20220523041006-00294.warc.gz\"}"} | null | null |
\section{Introduction}
The phenomenon of MR: a process where MFLs rearrange while magnetic energy gets converted into heat and kinetic energy of the plasma, is believed to initiate flares and CMEs. Occurrence of MRs strongly depends on the MFL topology and hence it is imperative to explore their interdependence, the focus of the paper.
For the purpose, we select the well documented active region NOAA 11158 and extrapolate MFLs by using magnetograms from HMI/SDO at 01:12 UT on 15th February, 2011, which is roughly half an hour before the peak of an X2.2 class flare (1:45 UT) followed by an earth-directed CME. The coronal MFLs are constructed by NFFF extrapolation technique (\cite[Hu et al. 2008]{2008ApJ...679..848H}), resolving a physical domain of extents $\approx$ $268\times134\times134$ (in Mm)
by a computational domain having
$128\times64\times64$ grids in the $x$, $y$ and $z$ respectively. The normalized deviation of the extrapolated field at the photosphere from its magnetogram value: En=0.227; which renders the extrapolation reasonably accurate. Figure (1) overlay extrapolated MFLs on the AR image in 94 {\AA} channel. Evident is the overall good match between the MFLs and the coronal loops. Importantly, Quasi Separatrix Layers (QSLs) regions having large gradient of MFL connectivity are potential sites for MRs
(\cite[D{\'e}moulin 2006]{2006AdSpR..37.1269D}). QSLs are also found to be present whereas magnetic flux ropes are completely absent. The presence of QSLs are further confirmed by large values of the squashing factor (not shown). To explore such MRs, MHD simulations are performed using the parallelized three dimensional numerical model EULAG-MHD. The computations are carried out on the Vikram-100, the 100TF computational facility at the Physical Research Laboratory.
\begin{figure}[ht]
\centering
\includegraphics[width=.36\linewidth]{field1.jpg}
\caption{ Extrapolated MFLs plotted over 94 {\AA} image from AIA/SDO for AR
11158 at 01:12 UT. The dimension of the image is ($128\times64$) respectively.}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}[]{0.45\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{comb3.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}[]{0.42\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{fig1.jpg}
\caption{}
\end{subfigure}
\caption{(a) evolution of MFLs at t=60, 1400, 2100, 2800 depicting the formation of flux rope. (b) flipping of MFLs observed near the qsls at time t=1, 20, 40, 60.}
\label{f:obs}
\end{figure}
\section{Simulation results}
The plasma is idealized as incompressible, viscous, thermally homogeneous and having perfect electrical conductivity. The MRs, here, are simulated in the sense of Implicit Large Eddy Simulations; induced by a residual dissipation which is generated when scales get under-resolved. The panel (a) Figure (2) plots a MFL system having QSLs marked by bifurcating field lines. Subsequent evolution of the MFLs are documented in the panels (a) to (b). The MFLs are readily seen to shift their footpoint connectivity from the left of the major bifurcation to its right evident by a temporal increase in number of MFLs on the right. Such slipping reconnections are known to onset blowout jets, which are observed in the collocated region and corroborates to the efficacy of the simulation. However, the post-flare CME indicates the possible presence of a flux-rope which, are not captured at the simulation and is left as a future work.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,223 |
Friedrich Schröder Sonnenstern (* 11. September 1892 in Kaukehmen bei Tilsit als Emil Friedrich Schröder; † 10. Mai 1982 in Berlin) war ein deutscher Zeichner und Maler. Er gilt als einer der wichtigsten Vertreter der Outsider Art.
Leben
Friedrich Schröder war eines von 13 Kindern, von denen allerdings zwei unmittelbar nach der Geburt starben. Sein frühes Leben war gekennzeichnet von Aufenthalten in Erziehungs- und Irrenanstalten, letzteres wegen angeblichen Jugendirreseins (Dementia praecox), was schließlich zu seiner Entmündigung führte. Als er 1919 nach Berlin floh, beschäftigte er sich mit Okkultismus, Wahrsagerei und Heilmagnetismus. Er gründete eine Sekte und verteilte seine Einnahmen in Form von Brötchen (Schrippen) bevorzugt an Kinder, was ihm den Titel "Schrippenfürst von Schöneberg" einbrachte.
1933 wurde Sonnenstern – den Namen hatte er sich um 1928 zugelegt (Eliot Gnas von Sonnenstern) – in die Provinzial-Irren- und Heilanstalt Neustadt in Schleswig-Holstein eingewiesen, wo er den Künstler Hans Ralfs kennenlernte, der ihn zum Zeichnen erster Bilder animierte. Nach der Entlassung folgte ein dreijähriger Gefängnisaufenthalt, anschließend der kurzzeitige Dienst im Luftwaffendepot und die Abschiebung ins Arbeitslager Himmelmoor bei Quickborn. 1942 gelang ihm die Flucht nach Berlin. Unter schwierigsten Umständen überlebte er die letzten Jahre des Zweiten Weltkriegs und begann ab 1949 intensiv zu zeichnen.
Die Surrealismus-Ausstellung in Paris 1959 – L'Exposition InteRnatiOnale du Surréalisme (15. Dezember 1959 – 29. Februar 1960) – unter der Regie von André Breton und Marcel Duchamp feierte ihn als den beeindruckendsten Künstler des 20. Jahrhunderts, zu seinen Fans und Käufern zählen Henry Miller, Picasso, Max Ernst und der spätere französische Staatspräsident Georges Pompidou. Es folgen international aufsehenerregende Ausstellungen in Hamburg, Tokio, Mailand etc. Schröder-Sonnenstern zählte ab Anfang der 1970er Jahre irrtümlicherweise zur Künstlergruppe der Berliner Malerpoeten, obwohl er damit nichts zu tun hatte. Nach der Ausstellung in Paris (1959/60) kam er den Aufträgen nicht mehr nach, ließ von Gehilfen seine Bilder ausmalen und führte Details, Feinarbeiten und Korrekturen eigenhändig aus – bis die Gehilfen, auf vorsignierten Kartons Schröder-Sonnenstern-Motive kopierten, ausmalten, verkauften und ihn schließlich zum Opfer von Fälschercliquen degradierten – aber nur scheinbar, denn er war sich dessen durch und durch bewusst. Als dies bekannt wurde, ließ ihn der Kunstmarkt konsequent fallen. Seriöse Galeristen und Sammler wandten sich von ihm ab.
Friedrich Schröder-Sonnenstern war Mitglied im Deutschen Künstlerbund.
Mit dem Tod seiner Lebensgefährtin Martha Möller 1964 verlor Sonnenstern zunehmend den Halt im Leben. Er musste aus der Wohnung in der Schöneberger Crellestraße 14 ausziehen. Er wurde zum Alkoholiker und erneut in eine Nervenklinik eingeliefert. Zurückgezogen, fast vergessen und verarmt starb er 1982 im Alter von 89 Jahren in Berlin.
Sein Grab befindet sich auf dem Alten Zwölf-Apostel-Kirchhof in Berlin-Schöneberg. Die gesockelte Grabstele zeigt an der Vorderseite ein Marmorrelief sowie Inschriften und einen Sonnenstern. Auf dem Grabfeld stehen zudem zwei von Otto Drengwitz geschaffene Skulpturen.
Nachleben
2013 wurde seine Kunst wiederentdeckt und auf der Kunstbiennale in Venedig 2013 ausgestellt.
Eigenheiten seiner Werke
Seine Bilder zeigen bizarre, teils erotische, teils alptraumhafte Kreaturen, mit gewagten Kombinationen aus Mensch und Tier. Als eine Besonderheit sieht er die Darstellungen der Gesichtsteile wie Nase, Kinn und Ohr, welche er als "Männergeschlechtsteile" deutete. Er galt fälschlicherweise lange Jahre als Vertreter einer "Kunst der Geisteskranken", wurde aber im Nachhinein von Jean Dubuffet rehabilitiert, der Schröder-Sonnenstern nicht als Vertreter der Art brut bezeichnete, sondern als Vertreter der sogenannten, nicht ganz einfach abzugrenzenden Outsider Art. Schröder-Sonnenstern konterte den Vorwurf, irre zu sein mit den Worten: "Ich bin nicht verrückt, verrückt sind die, die meine Bilder nachmalen."
Literatur
Jes Petersen (Hrsg.): Die Pferdearschbetrachtung des Friedrich Schröder-Sonnenstern. München 1972.
Gerd Presler: Friedrich Schröder-Sonnenstern. In: Gerd Presler: L'Art Brut. Kunst zwischen Genialität und Wahnsinn. (= DuMont Taschenbücher; 111). DuMont, Köln 1981, ISBN 3-7701-1307-1, S. 140–145.
Jes Petersen (Hrsg.): Friedrich Schröder-Sonnenstern: Seelenerkennungsdienst. Berlin 2006.
Peter Gorsen: Friedrich Schröder-Sonnenstern. Eine Interpretation. Von Sydow-Zirkwitz, Frankfurt am Main 1962.
Hartmut Kraft: Grenzgänger zwischen Kunst und Psychiatrie. Deutscher Ärzteverlag, Köln 2005.
Alfred Bader: Geisteskranker oder Künstler. Der Fall Friedrich Schröder-Sonnenstern. Bern, Stuttgart 1972.
Jes Petersen: Friedrich der Einzige. Zum Tod von Friedrich Schroeder Sonnenstern. In: Berliner Kunstblatt. Nr. 35, 1982.
Klaus Ferentschik, Peter Gorsen: Friedrich Schröder-Sonnenstern und sein Kosmos. Parthas Verlag, Berlin 2013, ISBN 978-3-86964-069-3.
Joachim Dehne: Der Fall Schröder Sonnenstern. In: Durch welche Kriterien lassen sich moderne und schizophrene Malerei überzeugend abgrenzen? (Dissertation an der Universität Düsseldorf), 1967, teilweise abgedruckt in: Das Kunstwerk 9-10/XX, Düsseldorf, 1967.
Weblinks
Bilder in der Galerie J Möller
Outsider Bildwelten, Sammlung Demirel
Robin Pape, Burkhart Brückner: Biographie von Friedrich Emil Schröder-Sonnenstern. In: Biographisches Archiv der Psychiatrie (BIAPSY).
Einzelnachweise
Maler (Deutschland)
Grafiker (Deutschland)
Mitglied im Deutschen Künstlerbund
Bildender Künstler (Berlin)
Künstler (Art brut)
Deutscher
Geboren 1892
Gestorben 1982
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,568 |
Home Favorites Explore Countries APP
FMFLIX
Fmflix - Listen to FM Radio on Internet
Radio Pop
Radio Rock
KVIP Radio
From the beginning, KVIP was never intended to be "entertainment," although much of its programming is indeed entertaining. KVIP is a non-profit ministry, dedicated exclusively to a specific purpose: ministry to the people of the body of Christ, as an "arm" of the church, and a tool of evangelism to those who are searching and open to the saving message of the Gospel.
Recommended Stations:
977 Today's Hits
pop top40 From the Black Eyed Peas to Usher, the Hitz Channel plays Todays Best Music, not just some of it. You'll also get a taste of the old skool…
rock classic rock 80s 70s 60s Classic Rock Florida with Classic Rock Hits from the 60s, 70s, 80s through today! Making Memories from Days of Future Gone By! The music…
FOX News Talk
news talk From the fair and balanced network comes the power of FOX News on Radio! News at the top and bottom of every hour. FOX News Talk/Channel/Business Programming All…
rock r'n'b pop adult contemporary Mix 92.9 – WJXA is a broadcast radio station in Nashville, Tennessee, United States, providing Adult Contemporary Pop and Rock music.
pop news talk reggae spanish latin merengue METRO FM, es un concepto radial de mayor penetración en el mercado Latino en New York. La música en español que diariamente se…
La Mega 97.9
news reggae spanish La Mega 97.9 – WSKQ-FM is a broadcast radio station in New York City, New York, United States, providing Tropical, Salsa, Merenge and Reggeaton music.
Music fm Jazz radio Radio pop Radio rock Music fm online About FAQ Privacy Policy Terms Blog
© 2020 FMFLIX All rights reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,273 |
from rdflib.namespace import DefinedNamespace, Namespace
from rdflib.term import URIRef
class TIME(DefinedNamespace):
"""
OWL-Time
Generated from: http://www.w3.org/2006/time#
Date: 2020-05-26 14:20:10.531265
"""
# http://www.w3.org/2000/01/rdf-schema#Datatype
generalDay: URIRef # Day of month - formulated as a text string with a pattern constraint to reproduce the same lexical form as gDay, except that values up to 99 are permitted, in order to support calendars with more than 31 days in a month. Note that the value-space is not defined, so a generic OWL2 processor cannot compute ordering relationships of values of this type.
generalMonth: URIRef # Month of year - formulated as a text string with a pattern constraint to reproduce the same lexical form as gMonth, except that values up to 20 are permitted, in order to support calendars with more than 12 months in the year. Note that the value-space is not defined, so a generic OWL2 processor cannot compute ordering relationships of values of this type.
generalYear: URIRef # Year number - formulated as a text string with a pattern constraint to reproduce the same lexical form as gYear, but not restricted to values from the Gregorian calendar. Note that the value-space is not defined, so a generic OWL2 processor cannot compute ordering relationships of values of this type.
# http://www.w3.org/2002/07/owl#Class
DateTimeDescription: URIRef # Description of date and time structured with separate values for the various elements of a calendar-clock system. The temporal reference system is fixed to Gregorian Calendar, and the range of year, month, day properties restricted to corresponding XML Schema types xsd:gYear, xsd:gMonth and xsd:gDay, respectively.
DateTimeInterval: URIRef # DateTimeInterval is a subclass of ProperInterval, defined using the multi-element DateTimeDescription.
DayOfWeek: URIRef # The day of week
Duration: URIRef # Duration of a temporal extent expressed as a number scaled by a temporal unit
DurationDescription: URIRef # Description of temporal extent structured with separate values for the various elements of a calendar-clock system. The temporal reference system is fixed to Gregorian Calendar, and the range of each of the numeric properties is restricted to xsd:decimal
GeneralDateTimeDescription: URIRef # Description of date and time structured with separate values for the various elements of a calendar-clock system
GeneralDurationDescription: URIRef # Description of temporal extent structured with separate values for the various elements of a calendar-clock system.
Instant: URIRef # A temporal entity with zero extent or duration
Interval: URIRef # A temporal entity with an extent or duration
MonthOfYear: URIRef # The month of the year
ProperInterval: URIRef # A temporal entity with non-zero extent or duration, i.e. for which the value of the beginning and end are different
TRS: URIRef # A temporal reference system, such as a temporal coordinate system (with an origin, direction, and scale), a calendar-clock combination, or a (possibly hierarchical) ordinal system. This is a stub class, representing the set of all temporal reference systems.
TemporalDuration: URIRef # Time extent; duration of a time interval separate from its particular start position
TemporalEntity: URIRef # A temporal interval or instant.
TemporalPosition: URIRef # A position on a time-line
TemporalUnit: URIRef # A standard duration, which provides a scale factor for a time extent, or the granularity or precision for a time position.
TimePosition: URIRef # A temporal position described using either a (nominal) value from an ordinal reference system, or a (numeric) value in a temporal coordinate system.
TimeZone: URIRef # A Time Zone specifies the amount by which the local time is offset from UTC. A time zone is usually denoted geographically (e.g. Australian Eastern Daylight Time), with a constant value in a given region. The region where it applies and the offset from UTC are specified by a locally recognised governing authority.
# http://www.w3.org/2002/07/owl#DatatypeProperty
day: URIRef # Day position in a calendar-clock system. The range of this property is not specified, so can be replaced by any specific representation of a calendar day from any calendar.
dayOfYear: URIRef # The number of the day within the year
days: URIRef # length of, or element of the length of, a temporal extent expressed in days
hasXSDDuration: URIRef # Extent of a temporal entity, expressed using xsd:duration
hour: URIRef # Hour position in a calendar-clock system.
hours: URIRef # length of, or element of the length of, a temporal extent expressed in hours
inXSDDate: URIRef # Position of an instant, expressed using xsd:date
inXSDDateTimeStamp: URIRef # Position of an instant, expressed using xsd:dateTimeStamp
inXSDgYear: URIRef # Position of an instant, expressed using xsd:gYear
inXSDgYearMonth: URIRef # Position of an instant, expressed using xsd:gYearMonth
minute: URIRef # Minute position in a calendar-clock system.
minutes: URIRef # length, or element of, a temporal extent expressed in minutes
month: URIRef # Month position in a calendar-clock system. The range of this property is not specified, so can be replaced by any specific representation of a calendar month from any calendar.
months: URIRef # length of, or element of the length of, a temporal extent expressed in months
nominalPosition: URIRef # The (nominal) value indicating temporal position in an ordinal reference system
numericDuration: URIRef # Value of a temporal extent expressed as a decimal number scaled by a temporal unit
numericPosition: URIRef # The (numeric) value indicating position within a temporal coordinate system
second: URIRef # Second position in a calendar-clock system.
seconds: URIRef # length of, or element of the length of, a temporal extent expressed in seconds
week: URIRef # Week number within the year.
weeks: URIRef # length of, or element of the length of, a temporal extent expressed in weeks
year: URIRef # Year position in a calendar-clock system. The range of this property is not specified, so can be replaced by any specific representation of a calendar year from any calendar.
years: URIRef # length of, or element of the length of, a temporal extent expressed in years
# http://www.w3.org/2002/07/owl#DeprecatedClass
January: URIRef # January
Year: URIRef # Year duration
# http://www.w3.org/2002/07/owl#DeprecatedProperty
inXSDDateTime: URIRef # Position of an instant, expressed using xsd:dateTime
xsdDateTime: URIRef # Value of DateTimeInterval expressed as a compact value.
# http://www.w3.org/2002/07/owl#FunctionalProperty
hasTRS: URIRef # The temporal reference system used by a temporal position or extent description.
# http://www.w3.org/2002/07/owl#ObjectProperty
after: URIRef # Gives directionality to time. If a temporal entity T1 is after another temporal entity T2, then the beginning of T1 is after the end of T2.
dayOfWeek: URIRef # The day of week, whose value is a member of the class time:DayOfWeek
hasBeginning: URIRef # Beginning of a temporal entity.
hasDateTimeDescription: URIRef # Value of DateTimeInterval expressed as a structured value. The beginning and end of the interval coincide with the limits of the shortest element in the description.
hasDuration: URIRef # Duration of a temporal entity, event or activity, or thing, expressed as a scaled value
hasDurationDescription: URIRef # Duration of a temporal entity, expressed using a structured description
hasEnd: URIRef # End of a temporal entity.
hasTemporalDuration: URIRef # Duration of a temporal entity.
hasTime: URIRef # Supports the association of a temporal entity (instant or interval) to any thing
inDateTime: URIRef # Position of an instant, expressed using a structured description
inTemporalPosition: URIRef # Position of a time instant
inTimePosition: URIRef # Position of a time instant expressed as a TimePosition
inside: URIRef # An instant that falls inside the interval. It is not intended to include beginnings and ends of intervals.
intervalAfter: URIRef # If a proper interval T1 is intervalAfter another proper interval T2, then the beginning of T1 is after the end of T2.
intervalBefore: URIRef # If a proper interval T1 is intervalBefore another proper interval T2, then the end of T1 is before the beginning of T2.
intervalContains: URIRef # If a proper interval T1 is intervalContains another proper interval T2, then the beginning of T1 is before the beginning of T2, and the end of T1 is after the end of T2.
intervalDisjoint: URIRef # If a proper interval T1 is intervalDisjoint another proper interval T2, then the beginning of T1 is after the end of T2, or the end of T1 is before the beginning of T2, i.e. the intervals do not overlap in any way, but their ordering relationship is not known.
intervalDuring: URIRef # If a proper interval T1 is intervalDuring another proper interval T2, then the beginning of T1 is after the beginning of T2, and the end of T1 is before the end of T2.
intervalEquals: URIRef # If a proper interval T1 is intervalEquals another proper interval T2, then the beginning of T1 is coincident with the beginning of T2, and the end of T1 is coincident with the end of T2.
intervalFinishedBy: URIRef # If a proper interval T1 is intervalFinishedBy another proper interval T2, then the beginning of T1 is before the beginning of T2, and the end of T1 is coincident with the end of T2.
intervalFinishes: URIRef # If a proper interval T1 is intervalFinishes another proper interval T2, then the beginning of T1 is after the beginning of T2, and the end of T1 is coincident with the end of T2.
intervalIn: URIRef # If a proper interval T1 is intervalIn another proper interval T2, then the beginning of T1 is after the beginning of T2 or is coincident with the beginning of T2, and the end of T1 is before the end of T2, or is coincident with the end of T2, except that end of T1 may not be coincident with the end of T2 if the beginning of T1 is coincident with the beginning of T2.
intervalMeets: URIRef # If a proper interval T1 is intervalMeets another proper interval T2, then the end of T1 is coincident with the beginning of T2.
intervalMetBy: URIRef # If a proper interval T1 is intervalMetBy another proper interval T2, then the beginning of T1 is coincident with the end of T2.
intervalOverlappedBy: URIRef # If a proper interval T1 is intervalOverlappedBy another proper interval T2, then the beginning of T1 is after the beginning of T2, the beginning of T1 is before the end of T2, and the end of T1 is after the end of T2.
intervalOverlaps: URIRef # If a proper interval T1 is intervalOverlaps another proper interval T2, then the beginning of T1 is before the beginning of T2, the end of T1 is after the beginning of T2, and the end of T1 is before the end of T2.
intervalStartedBy: URIRef # If a proper interval T1 is intervalStarted another proper interval T2, then the beginning of T1 is coincident with the beginning of T2, and the end of T1 is after the end of T2.
intervalStarts: URIRef # If a proper interval T1 is intervalStarts another proper interval T2, then the beginning of T1 is coincident with the beginning of T2, and the end of T1 is before the end of T2.
monthOfYear: URIRef # The month of the year, whose value is a member of the class time:MonthOfYear
timeZone: URIRef # The time zone for clock elements in the temporal position
unitType: URIRef # The temporal unit which provides the precision of a date-time value or scale of a temporal extent
# http://www.w3.org/2002/07/owl#TransitiveProperty
before: URIRef # Gives directionality to time. If a temporal entity T1 is before another temporal entity T2, then the end of T1 is before the beginning of T2. Thus, "before" can be considered to be basic to instants and derived for intervals.
# http://www.w3.org/2006/time#DayOfWeek
Friday: URIRef # Friday
Monday: URIRef # Monday
Saturday: URIRef # Saturday
Sunday: URIRef # Sunday
Thursday: URIRef # Thursday
Tuesday: URIRef # Tuesday
Wednesday: URIRef # Wednesday
# http://www.w3.org/2006/time#TemporalUnit
unitDay: URIRef # day
unitHour: URIRef # hour
unitMinute: URIRef # minute
unitMonth: URIRef # month
unitSecond: URIRef # second
unitWeek: URIRef # week
unitYear: URIRef # year
_NS = Namespace("http://www.w3.org/2006/time#")
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,282 |
{"url":"https:\/\/math.stackexchange.com\/questions\/882256\/why-does-the-imaginary-number-i-satisfy-i-times-0-0","text":"# Why does the imaginary number $i$ satisfy $i\\times 0=0$?\n\nWhy does the imaginary number $i$ satisfy $i\\times 0=0$? I mean, we don't really know what $i$ is. How could we be sure about that? I think there's a reason behind why mathematicians decided that.\n\n\u2022 We've understood complex numbers for a few centuries now; the meaning of mathematical word \"imaginary\" hasn't had any relation to the meaning of the English word \"imaginary\" for a long time. \u2013\u00a0user14972 Jul 30 '14 at 5:11\n\u2022 How is closing this helpful? It is clear what the question is, and it is a good un'! It is also not obvious to me how to prove $0\\times i=0$ given simply that $i^2=-1$. It is a theorem, not something we have decided. Thus, an answer will prove this theorem! (As, for example, user13157 has done.) \u2013\u00a0user1729 Jul 30 '14 at 10:07\n\u2022 @Did Hmm, I don't know. How do you define $\\mathbb{C}$ if not as \"the real numbers with an element $i$ such that $i^2=-1$\"? (Implicitly, we assume brackets work as we like.) Then the result follows according to user13157's proof (and I remember people struggling with the analogous proof in a first course on rings - I do not think \"trivial\" is really the best word!). \u2013\u00a0user1729 Jul 30 '14 at 10:23\n\u2022 @Did I have three relevant books on my desk: Calculus: A complete Course by Robert A. Adams, Engineering Mathematics by K.A.Stroud, and Fundamentals of University Mathematics by McGregor, Nimmo and Stothers. The first two, Adams and Stroud, go for the \"$i^2=-1$ and complex numbers have the form $ai+b$, $a, b\\in\\mathbb{R}$\" definition, while McGregor, Nimmo and Stothers start with the complex plane and work backwards. So...I would think that the \"extension\" was quite common. No? \u2013\u00a0user1729 Jul 30 '14 at 10:52\n\u2022 @robjohn If you want to learn more about the general way in which complex numbers were intorduced to Did, look up the \"third\" book I mention in my comments above, Fundamentals of University Mathematics by McGregor, Nimmo and Stothers. The main advantage of this method, as I see it, is that you start with the complex plane. So it is more geometric and less algebraic. \u2013\u00a0user1729 Aug 6 '14 at 8:29\n\nIgnore this answer if you've never heard of matrix multiplication. Better yet, learn matrix multiplication and then read this answer!\n\nThe answer to your question depends on what definition of complex numbers you're using.\n\nFor example, I like to think of the complex numbers as matrices of the form $$\\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix}$$ where $a$ and $b$ are real numbers. This allows us to define two complex numbers \\begin{align*} \\mathbf 0 &= \\begin{pmatrix} 0 & 0\\\\ 0 & 0 \\end{pmatrix} & i &= \\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix} \\end{align*} Hence we have the identity $$\\mathbf 0\\cdot i = \\begin{pmatrix} 0 & 0\\\\ 0 & 0 \\end{pmatrix} \\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix} = \\begin{pmatrix} 0\\cdot 0+0\\cdot 1 & 0\\cdot(-1)+0\\cdot 0\\\\ 0\\cdot 0+0\\cdot 1 & 0\\cdot (-1)+0\\cdot 0 \\end{pmatrix} = \\begin{pmatrix} 0 & 0\\\\ 0 & 0 \\end{pmatrix} = \\mathbf0$$\n\nOne of the nice things about defining the complex numbers this way is that we avoid the confusing equation $i=\\sqrt{-1}$. We also don't have to resort to using silly words like \"imaginary.\"\n\nNote, however, that we do have $$i^2 = \\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix}^2 = \\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix}\\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix} = \\begin{pmatrix} -1 & 0 \\\\ 0 & -1 \\end{pmatrix}$$ So, if we define the complex number $$\\mathbf{1}= \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}$$ then we recover the formula $$i^2=-\\mathbf{1}$$ To convince yourself that our definition of the complex number $\\mathbf1$ is not arbitrary, note that $\\mathbf 1$ enjoys the property $$\\mathbf 1 \\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix} = \\begin{pmatrix} 1 & 0\\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix} = \\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix}$$ This is remarkably similar to the celebrated equation $1\\cdot a=a$. It's also worth noting that the complex number $\\mathbf0$ satisfies $$\\mathbf0+ \\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix} = \\begin{pmatrix} 0&0\\\\0&0 \\end{pmatrix}+ \\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix} = \\begin{pmatrix} 0+a & 0-b \\\\ 0+b & 0+a \\end{pmatrix} = \\begin{pmatrix} a & -b \\\\ b & a \\end{pmatrix}$$ which is similar to our usual equation $0+a=a$.\n\nThe point here is that complex-arithmetic \"feels\" like ordinary arithmetic but is indeed different. As @Hurkyl points out, algebraic gadgets that behave like this are called rings.\n\nJust like real numbers, we can write: $0z = (0+0)z = 0z + 0z$. Canceling like terms on each side allows us to see that $0 = 0z$. Thus, this is really a property of zero itself.\n\nAlso, as others have mentioned, complex numbers (including imaginaries) are fairly well understood. However, one important nuance is that @iHubble's definition quickly leads to contradictions (such as using properties of square roots to show that $-1 = i^2 = \\sqrt{-1}\\sqrt{-1} = \\sqrt{(-1)^2} = 1$). It is best to instead always define $i$ as a number such that $i^2 = -1$.\n\n\u2022 It's not a \"property of zero itself\". Rather, it is a property of any algebraic structure (e.g. ring) which satisfies the laws that you employed in your proof, e.g. the distributive law, and additive cancellation law. \u2013\u00a0Bill Dubuque Aug 5 '14 at 13:10\n\u2022 @BillDubuque: I think he means it is a property of $0$ in any ring, in this case, $\\mathbb{C}$. \u2013\u00a0robjohn Aug 5 '14 at 22:15\n\u2022 @robjohn Possibly. The point of my comment was to help clarify that, i.e. to whittle it down to the algebraic essence of the matter (which may not be so clear to readers who have not had much experience with abstract algebra) \u2013\u00a0Bill Dubuque Aug 5 '14 at 22:25\n\nThe complex numbers are defined to satisfy all of the usual ring axioms: i.e. all of the usual identities involving $0,1,+,-,\\times,\\div$ hold for complex numbers. $0z=0$ is one of those identities.\n\nTechnically, $0\\,i\\not\\in\\mathbb{R}$; instead, $0\\,i\\in\\mathbb{C}$. Thus, being (overly?) pedantic, $0\\times i\\ne0$; instead $0\\times i=0+0\\,i$. However, being the zero element of $\\mathbb{C}$, we usually abbreviate $0+0\\,i$ as $0$, not meaning an element of $\\mathbb{R}$ but an element of the subset of $\\mathbb{C}$ that is homeomorphic to $\\mathbb{R}$, that is $\\{x+0\\,i:x\\in\\mathbb{R}\\}$. With this abbreviation, we do have $0\\times i=0$\n\nThis comment by AlexB says\n\nThe thing is that when one defines the integers, they come with a canonical embedding of $\\mathbb{N}$. Similarly, the rationals come with a canonical embedding of $\\mathbb{Z}$ and so on. So it does make sense to speak of subsets. Indeed, it becomes so cumbersome to distinguish between genuine subsets and images under an embedding that one fairly quickly drops this distinction in practice, unless one really has to thing about the foundations.\n\nThe question \"Why does the imaginary number $i$ satisfy $i\\times0=0$?\" seems to me to be fairly foundational, so I think that my approach above seems appropriate.\n\nOnce we have established why $i\\times0=0$, then we can move on as Arturo Magidin says in this answer:\n\nSo even though they are actually very different sets, we have copies of each sitting inside the \"next one\", copies that respect all the structures we are interested in, so we can still think of them as being \"subsets\".\n\n\u2022 would the downvoter care to comment? \u2013\u00a0robjohn Aug 6 '14 at 5:08\n\u2022 I didn't downvote, but I am not convinced by your pedantry. (Although it certainly made me think!) Are you also claiming that $i^2=-1\\not\\in\\mathbb{R}$? (Mainly I am not convinced because, if we adopt the language of rings which seems to be the fashion in these answers, $0i$ is contained in the subring $\\mathbb{R}$, $0i\\in\\mathbb{R}$. No?) \u2013\u00a0user1729 Aug 6 '14 at 8:22\n\u2022 @user1729: $\\mathbb{R}$ is one-dimensional. $\\{x+0\\,i:x\\in\\mathbb{R}\\}$ is a one-dimensional subspace of $\\mathbb{C}$. They are isomorphic, but not the same. It is possible that after $\\mathbb{C}$ is introduced, $\\mathbb{R}$ could be redefined to be this one-dimensional subspace of $\\mathbb{C}$, but this feels circular to me. \u2013\u00a0robjohn Aug 6 '14 at 12:17\n\u2022 Another downvote without comment? Consider this: is $(0,0)\\in\\mathbb{R}$? How can $\\mathbb{R}$ be a subset of $\\mathbb{C}$ when we can't even define $\\mathbb{C}$ without $\\mathbb{R}$ so that either $\\mathbb{C}=\\{x+iy:x,y\\in\\mathbb{R}\\}$ or $\\mathbb{C}=\\{(x,y):x,y\\in\\mathbb{R}\\}$. There seems to be a confusion between $\\mathbb{R}$ and the subspace of $\\mathbb{C}$ that is homeomorphic to $\\mathbb{R}$. \u2013\u00a0robjohn Aug 7 '14 at 0:52\n\u2022 There is a well-established tradition in mathematics of identification that permeates it at all levels. This practice is endlessly convenient and enlightening. E.g. we identify things with other things in order to get $\\Bbb N\\subset\\Bbb Z\\subset\\Bbb Q\\subset\\Bbb R\\subset\\Bbb C$. It's true that on a machine-language level, pedantry is useful for rigorous constructions, but after the constructions are done or trivial or subconscious and we've moved on, it is no longer useful but distracting and gets in the way of intuitively seeing the big picture and what's \"morally correct.\" \u2013\u00a0blue Aug 7 '14 at 1:38","date":"2021-07-30 02:26:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9216729998588562, \"perplexity\": 314.2990871932964}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046153899.14\/warc\/CC-MAIN-20210729234313-20210730024313-00080.warc.gz\"}"} | null | null |
Fra Carnevale, (kolem 1420-1425, Urbino, Itálie – 1484, Urbino, Itálie) byl italský renesanční malíř aktivní v období italské renesance, tzv. quattrocenta. Působil hlavně v Urbinu. Carnivale je obecně považován za jednoho z nejzáhadnějších umělců. Existuje pouze devět děl, která mu lze s konečnou platností přičíst. Většina z nich byla uměleckými kritiky v různých obdobích jako autentická díla tohoto umělce zpochybňována.
Používal řadu jmen: Bartolomeo di Giovanni Corradini, Bartolomeo Coradini a Fra 'Carnevale.
Životopis
Fra Carnevale se narodil se v Urbinu a do řádu dominikánů vstoupil v roce 1449 pod jménem Fra 'Carnevale nebo Carnovale. Byl žákem ferrarského malíře Antonia Albertiho. Farquhar tvrdí, že byl učitelem Giovanni Santiho, otce Raffaela Santiho. V letech 1445-1446 pracoval v ateliéru Filippa Lippiho ve Florencii. Poté, někdy před rokem 1450, se vrátil do Urbina a vstoupil do řádu dominikánů. Historici prokázali jeho aktivity v Urbinu letech 1456 až 1488. Během této doby se učil u Fra Jacopa Veneta. Byl pověřen namalovat oltářní obraz v del Corpus Domini, ale jeho práce byla ukončena v roce 1456. Dokument z roku 1467 zaznamenává platbu za oltářní obraz v kostele Santa Maria della Bella. Ze záznamů také víme, že byl kurátorem v San Cassiano del Cavallino a připojil se k Bratrstvu Svatého Kříže (Confraternità di Santa Croce).
Po celá staletí existovala pouze jediná zmínka o tomto malíři a to v bibliografii "Životy nejvýznačnějších malířů, sochařů a architektů" Giorgia Vasariho. Zde Vasari označil Carnevaleho jako "Carnovale da Urbino", malíře oltářního obrazu v kostele Santa Maria della Bella v Urbinu. Píše, že byl ovlivněn italským malířem a architektem Donatem Bramantem a jeho prací v Bazilice Svatého Petra ve Vatikánu. Filippo Baldinucci ve svém Dictionary of Masters of Disegno (Slovník mistrů kresby) popsal Fra Carnevaleho jako žáka, který byl pověstný svým excelentním zvládnutím umění perspektivy. Italský historik umění a archeolog Luigi Lanzi (14. června 1732 - 30. března 1810) ve svém Storia Pittorica dela Italia (Historie Itálie v obrazech) píše o Fra Carnevalem : "Bramante a Raffael studovali jeho práci, protože v Urbinu nebyl nikdo lepší." Ačkoli byl poměrně nemilosrdný při posuzování perspektivy použité na oltářním obraze, pochválil jeho architekturu. Carnivale byl také architektem portálů kostela San Domenico v Urbinu a ve svých obrazech položil základy využití perspektivy s důrazem na architekturu.
Carnevale se obklopil prominentními členy místní společnosti, včetně právníka Guida Bonclericiho, generálního vikáře biskupa Giovanniho Battisty Melliniho, Ottaviana Ubaldiniho, který byl vlivnou osobou u dvora v Urbinu a Matteo di Cataneisem, který měl také blízko k vévodům. Jeho obrazy odrážejí tuto zkušenost s elitářskou kulturou společnosti více než by se dalo očekávat od člena významného náboženského řádu. Byl duchovním asketou, v rámci svých zakázek však dostal jméno "carnevale", což znamená "půjčil".
Ovlivněn
Carnevaleova první díla ukázala vliv Dominica Veneziana. Na základě platby, kterou přijal Fra Carnevale jménem Antonia Albertiho, se však předpokládá, že se u tohoto malíře učil ve 30. letech 14. století. Proto přijíždí do Florencie v roce 1445 "jako žák, už ne jako učeň, což naznačuje, že jeho učení už bylo ukončeno, pravděpodobně se učil u mnicha Jacopa Veneta; s Antoniem Albertim ho spojuje jiný dokument.
Po návratu do Urbina zahájil Carnevale další architektonický projekt a přivedl umělce z Florencie, jako jsou Maso di Bartolomeo a Luca della Robbia.
Jeho styl ztvárnění obličeje a techniky malby záhybů tkanin jsou známy i z děl Piera della Francesca. Přestože jsou jeho obrazy obecně považovány za perspektivně nepřesné, v jeho prospěch mluví použití motivu architektonického pozadí. Preciznost jeho stylu je založena na jeho znalostech architekta. Carnevaleovi jsou také připisovány architektonické návrhy katedrály v Urbinu.
Obrazy
Obraz Ideální město, o kterém se v knihách o teorii a historii městského designu velmi často hovoří, a který je umístěn ve Waltersově muzeu umění v Baltimoru, je jedním ze tří obrazů podobného stylu a je přičítán Fra Carnevalemu. Obraz je však některými historiky přisuzován Francescovi di Giorgio Martini, částečně pro větší význam tohoto umělce u vévodského dvora v Urbinu a také proto, že obraz ukazuje architektonická témata, která jsou publikovaná v pojednání Leona Battisty Albertiho. Tento obraz ukazuje Carnevaleho silný cit pro architekturu a její znalost. Lineární perspektiva a trojrozměrné detaily fasád budov jsou dokonalé, a to vše je ve stylu Carnevaleho práce.
Pouze jedno z děl Fra Carnevale je na původním místě: v Urbinu vymaloval Carnevale niku v Palazzo Ducale, dóžecím paláci Federica da Montelfeltro. Mezi osm dalších děl připisovaných Carnevalovi patří oltářní obraz v Santa Maria della Bella Zrození Panny Marie, Představení Panny Marie v chrámu, Zvěstování, vytvořeném ve spolupráci s Lippim, Korunování Panny Marie, Ukřižování, Svatý Jan Křtitel na poušti a Portrét člověka. Carnevaleho autorství některých z těchto prací je často zpochybňováno, pokud jde o Carnevaleovu ruku - co se týká obrazů Korunování Panny Marie, Ukřižování, Svatý Jan Křtitel v poušti a Portrét člověka, u těchto děl mají historici rozporuplné názory. Nicméně výstava Carnevaleových děl v Miláně v roce 2004 definitivně připsala těchto devět děl Carnevalemu.
Zrození Panny Marie a Představení Panny Marie v chrámu nyní připisované Carnevalemu, jsou nyní umístěny v Metropolitním muzeu v New Yorku a v Muzeu výtvarného umění v Bostonu. Panely byly součástí oltářního obrazu objednaného pro kostel Santa Maria della Bella v Urbinu v roce 1467. Části oltářního obrazu se staly známým jako "panely Barberini", když je v roce 1632 Antonio Barberini odvezl do Říma. Oba obrazy jsou kompozičně moderní a ukazují vliv humanismu a antických ideálů. Architektonické aspekty jsou opět silné a v popředí a v reliéfech budov v římském stylu jsou ukázány velké detaily, odkazující na architekturu vévodského paláce v Urbinu. Další charakteristickou vlastností Carnevaleho je hra světla a stínu na řasení oděvu a na budovách. Zrození Panny Marie je nekonvenční v kompozici, protože dítě není v popředí a není ústředním bodem obrazu, zatímco v Představení Panny Marie v chrámu Carnevale umístil do popředí postavy tradičních židovských kněží.
V roce 1930 se oba obrazy objevily na výstavě italského umění v Royal Academy of Arts v Londýně. V roce 2004 byly odeslány na výstavu Carnevaleových prací v Miláně. V té době bylo Carnevalemu definitivně přičítáno pouze devět obrazů. Uvažuje se ovšem, že další jeho díla existují v soukromých sbírkách.
Zvěstování je v současné době v National Gallery of Art ve Washingtonu, DC. Na tomto obraze stejně jako na svých jiných dílech Carnevale použil jasné barvy u budov a na oblečení s dokonale ztvárněnou plastičností šatu. Kompozičně netradíční je umístění postav na ulici.
Portrét člověka, jedno z kontroverzních děl, vypadá téměř jako římský reliéf jedné z budov v dalších Carnevalových obrazech, ale s přidanou barvou. Hluboké, jako vytesané detaily vlasů připomínají malířský styl použitý pro těžkou zvlněnou látku na "panelech Barberini". Detail svalů a žil na krku je velmi naturalistický. Tento obraz však postrádá obvyklou živost barvy, která je známá v Carnevaleho dílech.
O obrazech Svatý Jan Křtitel na poušti, Ukřižování, Svatý Petr a Svatý František se vedla diskuze zda byly součástí jednoho oltářního obrazu. Zkoumání provedené pro výstavu v Metropolitním muzeu ukazuje, že navzdory rozdílům v rozměrech byla tato čtyři díla skutečně součástí polyptychu. Ukřižování bylo dříve přičítáno Giovannimu Boccatimu, Pierovi della Francescovi či Domenico Venezianovi, ale protože dílo odkazovalo více na lombardské umělecké styly než florentskou kultura Lippiho nebo benátskou školu, bylo nakonec přisouzeno Carnevalemu. Při zkoumání těchto čtyř těchto prací vidíme přímo modelové zpracování drapérie s jejími hlubokými stíny a citlivost ve způsobu, jakým jsou zvýrazněny barvy. Dokonce i skály v pozadí Ukřižování se zdají být látkou, nikoliv pevným kamenem.
Kresby
Jeden výkres, který je nyní přičítán Fra Carnevalemu je zřejmě kresbou ze stránky Vasariho Libro de 'Disegni (Kniha kreseb). Tato kresba byla původně přičítána Pierovi della Francesca. Zdá se však, že kresba pochází z Florencie z doby dospívání Fra Carnevale. Kresba zobrazuje postavu mladíka v bez oděvu, který v hrdém postoji ukazuje znetvořený vzhled. Stylisticky jeho mládí ukazuje našpulený výraz, který je ochrannou známkou Carnevaleho kresby, stejně jako standardizovaná forma Carnevaleovy charakterizace postavy a detailů šatů postavy. Fra Carnevale, známý spíše pro svou schopnost přesvědčivě využívat stín a světlo, nedokázal u svých lidských subjektů vytvořit stejně přesvědčivou fyziologii. Jeho postavy vypadají pokřivené a výsledek je spíše plochý než živý.
Libro de 'Disegni obsahuje další kresbu, kterou pravděpodobně dokončil Fra Carnevale. Toto zobrazení jedenácti mužských aktů bylo původně přičítáno Domenico Venezianovi. Dílo bylo pravděpodobně dokončeno mezi lety 1445 a 1450, kdy se Fra Carnevale učil u Filippa Lippiho.
Autorství další sady kreseb Fra Carnevaleho je kontroverznější. Kresby vycházejí z obrazu Filippa Lippiho Korunování Panny. Existují dvě reprodukce tohoto díla, které možná zhotovil Fra Carnevale; kompozice je stojící žena a klečící mnich. Vzhledem k tomu, že se jedná o reprodukce obrazů jiného umělce, je připisování Fra Carnevalemu nejisté.
Galerie
Odkazy
Reference
Literatura
Matteo Ceriana, Keith Christiansen, Emanuela Daffra, Andrea De Marchi, Fra Carnevale. Un artista rinascimentale da Filippo Lippi a Piero della Francesca, catalogo della mostra curata dalla Pinacoteca di Brera, Edizioni Olivares, 2004, ISBN 88-85982-85-9
Externí odkazy
Biography in Italian.
Italští malíři
Dominikáni
Narození v roce 1430
Osoby s nejistým datem narození
Narození v Urbinu
Úmrtí v roce 1498
Osoby s nejistým datem úmrtí
Úmrtí v Urbinu
Muži | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,856 |
Kurt Geiger sold to Cinven
Veebs Sabharwal
Private equity firm Cinven has bought Kurt Geiger as part of its Christmas shopping, for an undisclosed sum.
Cinven picked up the footwear and accessories retailer from Sycamore partners. The Kurt Geiger package includes a portfolio of brands (such as Kurt Geiger London, KG, Miss KG and Carvela), 80 standalone global stores, partnerships with international brands and 240 concessions within renowned department stores including Harrods and Selfridges.
The UK footwear market is growing. Currently valued at around £8bn, it's a sector with 3% annual forecast growth.
The buyout firm outlined that Kurt Geiger has a proven omni-channel business model, operating thirty party concessions, owned and international franchised stores together with high-growth ecommerce sites.
The takeover was seen as a "highly attractive investment opportunity" based on further expansion opportunities, both through in-market consolidation, digitisation strategy and international growth, including further rollouts in Asia and Australia.
There is also potential to expand the upmarket retailer's own brand offering into adjacent categories such as the children's footwear segment.
Cinven also said that there is a proven track record of consistent growth during economic cycles, confirming the "highly regarded management team" (CEO Neil Clifford, CFO Dale Christilaw and Creative Director Rebecca-Farrar-Hockley) will continue in their key roles under new ownership.
"Kurt Geiger represents an exciting opportunity to acquire the UK's leading women's footwear and accessories company with significant international growth and consolidation opportunities," commented Maxim Crewe, Partner at Cinven. "The business enjoys an attractive specialist retail model with multiple brands and routes to market. It is led by a strong, experienced and committed management team and we are looking forward to working with them to achieve the next phase of growth."
Kurt Geiger is the UK's largest shoe retailer by sales. In the year ended 31 December 2014, it generated sales of £260m and continues to invest in new and existing outlets throughout the UK.
Cinven
Jigsaw board reshuffles with spate of appointments
Kurt Geiger owner mulls selling retailer to US group for £450m
Liam Gallagher's Pretty Green expected to appoint administrators
Brighton's £100m Hanningtons Lane development opens next month
Welcome to Charged!
Superdry co-founder: "I can't just sit back & watch 30 years of my life be gently eroded"
CHARGED: Amazon workers strike in Germany over "discount on incomes"
BRC acquires Oxford Summer School
Usdaw to protest Boohoo AGM today | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,538 |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/precalculus\/precalculus-6th-edition\/chapter-1-equations-and-inequalities-1-4-quadratic-equations-1-4-exercises-page-122\/55","text":"Chapter 1 - Equations and Inequalities - 1.4 Quadratic Equations - 1.4 Exercises: 55\n\n$\\color{blue}{\\left\\{1-2i, 1+2i\\right\\}}$\n\nWork Step by Step\n\nSubtract $2x-5$ to both sides of the equation to obtain: $x^2-(2x-5)=2x-5-(2x-5) \\\\x^2-2x-(-5)=0 \\\\x^2-2x+5=0$ RECALL: The quadratic equation $ax^2+bx+c=0$ can be solved using the quadratic formula: $x=\\dfrac{-b\\pm\\sqrt{b^2-4ac}}{2a}$ The given equation has: $a=1, b=-2 c=5$ Substitute these values into the quadratic formula to obtain: $x=\\dfrac{-(-2) \\pm \\sqrt{(-2)^2-4(1)(5)}}{2(1)} \\\\x=\\dfrac{2\\pm\\sqrt{4-20}}{2} \\\\x=\\dfrac{2\\pm\\sqrt{-16}}{2} \\\\x=\\dfrac{2\\pm\\sqrt{16(-1)}}{2} \\\\x=\\dfrac{2\\pm 4i}{2} \\\\x=1 \\pm 2i$ Thus, the solution set is $\\color{blue}{\\left\\{1-2i, 1+2i\\right\\}}$.\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2018-07-17 23:42:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6018734574317932, \"perplexity\": 371.1225204647153}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676589932.22\/warc\/CC-MAIN-20180717222930-20180718002930-00577.warc.gz\"}"} | null | null |
Als graue Literatur, gelegentlich auch graue Materialien, bezeichnet man in der Bibliothekswissenschaft Publikationen, die nicht vom kommerziellen Verlagswesen kontrolliert werden und nicht im Buchhandel erhältlich sind. Sie werden meist von Institutionen oder Organisationen veröffentlicht. Dabei handelt es sich vor allem um: Regierungsstellen, Behörden, nationale und internationale Organisationen, Forschungseinrichtungen, Hochschulen, Schulen, Museen, Bibliotheken, Firmen, Verbände, Vereine, Parteien und Gewerkschaften. Beispiele für solche graue Literatur sind: Forschungsberichte, Privatdrucke, Firmenschriften, Kongressberichte und bestimmte akademische Schriften. Privatdrucke werden teilweise auch im Auftrag und auf Kosten von Privatpersonen ohne kommerzielle Absicht hergestellt. Die Auflage ist oft sehr klein.
Deutsche Titel werden in Deutschland in der Deutschen Nationalbibliografie, Reihe B, veröffentlicht. Internetpublikationen werden dabei nicht vollständig von der Deutschen Nationalbibliografie erfasst.
Texte, die der grauen Literatur zuzuordnen sind, werden heute in hohem Maß in Form von elektronischen Veröffentlichungen publiziert.
Graue Literatur in der Forschung
Viele wissenschaftliche Arbeiten bleiben unveröffentlicht und sind nur direkt über die entsprechenden Institute erhältlich. Gründe dafür können sein, dass wissenschaftliche Mindestanforderungen (z. B. die statistische Signifikanz, Angemessenheit der Methodik, Qualität der Präsentation) nicht erreicht werden oder dass die Inhalte der Arbeit ideologischen Vorstellungen nachgehen und keinen Verlag finden.
Wenn man sich einen Überblick über den Stand der Forschung zu einem Themenbereich verschaffen will, bedient man sich häufig sogenannter Metaanalysen und Überblicksarbeiten (Review-Artikel). Bei Metaanalysen werden mehrere Statistiken mit kleineren Stichproben zu einer großen Stichprobe zusammengefasst und über deren Ergebnisse ein Mittelwert gebildet. Bei Überblicksarbeiten werden mehrere Forschungsarbeiten zu einem Thema zusammengefasst. Hier werden die Arbeiten allerdings nicht statistisch verarbeitet, sondern inhaltlich zueinander in Beziehung gesetzt und diskutiert.
Sofern nur veröffentlichte Arbeiten in Metaanalysen und Überblicksarbeiten einbezogen werden, können die wissenschaftlichen Ergebnisse zu einem Themenbereich übereinstimmender erscheinen, als sie tatsächlich sind. Im Extremfall könnten nicht existierende Unterschiede zwischen Gruppen oder beobachtete Zusammenhänge nur durch Zufall beobachtet worden sein, während Untersuchungen, in denen nichts dergleichen beobachtet werden konnte, niemals veröffentlicht wurden. Für Beobachtungen, die eigentlich durch Zufall erklärbar sind, würde dann fälschlicherweise eine Korrelation festgestellt und möglicherweise sogar ein falscher Kausalzusammenhang abgeleitet. Wenn einige unpopuläre Meinungen durch Zensur nicht zu Wort kommen, entsteht fälschlicherweise der Eindruck von Einhelligkeit, da Meinungsverschiedenheiten nicht berücksichtigt werden.
Dieser falsche Eindruck wird als Publikationsbias bezeichnet. Um einem möglichen Publikationsbias entgegenzuwirken, sollten unveröffentlichte Arbeiten mit einbezogen werden. "Das Ergebnis einer Metaanalyse ist selbstverständlich von der Auswahl der einbezogenen Primäruntersuchungen abhängig." Dasselbe gilt analog für Überblicksarbeiten (Reviews).
Darüber hinaus können auch Schriften, die wissenschaftlichen Ansprüchen nicht genügen (und womöglich auch gar keinen wissenschaftlichen Anspruch erheben wollten), für die Forschung relevant sein, wenn sie etwa Informationen enthalten, die sonst nirgendwo publiziert sind, oder nur dem Autor zugänglich waren. Dies kann z. B. bei familiengeschichtlichen Abhandlungen der Fall sein, wenn der Verfasser auf Material in Privatbesitz zurückgreifen konnte.
Beispiele
Programmhefte
Tagungs- und Kongressberichte
Vorlesungsverzeichnisse
Präsentationen
technische Berichte
Lehrmaterialien
Institutsschriften
Preprints
E-Zine und Untergrundmagazine
Kataloge, Berichte und Pläne
Fanzines
Flugblätter
Firmenschriften
Gelegenheitsschriften
Schülerzeitungen
Websites, sofern sie z. B. keine zuvor auch in Buchform veröffentlichten Texte enthalten
nicht im Buchhandel erhältliche Habilitationsschriften, Dissertationen, Seminararbeiten, Diplomarbeiten und andere Texte aus dem universitären Umfeld, die direkt oder über andere Vertriebskanäle an ihre Zielgruppe gelangen.
Samisdat, nicht systemkonforme Literatur in der UdSSR und später auch in anderen realsozialistischen Staaten, die handschriftlich, "abgetippt" oder fotokopiert bzw. vervielfältigt und auf nichtoffiziellen Kanälen weitergegeben wurde.
Weblinks
Grey Literature International Steering Committee (2007): Richtlinien für die Erstellung wissenschaftlicher und technischer Berichte: Verfassen und Verbreiten grauer Literatur.
Grey Literature Network Service
WHOIS in the field of Grey Literature
A Selection of Web-based Resources in Grey Literature
OpenGrey, System for Information on Grey Literature in Europe
Einzelnachweise
4. Klaus Gantert: Bibliothekarisches Grundwissen. 9. Auflage. De Gruyter, Berlin 2016, ISBN 978-3-11-032145-6. S. 78
Publikation | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,140 |
{"url":"https:\/\/www.dpmms.cam.ac.uk\/~or257\/publications.htm","text":"### Publications\n\nHomological stability for automorphism groups ( ArXiV), with Nathalie Wahl.\nAdvances in Mathematics 318 (2017) 534-626.\n\nHomological stability for moduli spaces of high dimensional manifolds. I ( ArXiV), with S\u00f8ren Galatius.\nJournal of the American Mathematical Society 31 (1) (2018) 215-268.\n(This supersedes our earlier preprint arXiv:1203.6830.)\n\nCohomology of automorphism groups of free groups with twisted coefficients ( ArXiV).\nSelecta Mathematica, to appear.\n(This supersedes the earlier preprint arXiv:1012.1433.)\n\nHomological stability for moduli spaces of high dimensional manifolds. II ( ArXiV), with S\u00f8ren Galatius.\nAnnals of Mathematics 186 (1) (2017) 127-204.\n\nInfinite loop spaces and positive scalar curvature (pdf) ( ArXiV), with Boris Botvinnik and Johannes Ebert.\nInventiones mathematicae 209 (3) (2017), 749-835.\n\nAn upper bound for the pseudoisotopy stable range ( ArXiV).\nMathematische Annalen, 368 (3) (2017), 1081-1094.\n\nTautological rings for high dimensional manifolds ( ArXiV), with S\u00f8ren Galatius and Ilya Grigoriev.\nCompositio Mathematica 153 (4) (2017) 851-866.\n\nHomological stability for spaces of embedded surfaces ( ArXiV), with Federico Cantero.\nGeometry & Topology 21 (2017) 1387-1467.\n\nAbelian quotients of mapping class groups of highly connected manifolds ( ArXiV), with S\u00f8ren Galatius.\nMathematische Annalen 365 (1) (2016) 857-879.\n\nResolutions of moduli spaces and homological stability ( ArXiV).\nJournal of the European Mathematical Society 18 (2016) 1-81.\n\nTorelli spaces of high-dimensional manifolds ( ArXiV), with Johannes Ebert.\nJournal of Topology 8 (1) (2015) 38-64.\n\nStable moduli spaces of high-dimensional manifolds ( ArXiV), with S\u00f8ren Galatius.\nActa Mathematica 212 (2014), no. 2, 257-377.\nErratum\n\nDetecting and realising characteristic classes of manifold bundles ( ArXiV), with S\u00f8ren Galatius.\nAlgebraic Topology: Applications and New Directions (Stanford, CA, 2012), Contemp. Math.\n\nGeneralised Miller-Morita-Mumford classes for block bundles and topological bundles ( ArXiV), with Johannes Ebert.\nAlgebraic & Geometric Topology 14 (2014) 1181-1204.\n\n\"Topological chiral homology\" and configuration spaces of spheres.\nMorfismos, (MIMS proceedings issue), Vol. 17 No 2 (2013) 57-70.\n\nHomology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces ( ArXiV).\nJournal of Topology 7 (1) (2014) 155-186.\nErratum\n\n\"Group-Completion\", local coefficient systems, and perfection ( Preprint).\nQuarterly Journal of Mathematics 64 (3) (2013) 795-803.\n\nThe space of immersed surfaces in a manifold ( ArXiV).\nMathematical Proceedings of the Cambridge Philosophical Society 154 (3) (2013) 419-438.\n\nRelations among tautological classes revisited ( ArXiV).\nAdvances in Mathematics 231 (3-4) (2012) 1773-1785.\n\nThe Picard group of the moduli space of r-Spin Riemann surfaces ( ArXiV).\nAdvances in Mathematics 231 (1) (2012) 482-515.\n\nStable cohomology of the universal Picard varieties and the extended mapping class group ( ArXiV), with Johannes Ebert.\nDocumenta Mathematica 17 (2012) 417-450.\n\nHomological stability for unordered configuration spaces ( ArXiV).\nQuarterly Journal of Mathematics 64 (1) (2013) 303-326.\n\nEmbedded cobordism categories and spaces of submanifolds ( ArXiV).\nInternational Mathematics Research Notices 3 (2011) 572-608.\n\nMonoids of moduli spaces of manifolds ( ArXiV), with S\u00f8ren Galatius.\nGeometry & Topology 14 (2010) 1243-1302.\nErratum\n\nThe homology of the stable non-orientable mapping class group ( ArXiV).\nAlgebraic & Geometric Topology 8 (2008) 1811-1832.\nErratum\n\nOn the divisibility of characteristic classes of non-orientable surface bundles ( ArXiV), with Johannes Ebert.\nTopology and its Applications 156 (2008) 246-250.\n\n### Other documents\n\nA combinatorial identity, 2012.\n\nCohomology of Aut(F_n) with twisted coefficients, for the \"Topologie\" Oberwolfach report, 2016.\n\nStable moduli spaces of high dimensional manifolds, for the \"Topologie\" Oberwolfach report, 2014.\n\nHomological stability for moduli spaces of manifolds, for the \"Topologie\" Oberwolfach report, 2012.\n\nMonoids of moduli spaces of manifolds, II, for the \"Topologie\" Oberwolfach report, 2010.\n\nMonoids of moduli spaces of manifolds, for the \"Manifold Perspectives\" Oberwolfach report, 2009.\n\n Spectrum E2 page Module definition $\\mathbf{MTO}(1)$ Here Here $\\mathbf{MTO}(2)$ Here Here $\\mathbf{MTSO}(2)$ Here Here $\\mathbf{MTSpin}(4)$ Here Here\nI have computed the (mod 2) Adams E2 terms for the stable homotopy groups of some Madsen-Tillmann spectra. Recall that $\\mathbf{MTO}(2)$ and $\\mathbf{MTSO}(2)$ are respectively the homotopy-types of the non-orientable and oriented stable mapping class groups, after plus-construction. This uses Robert Bruner's program for calculating Ext groups over the Steenrod algebra. I wrote a MAGMA program to give the description in terms of generators (up to a given degree) of these modules over the Steenrod alegebra, in a format admissible for the program mentioned above. There is also a chart for the beginning of $\\mathbf{MTSO}(2)$ with some differentials filled in here, which are those which can be deduced from John Rognes' \"Two-primary algebraic K-theory of pointed spaces\", Topology 41 ( PDF).","date":"2018-01-20 11:16:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7817829847335815, \"perplexity\": 3258.5594003238684}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084889567.48\/warc\/CC-MAIN-20180120102905-20180120122905-00252.warc.gz\"}"} | null | null |
Q: Android WifiManager.addNetwork() returns -1 I am writing an android app which will connect to a specific WPA access point, when connected, it will issue a http call. It will not save the network config.
I have read almost every post on stack overflow on connecting to wifi network but can't find the answer which works for me. Here is the code I am using to connect..
WifiConfiguration wc = new WifiConfiguration();
wc.allowedAuthAlgorithms.clear();
wc.allowedGroupCiphers.clear();
wc.allowedPairwiseCiphers.clear();
wc.allowedProtocols.clear();
wc.allowedKeyManagement.clear();
wc.SSID = "\"".concat("<ssid>").concat("\"");
wc.preSharedKey = "\"".concat("<password>").concat("\"");
wc.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.TKIP);
wc.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.CCMP);
wc.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.WEP40);
wc.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.WEP104);
wc.allowedKeyManagement.set(WifiConfiguration.KeyMgmt.WPA_PSK);
wc.allowedPairwiseCiphers.set(WifiConfiguration.PairwiseCipher.CCMP);
wc.allowedPairwiseCiphers.set(WifiConfiguration.PairwiseCipher.TKIP);
wc.allowedProtocols.set(WifiConfiguration.Protocol.RSN); // For WPA2
wc.allowedProtocols.set(WifiConfiguration.Protocol.WPA); // For WPA
wc.priority = 0;
//wc.hiddenSSID = true;
wc.status = WifiConfiguration.Status.ENABLED;
// connect to and enable the connection
WifiManager wifiManager = (WifiManager) getSystemService(this.WIFI_SERVICE);
int netId = wifiManager.addNetwork(wc);
boolean wifiEnabled = wifiManager.enableNetwork(netId, true);
wifiManager.setWifiEnabled(true);
Log.d("opener", "addNetwork returned " + netId);
if (netId > 0) {
wifiId = netId;
}
However netId is always -1. I have tried it on two different phones (ICS:HTC Rezound and GingerBread:Motorola DroidX). Both show exactly same results.
What am I doing wrong?
Edit: I tried same code with WPA2 access point and got very weird results. When this code was run, first time it would return -1, but if I call same method second time with a delay of 1 second, it would return valid netId. So the questions are
*
*why does above code not connect to wpa ?
*in wpa2, why do I need to call above method twice to get connected ? Edit: I observed that I had to connect multiple times to get connected. Sometimes it would take 3-4 times to connect. So for now I am looping until adding config returns >0 id.
A: The problem is that you're trying to add the network configuration that already exists. When you call:
int netId = wifiManager.addNetwork(wc);
it will fail (return -1) if the network configuration with the same SSID already exists. So, what you need to do is to check if netId is -1 and if it is, traverse through the configured networks searching for the network with same SSID and once it's found, return the networkId.
Kotlin:
var netId = wifiManager.addNetwork(conf)
if (netId == -1) netId = wifiManager.configuredNetworks?.let {
it.firstOrNull { it.SSID.trim('"') == ssid.trim('"') }?.networkId ?: -1
}
wifiManager.enableNetwork(netId, true)
A: I had the same problem. I found out that your wifi must be on while you are calling addNetwork.
A: I just had this very same problem. Seemingly everything was okay, but then - it wasn't.
What I found is this:
*
*Android WifiStateMachine will not allow you to add or modify networks unless supplicant is running and connected-to. This affects services running on start-up and services running even if WIFI is off.
*Android WifiConfigStore tracks owners of the wifi network by UID. It means that you may not be able to modify network created by another process.
The proper way to add a network is:
*
*Check if WIFI network is enabled. If not, call WifiManager.setWifiEnabled(true).
*Wait until WifiManager.pingSupplicant() returns true.
*Create and fill a new WifiConfiguration, then pass it to WifiManager.addNetwork(). Make sure the value returned is not (-1).
*(optional) use value returned by addNetwork() as an argument to WifiConfiguration.enableNetwork() call (unless it's -1). Note, that the boolean parameter means disableOthers and should be false, unless you have right to modify all other networks: it may fail internally, if you set it to true.
This should allow you to add (and connect) programmatically to a new Wifi network.
A: Possibly a bit late but try this to connect to Open/WPA/WPA2/WEP secured networks
WifiConfiguration wifiConfig = new WifiConfiguration();
wifiConfig.SSID = selectedNetwork.SSID();
wifiConfig.status = WifiConfiguration.Status.DISABLED;
wifiConfig.priority = 40;
// Dependent on the security type of the selected network
// we set the security settings for the configuration
if (/*Open network*/) {
// No security
wifiConfig.allowedKeyManagement.set(WifiConfiguration.KeyMgmt.NONE);
wifiConfig.allowedProtocols.set(WifiConfiguration.Protocol.RSN);
wifiConfig.allowedProtocols.set(WifiConfiguration.Protocol.WPA);
wifiConfig.allowedAuthAlgorithms.clear();
wifiConfig.allowedPairwiseCiphers.set(WifiConfiguration.PairwiseCipher.CCMP);
wifiConfig.allowedPairwiseCiphers.set(WifiConfiguration.PairwiseCipher.TKIP);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.WEP40);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.WEP104);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.CCMP);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.TKIP);
} else if (/*WPA*/ || /*WPA2*/) {
//WPA/WPA2 Security
wifiConfig.allowedProtocols.set(WifiConfiguration.Protocol.RSN);
wifiConfig.allowedProtocols.set(WifiConfiguration.Protocol.WPA);
wifiConfig.allowedKeyManagement.set(WifiConfiguration.KeyMgmt.WPA_PSK);
wifiConfig.allowedPairwiseCiphers.set(WifiConfiguration.PairwiseCipher.CCMP);
wifiConfig.allowedPairwiseCiphers.set(WifiConfiguration.PairwiseCipher.TKIP);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.WEP40);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.WEP104);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.CCMP);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.TKIP);
wifiConfig.preSharedKey = "\"".concat(password).concat("\"");
} else if (/*WEP*/) {
// WEP Security
wifiConfig.allowedKeyManagement.set(WifiConfiguration.KeyMgmt.NONE);
wifiConfig.allowedProtocols.set(WifiConfiguration.Protocol.RSN);
wifiConfig.allowedProtocols.set(WifiConfiguration.Protocol.WPA);
wifiConfig.allowedAuthAlgorithms.set(WifiConfiguration.AuthAlgorithm.OPEN);
wifiConfig.allowedAuthAlgorithms.set(WifiConfiguration.AuthAlgorithm.SHARED);
wifiConfig.allowedPairwiseCiphers.set(WifiConfiguration.PairwiseCipher.CCMP);
wifiConfig.allowedPairwiseCiphers.set(WifiConfiguration.PairwiseCipher.TKIP);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.WEP40);
wifiConfig.allowedGroupCiphers.set(WifiConfiguration.GroupCipher.WEP104);
if (getHexKey(password)) wifiConfig.wepKeys[0] = password;
else wifiConfig.wepKeys[0] = "\"".concat(password).concat("\"");
wifiConfig.wepTxKeyIndex = 0;
}
// Finally we add the new configuration to the managed list of networks
int networkID = wifiMan.addNetwork(wifiConfig);
Obviously set your own variables for the different security types as applicable. The key (pardon the pun) to connecting to a WEP secured network is the getHexKey method as below.
/**
* WEP has two kinds of password, a hex value that specifies the key or
* a character string used to generate the real hex. This checks what kind of
* password has been supplied. The checks correspond to WEP40, WEP104 & WEP232
* @param s
* @return
*/
private static boolean getHexKey(String s) {
if (s == null) {
return false;
}
int len = s.length();
if (len != 10 && len != 26 && len != 58) {
return false;
}
for (int i = 0; i < len; ++i) {
char c = s.charAt(i);
if ((c >= '0' && c <= '9') || (c >= 'a' && c <= 'f') || (c >= 'A' && c <= 'F')) {
continue;
}
return false;
}
return true;
}
A: I have found that if the shared key is less than 8 characters, it will return -1.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,990 |
{"url":"https:\/\/tex.stackexchange.com\/questions\/365445\/how-to-un-indent-theorems","text":"# how to un-indent theorems\n\nI'm about to deposit my thesis. The grad school administrators set out formatting requirements and someone from years past wrote some tex files (a document class?) that I'm using to make sure my thesis complies with the formatting requirements. The only problem is that whatever this person wrote indents every theorem, definition, lemma etc. So that no matter what I do (actually \\noindent is the only thing I know to try), every new theorem looks like this:\n\nobtained from\n\n\\documentclass{brandiss}\n\\newtheorem*{main}{Theorem}\n\\newtheorem{thm}{Theorem}[section]\n\\begin{document}\n\\noindent Here's a non-indented line of text.\n\\begin{thm}\nHere's a Theorem\n\\end{thm}\n\\end{document}\n\n\nwhich looks really dumb. Especially when I state a theorem right after the first sentence of a new paragraph. Maybe this is impossible to answer without seeing the code for the document class (Actually its right here: http:\/\/www.brandeis.edu\/departments\/mathematics\/graduate\/current\/brandiss.html )\n\nbut I was wondering if anyone knows how to fix this. I recall a few weeks ago I was able to do something that unindented everything, but if I did that I'd have to manually indent every paragraph and its a pretty long document.\n\n\u2022 \\parindent=0pt? But then also the normal paragraph start will be unindented (ok in some part of the world, don't know over there). Anyway, the class seems just a slight overload over amsbook, so if you add a MWE with that class it would be useful to help us help you. Apr 19 '17 at 7:34\n\u2022 Could you post a minimal example code? Apr 19 '17 at 9:06\n\u2022 @Bernard Thanks, I've added a small bit of example code. the \"brandiss\" thing is a file available at the link to Brandeis.edu I gave. Apr 19 '17 at 9:23\n\u2022 @Rmano Thanks for your comment. Is this what you mean by MWE? Apr 19 '17 at 9:25\n\u2022 Yes. @Bernard, you can just change the class to amsbook and it shows the same thing. I think that tex.stackexchange.com\/questions\/35215\/using-the-amsbook-class is related. Apr 19 '17 at 9:34\n\nYou can try this (from this other question):\n\n\\documentclass{amsbook}\n\n\\usepackage{etoolbox}\n\\makeatletter\n\\patchcmd\\@thm\n{\\let\\thm@indent\\indent}{\\let\\thm@indent\\noindent}%\n{}{}\n\\makeatother\n\n\\newtheorem*{main}{Theorem}\n\\newtheorem{thm}{Theorem}[section]\n\\begin{document}\n\\noindent Here's a non-indented line of text.\n\\begin{thm}\nHere's a Theorem\n\\end{thm}\n\\end{document}\n\n\nI checked with amsbook but should work for your document class.\n\n\u2022 Do you have any ideas about how to un-indent proofs? (I use \\begin{proof} \\end{proof}). I did the dumbest thing possible and replaced \"thm\" in your code with \"proof\", but it didn't work. Apr 19 '17 at 10:01\n\u2022 nm. I worked something out from the post you linked to in the comments. Thanks again. Apr 19 '17 at 10:46","date":"2021-11-28 15:44:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7696727514266968, \"perplexity\": 1068.2728374821027}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358560.75\/warc\/CC-MAIN-20211128134516-20211128164516-00055.warc.gz\"}"} | null | null |
Tickets for Funny As Ish Comedy Tour are not hard to find. We have great Funny As Ish Comedy Tour tickets! If you are looking for the best Funny As Ish Comedy Tour tickets, we've got your seat!
Below you will find all dates for Funny As Ish Comedy Tour scheduled events.
If you have questions about Funny As Ish Comedy Tour tickets just call our ticket specialists and leave the rest to Jiffy Tickets! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,843 |
WNYC Studios Presents 'Black Folks': Eve Ewing, Madison McFerrin, Alexis Okeowo, Jay Smooth and Jamal Lewis
"The Souls of Black Folk" by W.E.B. Du Bois was published in 1903 as a celebration of black people. It was also an assessment of race and structural racism in America through an assemblage of essays, poetry, music and conversations. The book offered a kaleidoscopic and nuanced excavation of black identity and culture some 50 years after emancipation, at which time Du Bois famously declared, "the problem of the Twentieth Century is the problem of the color-line."
Join Rebecca Carroll for Black Folks, a WNYC Studios podcast pilot that draws on the work of Carroll and that of DuBois. A curated mosaic of black expression, Black Folks features a wide array of guests who will be thoughtful and provocative, at a time in history when black people must still assert that Black Lives Matter.
On the show: Eve Ewing, writer and sociologist of race and education; singer-songwriter Madison McFerrin; The New Yorker's Alexis Okeowo; Jay Smooth host of WBAI's "Underground Railroad" and the Ill Doctrine videoblog; and filmmaker/performance artist Jamal Lewis.
Watch live beginning 7pm
About Rebecca Carroll
As a cultural commentator, journalist and editor of special projects at WNYC, Carroll has in many ways been following in the Du Bois footsteps over the past two years with her own work at WNYC, including How I Got Over, Dear President: What You Need to Know About Race and The Newark Riots at 50, series that combine interviews, conversation and performance. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,138 |
Woodland Park School District Re-2 (WPSD) is a school district headquartered in Woodland Park, Colorado.
It originated from a two-story schoolhouse built in 1890. The Edlowe School District, the Eisweth School District, and two other school districts consolidated into Woodland Park School District in 1924. In Florissant School District and the Midland School District consolidated into Woodland Park schools after voters approved a school consolidation in a referendum in 1959.
Schools
References
External links
School districts in Colorado
Education in Teller County, Colorado
1890 establishments in Colorado
School districts established in 1890 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,620 |
Fungiacyathus sibogae är en korallart som först beskrevs av Alcock 1902. Fungiacyathus sibogae ingår i släktet Fungiacyathus och familjen Fungiacyathidae. Inga underarter finns listade i Catalogue of Life.
Källor
Stenkoraller
sibogae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,807 |
Sam Obisanya
Sam Obisanya (Toheeb Jimoh) is a character in the soccer comedy-drama, Ted Lasso. He plays right back defense for AFC Richmond. Having moved to the UK from Nigeria, he gets homesick while trying to find his place on the team. After becoming a leader in the dressing room, he pursues a relationship with team owner Rebecca Welton. At the conclusion of the second season, Sam buys a building to open his own Nigerian restaurant.
Emily Dickinson from Dickinson
Emily Dickinson's life is the stuff of literary class lessons and book foreword fodder. But in the hands of Alena Smith and Hailee Steinfeld, Dickinson's life takes on new glamour, pain, and style. The new approach brings Dickinson to life in a way few class syllabi could hope to. And it does it with a style all its own.
Given the show's unique style, there are a range of looks to choose from. To get the look featured here, however, you'll need to start with a red ballgown. After that, match red lipstick and flats to the gown, then style a brunette wig into the loose strands and updo Dickinson wears. Round out the look with a stunning leather journal for writing down all of your own brilliant ideas!
May 5, 2021 by Tom
Ted Lasso (Jason Sudeikis) is an American college football coach who gets hired to lead the Premier League soccer club AFC Richmond after he wins a championship with the Wichita State Shockers. Despite his unsophisticated demeanor and complete lack of experience with soccer, Coach Ted wins over owner Rebecca Welton and the rest of the team.
January 17, 2020 by Tom
Leanne Grayson from Servant
In M. Night Shyamalan's AppleTV+ series Servant, Leanne Grayson (Nell Tiger Free) is a mysterious young woman who is hired by Dorothy and Sean Turner to move in and take care of Baby Jericho. However all is not as it seems, as Leanne is actually responsible for caring for a reborn doll after the Turners' tragic loss, and strange happenings occur in their Philadelphia home.
Make Your Own: She-Ra
Make Your Own: Booker DeWitt from Bioshock Infinite
October 28, 2019 by Becky
Cosplay of CosBrawl 2020
August 24, 2020 by Tom
Costumes on Clearance: A Look at Leftover Costumes from Halloween 2020
November 16, 2020 by Tom
Cosplay at Retro Con 2022
How To Make Your Own Socks
September 3, 2019 by Otto
Billy Hargrove From Stranger Things
October 8, 2018 by Real Human Bean
Dustin Henderson | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,265 |
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>old swap</title>
<link rel="stylesheet" href="../../../demo/demo.css"/>
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/gridstack@1.0.0/dist/gridstack.min.css"/>
<script src="https://cdn.jsdelivr.net/npm/gridstack@1.0.0/dist/gridstack.all.js"></script>
</head>
<body>
<div class="container-fluid">
<h1>Swap collision demo from older 1.x builds</h1>
<div>
<a class="btn btn-primary" onClick="addNewWidget()" href="#">Add Widget</a>
<a class="btn btn-primary" onclick="toggleFloat()" id="float" href="#"></a>
<a class="btn btn-primary" onclick="toggleMax()" id="max" href="#"></a>
</div>
<br><br>
<div class="grid-stack">
<div class="grid-stack-item" data-gs-x="0" data-gs-y="0"><div class="grid-stack-item-content">0</div></div>
<div class="grid-stack-item" data-gs-x="0" data-gs-y="1"><div class="grid-stack-item-content">1</div></div>
<div class="grid-stack-item" data-gs-x="0" data-gs-y="2"><div class="grid-stack-item-content">2</div></div>
<div class="grid-stack-item" data-gs-x="1" data-gs-y="0"><div class="grid-stack-item-content">3</div></div>
<div class="grid-stack-item" data-gs-x="1" data-gs-y="1"><div class="grid-stack-item-content">4</div></div>
<div class="grid-stack-item" data-gs-x="1" data-gs-y="2"><div class="grid-stack-item-content">5</div></div>
</div>
</div>
<script type="text/javascript">
let floatButton = document.getElementById('float');
let maxButton = document.getElementById('max');
let count = 6;
let grid = GridStack.init({float: false, cellHeight: 70, maxRow: 3});
addNewWidget = function() {
let n = {
x: Math.round(12 * Math.random()),
y: Math.round(5 * Math.random()),
w: Math.round(1 + 3 * Math.random()),
h: Math.round(1 + 3 * Math.random()),
content: String(count++)
};
grid.addWidget(n);
};
toggleFloat = function() {
grid.float(! grid.getFloat());
floatButton.innerHTML = 'float: ' + grid.float();
};
floatButton.innerHTML = 'float: ' + grid.float();
toggleMax = function() {
grid.opts.maxRow = grid.engine.maxRow = grid.opts.maxRow ? 0 : 3;
maxButton.innerHTML = 'Max: ' + grid.opts.maxRow;
};
maxButton.innerHTML = 'Max: ' + grid.opts.maxRow;
</script>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,347 |
Nyctiophylax esli är en nattsländeart som beskrevs av Malicky 1993. Nyctiophylax esli ingår i släktet Nyctiophylax och familjen fångstnätnattsländor. Inga underarter finns listade i Catalogue of Life.
Källor
Fångstnätnattsländor
esli | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,879 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/264030\/improving-rmse-of-my-model","text":"# Improving RMSE of my model\n\nI'm trying to build a model based on some training set.\n\nThe training set contains 1460 observations, with 79 variables each (features).\n\nI'm using linear regression to build a model and after that building a step-regression model (forward selection algorithm), and still getting large value for RMSE.\n\nI'm very new to data analysis, so If someone can point me to what it is I'm doing wrong that would be great.\n\nData:\n\n79 Features of houses in some city (Lot Frontage, Pool area, Total area, etc.), And SalePrice label of that house.\n\n43 Of them are categorical variables identifying various types of dwellings, garages, etc.\n\nCleanup:\n\nIn order to clean my data - I'm performing the following steps:\n\nLots of the factor variables have logic order to their levels, e.g.:\n\nBsmtQual: Evaluates the height of the basement\n\nEx Excellent (100+ inches)\nGd Good (90-99 inches)\nTA Typical (80-89 inches)\nFa Fair (70-79 inches)\nPo Poor (<70 inches\nNA No Basement\n\n\nSo 1st, I'm converting them to numerical values, e.g.:\n\nBsmtQual: Evaluates the height of the basement\n\nEx records will be replaced with: 5\nGd records will be replaced with: 4\nTA records will be replaced with: 3\nFa records will be replaced with: 2\nPo records will be replaced with: 1\nNA records will be replaced with: 0\n\n\nThis step is in order to reduce amount of factor variables Afterwards, I'm using mice() library to impute missing values that are left in the data.\n\nModel building\n\n1. Running PCA on the training data\n2. Selecting K PCs based on cumulative scree plot\n3. Transforming test data into PCA\n4. Building linear regression model:\n\nlm.model <- lm(SalePrice ~ ., data = train.data)\n\n5. Using predict() with the lm.model on my test set.\n\nI'm getting very high RMSE for lm.model - 2.256228e+04, Although Rsquared param seems high - 0.91928\n\nI also tried running step regression based on the lm.model, but still I get high RMSE.\n\nAdditionally, PCA doesn't seem to make a big difference also.\n\nHow can I improve? What am I doing wrong?\n\nThanks\n\n\u2022 Is that a \"very high\" RMSE though? That's about a \\$20k error in sale price, which doesn't seem that bad compared to the variation in sale prices. What kind of RMSE were you expecting to get? \u2013\u00a0The Laconic Feb 25 '17 at 19:00\n\u2022 This is part of a Kaggle project (with no prizes) we are required to submit in academic course. For example, my model gets to the 3800 position out of 4600 submissions, which seems very poor. \u2013\u00a0Adiel Feb 25 '17 at 19:22","date":"2019-07-16 00:54:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.17900030314922333, \"perplexity\": 4220.264791713239}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195524290.60\/warc\/CC-MAIN-20190715235156-20190716021156-00406.warc.gz\"}"} | null | null |
Апо́стольське Передання́ — в широкому сенсі — частина священного Передання, в церковно-науковій літературі і патрології — передбачуване ядро цілого ряду літургіко-канонічних пам'яток, яке дає докладну картину устрою церковного життя III ст., Будучи важливим джерелом з історії ранньохристиянського богослужіння.
Оригінальний текст Апостольського Передання невідомий.
Пошуки твору, який називався би «Апостольське Передання», почалися з часу виявлення в 1551 р. в Римі при царюванні св. Лаврентія на Тибуртинский дорозі статуї якогось єпископа, на підставі якої були вигравірувані великодні таблиці і назви творів, що належать сщмч. Іполиту Римському ( 235?). Серед інших на підставі статуї вибиті слова: «Περὶ χαρισμάτων ἀποστολικὴ παράδοσις» (про харизму [дарування] Апостольського передання; цю фразу можна вважати назвою як першої, так і другої пам'ятки), однак жодний з відомих творів св. Іполита неможливо ототожнити зі згаданим «ἀποστολικὴ παράδοσις».
В кін. XIX - поч. XX ст., з відкриттям і публікацією великого числа літургіко-канонічних пам'ятників, дослідниками було відмічено значну схожість між окремими частинами 5 збірок: «Синодосу» (ін. Назва - «Єгипетські церковні постанови»; грец. Оригінал втрачено, відомі коптські - саідична (найдавніша, близько 500) і бохайрська - араб. і ефіоп. версії, що відрізняються одна від одної), Веронского палімпсесту (збірник лат. Перекладів декількох Літургіко-канонічних пам'ятників, бл. 494 - по Б. Ботту), «Епітоми» (V ст.?; Містить переробку VIII книги «Апостольських постанов»), «Заповіту Господа нашого Ісуса Христа» (V ст .; зберігся тільки сирійський переклад грецького тексту), « Канонів Іполита » (IV або V ст.; зберігся тільки араб. переклад грецького тексту). Ця схожість була пояснена Е. Шварцем і Р. Г. Конноллі шляхом виділення з «Синодосу» і Веронского палімпсесту якогось самостійного тексту (позначимо його як «κ»), таким чином, «Синодос» складається з «Канонів святих апостолів» ( III ст.), «κ» і переробленої VIII кн. «Апостольських постанов» (бл. 380); Веронський палімпсест - з «Дидаскалії апостолів» (III ст.), «Канонів свтихя апостолів» і «κ»; інші 3 збірника містять окремі фрагменти «κ». Цей текст «κ» і отримав назву «Апостольського передання», оскільки дослідники XX в. (Конноллі і ін.) Ототожнили його з втраченим «ἀποστολικὴ παράδοσις» свщмч. Іполита Римського. Вважається, що найближче до передбачуваного першоджерела знаходиться лат. текст Веронского палімпсесту (Hanssens. P. 171-216, 249-253), оскільки грецького оригіналу тексту «κ» не існує. Слід зазначити, що версії Апостольського передання, які виділяються з кожного збірника, не ідентичні, відмінності між ними в окремих розділах досить істотні. Критичну реконструкцію Апостольського передання зробив Б. Ботт. При цитуванні Апостольського передання зазвичай дотримуються нумерації глав по виданню Б. Ботта. Текстологічний аналіз показує, що єдина латинська редакція Апостольських передань (у Веронскому палімпсесті) є перекладом з грецького джерела III ст., має єгипецьке або сирійське походження.
Див. також
Святе Передання
Джерела
Християнські документи | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,149 |
@interface DPMButton : DPMComponent
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,781 |
{"url":"https:\/\/chemistry.stackexchange.com\/questions\/143070\/can-we-observe-the-shapes-of-mos","text":"# Can we observe the shapes of MOs? [duplicate]\n\nCondensed question formulation:\n\nIs there an experimental method to directly visualise the 3D form of a MO wave function, or at least the electron density associated with it?\n\nFull statement:\n\nMolecular orbital theory is a crowning achievement of modern Chemistry: it is by far the most accurate description of chemical bonding we possess and accounts for the vast majority of related phenomena.\n\nAn array of evidence exists to confirm predictions from MO theory. Ab initio computational methods such as Hartree-Fock are able to produce a very accurate description of orbital coefficients and energies, which match experimental data for the energies (via photoelectron spectroscopy etc.) or predicted reaction mechanisms\/FMO interactions (in kinetic runs).\n\nBut do we have any experimental evidence for the shape of MOs?\n\nI'd presume the most feasible way of observing some measure related to the MO wave function $$\\Psi$$ would be to follow a probabilistic approach and try imaging the electron density assumed to be given by $$|\\Psi|^2$$.\n\nOf course, X-ray diffraction first springs to mind; the notorious issue with that being the phase problem. In my superficial understanding: due to the loss of phase information, in going from the XRD pattern to the electron density isosurface (via ab initio methods), essentially we model some initial guess phase and try to fit it to the intensity data.\n\nSince usually the location of nuclei is of far greater practical interest, the formalisms developed to model the phases (e.g. Hansen-Coppens) tend to focus on charge densities localised on each atom (though they are not necessarily of spherical symmetry). This is useful in actually getting the form of the molecule, but produces a picture different to the expected electronic occupation of delocalised MOs.\n\nIs it possible to instead produce the delocalised picture if our ab initio phase guess is based on a RHF calculation? But even if possible, does this have any more legitimacy than the localised picture? I.e. it seems, as far as XRD is concerned, these are equally possible interpretations of the same intensity data where we just forced some phase information as convenient.\n\nSorry for the long description, this is an in depth presentation of my thoughts on the issue thus far. Any experimental technique\/fundamental aspect of MO behaviour I'm missing would be appreciated. But the bottom line is, as stated in the beginning, can we experimentally visualise the shape of MOs or at least their modulus squared?\n\n\u2022 There are two problems here: 1) Molecular orbitals coming from a one-electron model of the wavefunction, i.e. they are essentially an approximation. 2) Wavefunction, just like the molecular orbitals are not observables. In your question, you said that 2) is not an issue, as you accept electron densities, too, which are observables. Still, we have a problem with 1). While MOs are not real physical objects, a possible solution to your problem is to find physical properties that well approximated in a simple orbital picture \/ one MO Note: HF is far from an accurate description of chemistry. \u2013\u00a0Greg Nov 24 '20 at 17:02\n\u2022 chemistry.stackexchange.com\/questions\/5376\/\u2026 \u2013\u00a0Mithoron Nov 24 '20 at 21:29\n\u2022 chemistry.stackexchange.com\/questions\/57784\/\u2026 \u2013\u00a0Mithoron Nov 24 '20 at 21:35\n\u2022 Thanks for the suggestions and the quite helpful sign towards the right direction! Though not exactly the same question, the second link brought STM into my attention and eventually helped me find this paper on STM for individual MO imaging -which is the sort of technique I was asking about. \u2013\u00a0aureolin Nov 26 '20 at 10:26\n\u2022 > Molecular orbital theory is a crowning achievement of modern Chemistry This grand value judgement is matter of opinion, but ok. > it is by far the most accurate description of chemical bonding we possess well, that's just false. > Ab initio computational methods such as Hartree-Fock are able to produce a very accurate description of orbital coefficients and energies, which match experimental data for the energies (via photoelectron spectroscopy etc.) or predicted reaction mechanisms\/FMO interactions (in kinetic runs). \u2013\u00a0Lorents Nov 26 '20 at 12:59\n\nThe very question you pose is addressed thoroughly in this open access work:\nhttps:\/\/www.nature.com\/articles\/ncomms9287 .\nThe short answer to your question is that the electron density can be mapped using a technique akin to diffraction as described in their work. You mentioned a distinction between the electron density and the orbitals, and the orbital is not itself observable; but as you suggested, the electron density can be mapped.\n\n\u2022 Thank you very much, this is outstanding! I was not familiar with the ARPES technique, which seems to provide the exact answer I was looking for. Two quick additional questions (for anyone with better understanding in these methods than my half an hour earlier ignorant self): (a) is the angle-resolved part of ARPES which makes it not suffer from the phase problem of XRD? (b) do any further techniques as suitable as ARPES or STM (mentioned elsewhere in comments) for single MO imaging exist? \u2013\u00a0aureolin Nov 26 '20 at 10:30","date":"2021-06-13 20:49:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 2, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6637556552886963, \"perplexity\": 833.0115321908363}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487610841.7\/warc\/CC-MAIN-20210613192529-20210613222529-00240.warc.gz\"}"} | null | null |
Online Dating Support
Radio Show Guest Application
Home » love after 40 » How to Achieve Extraordinary Love
How to Achieve Extraordinary Love
Posted by Sandy Weiner in love after 40 | 0 comments
My podcast guests, Paige and Don Marrs, have achieved extraordinary love in their own lives and in the lives of their clients.
Happily married for over 30 years, Paige and Don Marrs have achieved extraordinary love. They've worked together since the day they joined their lives. They have co-authored two how-to memoirs, both of which teach through story: Grabbing Lightning: The Messy Quest for an Extraordinary Love, and Executive in Passage: When Life Lets You Know It's Time to Change, Let that Knowing Lead You. For the past decade they've offered their relationship program, The Love Conversation®, which teaches core principles and methods for making the quest for love less messy and has helped countless individuals create extraordinary love in their most cherished relationships.
Check out highlights of our podcast interview on Last First Date Radio, EP 384: How to Create Extraordinary Love and Cherished Relationships.
Tell us how you met and how you created extraordinary love.
Don: I was going through a difficult time in my career. I felt I had come to the end of the road. I closed my eyes, and I felt myself hurtling down a dark channel. I thought I was going to die, and suddenly, I was surrounded by love. I got vignettes about how many times I avoided love. The thing that keeps us from love is fear. I started a company based on love. I met Paige because she was a client.
Paige: I hired him. About 6 months later, we joined our lives. The quest to love is messy.
How do you help clients find the love that already exists?
Paige: We help clients see that love is their birthright. It's the human manifestation of something larger. They learn to open to it and become aware of it. A large part of the work is what blocks our experience with love.
Don: Love is different than we think it is. We think it's a dog, house, we love our husband/wife, and we go on vacation.
When we make love the focus, amazing things happen. It's a powerful energy that grows when you work out problems.
Describe Extraordinary Love.
Paige: One of the characteristics is we are both 150% committed to the fulfillment of the other person. We often hear clients speak of extraordinary love as the honeymoon stage or the fairy tale. There's a part of the honeymoon phase that's the gift of love. It's a low bar to get back to that. It's just a taste of extraordinary love. Love of one another and love of self grows and deepens and expands with time.
One of the core skills is the ability to face the normal things that come up and deal with them in a way that helps you become more intimate.
Don: We moved in together after 6 months, and we were in rapture. Within a few weeks, she said something that got me. I was a highly defensive man, and what she said sounded like criticism, and I spoke to her sharply. She crumbled, and I stormed out of the house. It reminded me of times in a marriage that had ended for me. The one who held the silence the longest won. I was in such pain, and Paige was different.
She said, "I'm sorry for my part." I said, "I'm sorry for my part." We embraced, and we learned we needed to get rid of the things that separate us and flood us. Keep restoring the connection.
Paige: We were committed to living our lives from love instead of fear. That's where our strength came from. It took us about five years to work through the rough stuff.
One of the things that developed in our relationship is I now know I'm lovable for who I am. I'm trustable for who I am. I am enough. Often when we jab at each other, it comes from a fear of not being lovable or enough.
How does one go about finding extraordinary love?
Don: By being in your heart and trusting there is such a thing as love, you will find it. Someone who's also looking will recognize you as you have that wonderful gift of generosity and kindness. You allow it to grow, and you grow it for each other.
Paige: After we were together a short time, girlfriends asked, "How did you find this?" When you watch figure skating on ice and see how fluid and graceful it is, they want relationships like that. Figure skaters have worked their butts off to be that fluid! There's an enormous amount of inner work before you meet the person and when you're together. Are you generous even when you don't feel like it? Are you able to act out of love and not smallness?
Don: It's work, but it's love work. The richness and bounty are incredible.
Paige: We are deeply engaged and active in our relationship.
Connect With Paige and Don
Website: https://theloveconversation.com
Check out Paige and Don's new book Grabbing Lightning: The Messy Quest for an Extraordinary Love available on Amazon (https://amzn.to/35SflZQ) and other online booksellers.
Facebook page: The love conversation
Please subscribe/rate and review the podcast here.
If you're feeling stuck in dating and relationships and would like to find love this year, sign up for a complimentary 1/2 hour breakthrough session with Sandy http://lastfirstdate.com/breakthrough
Join Your Last First Date on Facebook https://facebook.com/groups/yourlastfirstdate
What were some of the takeaways from this show? Please share below!
Tweets by @lastfirstdate1
Last First Date Radio
Listen to internet radio with Sandy Weiner on Blog Talk Radio
breaking up with grace
communication skills in dating
dating a dangerous man
dating a narcissist
dating in midlife
first date success
love after 40
online dating after 40
red flags in relationships
self-esteem in dating
single women over 40
understanding men over 40
Sandy Weiner | Copyright 2020 Last First Date, LLC | Last First Date: Dating Advice & Coaching with Sandy Weiner
Site Map | Privacy Policy | Contact | Return to top of page
We use cookies to improve the experience of users who browse our site and target marketing campaigns. Click "more information" to learn more about how we use data & how you can control it, or click "accept" to consent & continue browsing. More information | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,918 |
Q: AWS Security Group connection tracking failing for responses with a body in ASP.NET Core app running in ECS + Fargate In my application:
*
*ASP.NET Core 3.1 with Kestrel
*Running in AWS ECS + Fargate
*Services run in a public subnet in the VPC
*Tasks listen only in the port 80
*Public Network Load Balancer with SSL termination
I want to set the Security Group to allow inbound connections from anywhere (0.0.0.0/0) to port 80, and disallow any outbound connection from inside the task (except, of course, to respond to the allowed requests).
As Security Groups are stateful, the connection tracking should allow the egress of the response to the requests.
In my case, this connection tracking only works for responses without body (just headers). When the response has a body (in my case, >1MB file), they fail. If I allow outbound TCP connections from port 80, they also fail. But if I allow outbound TCP connections for the full range of ports (0-65535), it works fine.
I guess this is because when ASP.NET Core + Kestrel writes the response body it initiates a new connection which is not recognized by the Security Group connection tracking.
Is there any way I can allow only responses to requests, and no other type of outbound connection initiated by the application?
A: So we're talking about something like that?
Client 11.11.11.11 ----> AWS NLB/ELB public 22.22.22.22 ----> AWS ECS network router or whatever (kubernetes) --------> ECS server instance running a server application 10.3.3.3:8080 (kubernetes pod)
Do you configure the security group on the AWS NLB or on the AWS ECS? (I guess both?)
Security groups should allow incoming traffic if you allow 0.0.0.0/0 port 80.
They are indeed stateful. They will allow the connection to proceed both ways after it is established (meaning the application can send a response).
However firewall state is not kept for more than 60 seconds typically (not sure what technology AWS is using), so the connection can be "lost" if the server takes more than 1 minute to reply. Does the HTTP server take a while to generate the response? If it's a websocket or TCP server instead, does it spend whole minutes at times without sending or receiving any traffic?
The way I see it. We've got two stateful firewalls. The first with the NLB. The second with ECS.
ECS is an equivalent to kubernetes, it must be doing a ton of iptables magic to distribute traffic and track connections. (For reference, regular kubernetes works heavily with iptables and iptables have a bunch of -very important- settings like connection durations and timeouts).
Good news is. If it breaks when you open inbound 0.0.0.0:80, but it works when you open inbound 0.0.0.0:80 + outbound 0.0.0.0:*. This is definitely an issue due to the firewall dropping the connection, most likely due to losing state. (or it's not stateful in the first place but I'm pretty sure security groups are stateful).
The drop could happen on either of the two firewalls. I've never had an issue with a single bare NLB/ELB, so my guess is the problem is in the ECS or the interaction of the two together.
Unfortunately we can't debug that and we have very little information about how this works internally. Your only option will be to work with the AWS support to investigate.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,533 |
{"url":"https:\/\/brilliant.org\/discussions\/thread\/perfect-cuboid-unsolved\/","text":"# Perfect Cuboid (unsolved)\n\nGiven four equations\n\n$a^2+b^2=d^2$\n\n$a^2+c^2=e^2$\n\n$b^2+c^2=f^2$\n\n$a^2+b^2+c^2=g^2$\n\na,b,c,d,e,f and g are positive integers.\n\nAnd the problem is to find a triplet (hard) or prove that there are no pairs. (harder)\n\nThis problem is calles the perfect cuboid as if a,b and c are the sides of a cuboid, d,e and f are the face diagonnals and g is the space diagonnal. And all of these are integers.\n\nSo lets try to solve this?\n\nFor a start, I have shown that for integers x, y and z,\n\n$a = 2xz$\n\n$b = 2yz$\n\n$c = x^2+y^2-z^2$\n\n$d = 2z \\sqrt{x^2+y^2}$\n\n$e = \\sqrt {[(y+z)^2+x^2][(y-z)^2+x^2]}$\n\n$f = \\sqrt {[(x+z)^2+y^2][(x-z)^2+y^2]}$\n\n$g = x^2+y^2+z^2$\n\nThis is because for $a^2+b^2+c^2=g^2$, a, b, c and g must be in the form given above.\n\nThen, $d = \\sqrt{g^2-c^2}, e = \\sqrt{g^2-b^2}, f = \\sqrt{g^2-a^2}$\n\nNote by Aloysius Ng\n5\u00a0years, 2\u00a0months ago\n\nThis discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution \u2014 they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.\n\nWhen posting on Brilliant:\n\n\u2022 Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .\n\u2022 Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting \"I don't understand!\" doesn't help anyone.\n\u2022 Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.\n\nMarkdownAppears as\n*italics* or _italics_ italics\n**bold** or __bold__ bold\n- bulleted- list\n\u2022 bulleted\n\u2022 list\n1. numbered2. list\n1. numbered\n2. list\nNote: you must add a full line of space before and after lists for them to show up correctly\nparagraph 1paragraph 2\n\nparagraph 1\n\nparagraph 2\n\n[example link](https:\/\/brilliant.org)example link\n> This is a quote\nThis is a quote\n # I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\n# I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\nMathAppears as\nRemember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.\n2 \\times 3 $2 \\times 3$\n2^{34} $2^{34}$\na_{i-1} $a_{i-1}$\n\\frac{2}{3} $\\frac{2}{3}$\n\\sqrt{2} $\\sqrt{2}$\n\\sum_{i=1}^3 $\\sum_{i=1}^3$\n\\sin \\theta $\\sin \\theta$\n\\boxed{123} $\\boxed{123}$\n\nSort by:\n\nNo perfect cuboids have been found with side lengths under $10^{10}$. Not to discourage you, but things like these are unsolved for a reason.\n\n- 5\u00a0years, 2\u00a0months ago\n\nI know... Thats why im trying some other methods...\n\n- 5\u00a0years, 1\u00a0month ago\n\nI think it is conjecture that four dimensional Euler Bricks have no solution but a three dimensional Euler Brick has solutions smallest one is(44,117,240),where face diagonals are 125,244,267.This problem not fully similar to Euler Brick as Euler Brick's Diophantine equations was was the first the three equations you written not the fourth one a^2+b^2+c^2=g^2\n\n- 5\u00a0years, 2\u00a0months ago\n\nSorry, I confused the names between an Euler Brick and a perfect cuboid.\n\n- 5\u00a0years, 2\u00a0months ago\n\nYes it is okay,they are almost similar but not fully similar.\n\n- 5\u00a0years, 2\u00a0months ago\n\nAlso, see that d^2+e^2+f^2=2*g^2. So if a,b,c is a perfect cuboid combination, then it is not possible for d,e,f to also form a perfect cuboid combination. Where d,e,f are the sides of a new cuboid and d^2+e^2+f^2=L^2 such that L is the space diagonal.\n\n- 1\u00a0year, 11\u00a0months ago","date":"2020-02-18 08:26:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 22, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9721866846084595, \"perplexity\": 2715.8703814584824}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875143635.54\/warc\/CC-MAIN-20200218055414-20200218085414-00237.warc.gz\"}"} | null | null |
Routes and schedules Tickets Parking services For Riga guests E-services Services
What is e-ticket?
Types of e-tickets
E-ticket for companies
Check validity of your e-ticket!
Ticket trade outlets
Types and prices of tickets
One-month ticket
Ticket for a certain number of trips
Time ticket
Ticket sold in public transport
Free travel in Riga public transport
Prices of tickets in minibuses, express buses and night transport
Information for disabled persons
Application form for a fare reduction
Important information for disabled persons of group 1 and 2 and disabled persons under the age of 18
Information for disabled persons of group 1 and 2, disabled persons under the age of 18 and persons accompanying a disabled person of group 1 or a disabled person under the age of 18*.
When using Riga public transport:
disabled persons of group 1 and 2, disabled persons under the age of 18 and persons accompanying a disabled person of group 1 or a disabled person under the age of 18 are asked to take a FREE TICKET from the driver after presenting a disabled person's certificate;
disabled persons of group 1 and 2, and disabled persons under the age of 18 are asked to register each trip with a personalized e-ticket (it is issued free of charge in "Rīgas satiksme" customer service centres upon presenting an ID document and a disabled person's certificate).
The above changes in the order of using Riga public transport are related to the need of ensuring precise count of passengers**.
Information about disabled persons of group 3 is available here.
* Cabinet of Ministers Regulations No. 153 of 31.03.2015. "Regulations on the categories of passengers having right to use fare discounts on routes of the route network".
** Cabinet of Ministers regulations No.435 "The Procedure of Assessing and Compensating Losses and Expenditures Related to the Delivery of Public Transport Services and Setting the Tariff of the Public Transport Service" of 28.07.2015.
Other ticket types
Valid for one month from the first registration day.
Ticket for a certain period of time – 5, 3 days, as well as 24 hours
Single tickets for one trip or pets' tickets, which are sold by the driver only
We use cookies to make the homepage simpler. Please get familiar with our policy of using cookies.
About us Contacts
Other types of news
© Rīgas satiksme | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,390 |
/* global app: false */
app.controller('loginController', ['$scope', '$window', '$location', '$cookies', 'UserService', 'AuthService',
function($scope, $window, $location, $cookies, UserService, AuthService) {
// check cookie exist or not
var cookieObj = AuthService.getAccessToken();
if (cookieObj) {
$location.path('/usertimesheet');
}
// reset messages
var resetMessage = function() {
$scope.errorMessage = null;
$scope.successMessage = null;
};
// login user service
this.loginUser = function() {
$('#submit-btn').text('Please wait...').attr('disabled', 'disabled');
var employeesPromise = UserService.authUser($scope.user);
employeesPromise.then(function(res) {
if (res.success) {
var ttGlobals = $cookies.get('tt_globals');
if (ttGlobals) {
var path = JSON.parse(ttGlobals);
$location.path(path.redirectUrl);
AuthService.clearUrlTracker();
} else {
// $location.path('usertimesheet');
var host = location.host;
window.location = "http://" + host + "/usertimesheet";
}
} else {
$scope.errorMessage = 'Looks like wrong email or password';
$('#submit-btn').text('Sign In').removeAttr("disabled");
}
}, function() {
$scope.errorMessage = 'Something went wrong, please try again!';
$('#submit-btn').text('Sign In').removeAttr("disabled");
});
resetMessage();
};
this.toLocation = function(loc) {
$location.path('/' + loc);
};
}]);
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,805 |
Fort Gordon hosts annual Christmas House to help local families
By Sloane O'Cone
Published: Dec. 3, 2021 at 7:01 PM EST|Updated: Dec. 3, 2021 at 7:16 PM EST
AUGUSTA, Ga. (WRDW/WAGT) - It's a little warmer than we're used to this time of year, but that's not stopping all of the holiday fun. But Christmas time isn't always easy for everyone. Here's how you can help serve one family who's serving us this holiday season.
"We choose our family in the military – this is our family," said Lori Pflieger, vice president of Christmas House.
A 55-year tradition, the Christmas House is Santa's workshop at Fort Gordon.
"We have a statement that was put on one of the applications that says without you my children wouldn't have a Christmas so that's huge," said Pflieger.
MORE | How to prevent Christmas tree fires this holiday season
Service members Grade E-5 and below are helped by elf volunteers picking out toys to spread Christmas cheer. All for free, they get a select number of toys for under their tree.
"Being able to stabilize and have a place where they feel at home and they feel appreciated and loved is absolutely a wonderful opportunity both for them and the people who operate this amazing amazing place," said Truck Carlson, a helper for Santa.
In 2020, they served more than 450 children and 250 families. The gifts are for 11-year-olds down to babies. Toys are donated by the Association of the U.S. Army, churches and businesses.
But for Carlson, the magic might be real.
"This beard is not easy to pull off but when they pull on it and I go, 'oh' and they really believe I cannot my heart grows three sizes; I have a Grinch moment," said Carlson.
If you want to share the cheer, you can visit Fort Gordon Christmas House on Facebook or on its website to sign up to be a volunteer elf or find out how to donate.
MORE: | Holiday roundup: Tree lightings, parades and more are coming up | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,453 |
\section{Introduction
\IEEEPARstart{F}{itness} assignment is a component of many Evolutionary Algorithms (EAs).
It transforms the features of candidate solutions, such as their objective value(s), to scalar values which are then the basis for selection.
Frequency Fitness Assignment (FFA)~\cite{WWTWDY2014FFA,WWTY2014EEIAWGP} was developed to enable algorithms to escape from local optima.
In FFA, the fitness corresponding to an objective value is its encounter frequency so far in fitness assignment steps and is subject to minimization.
As we discuss in detail in Section~\ref{sec:ffa}, FFA turns a static optimization problem into a dynamic one where objective values that are often encountered will receive worse and worse fitness.
In this article, we uncover a so-far unexplored property of FFA:
It is invariant under any bijective transformation of the objective function values.
This is the strongest invariance known to us and encompasses all order-preserving mappings.
Other examples for bijective transformations include the negation, permutation, or even encryption of the objective values.
According to~\cite{OAAH2017IGOAAUPVIP}, \emph{invariance extends performance observed on a single function to an entire associated invariance class, that is, it generalizes from a single problem to a class of problems. Thus it hopefully provides better robustness w.r.t. changes in the presentation of a problem.}
FFA generalizes the performance of an algorithm on \onemax\ to all problems which are bijections thereof, including \jump\ and \trap.
While invariances are generally beneficial for optimization algorithms~\cite{W1989TGAASPWRBAORTIB,HRMSA2011IOIISWCAPFICANSP}, such strong invariance comes at a cost:
The idea that solutions of better objective values should be preferred to those with worse ones can no longer be applied, since many bijections are not order-preserving.
FFA only considers whether objective values are equal or not.
One would expect that this should lead to a loss of performance.
We find that the opposite is the case on several benchmarks evaluated in our study.
On those where FFA increases the number of function evaluations~(FEs) to find the optimum, i.e., the runtime, it seems to do so only linearly with the number of different objective values or the problem dimension~\scale, as both cases are indistinguishable in the investigated problems.
We plug FFA into the most basic EA~\cite{BFM1997HOEC}, the~\opoea\ with standard mutation at \mbox{rate~$1/\scale$}, and obtain the~\opofea.
We investigate its performance on several well-known problems, namely the \onemax, \leadingones, \twomax, \jump, \trap, and \plateau\ functions, the \wmodel, and \maxsat, all defined over bit strings of length~\scale.
We find that the resulting \opofea\ is slower on \onemax, \leadingones, and on the \plateau\ functions, while it very significantly reduces the runtime needed to solve the other problems.
Most notably, in our experiments, it has runtime requirements in the scale \mbox{of~$\scale^2\ln{\scale}$} on the \twomax, \trap\ and \jump\ problems, for which the expected runtime needed by the \opoea\ to find the global optimum is in~\bigOmegaOf{\scale^{\scale}}, \bigThetaOf{\scale^{\scale}}, and~\bigThetaOf{\scale^{\jumpWidthRaw}+\scale\ln{\scale}} (for jump width~\jumpWidth) FEs, respectively.
We confirm the invariance under bijections of the objective value by solving several benchmark problems with the \opofea\ by optimizing the \mbox{Md5~checksums}, i.e., cryptographic hashes, of their objective values and observing no change in algorithm behavior.
We also explore plugging FFA into a well-performing algorithm for a job shop scheduling problem, where it can improve the result quality under budget constraints.
In Section~\ref{sec:ffa}, we discuss the invariance property of FFA and how FFA can be plugged into the \opoea.
Related works are discussed in Section~\ref{sec:relatedWork}.
Our comprehensive experimental study is given in Section~\ref{sec:experiment}.
We conclude our article and give pointers to future work in Section~\ref{sec:conclusions}
\section{Frequency Fitness Assignment
\label{sec:ffa
This study investigates the impact of FFA when plugged into the maybe most basic EA, the \opoea.
The \opoea\ starts with a random bit string~\solspeli{c} of length~\scale.
Until the termination criterion is met, in each step, it applies the standard mutation operator, where each of the \scale~bits of~\solspeli{c} is flipped independently with \mbox{probability~$1/\scale$} and the result is a new string~\solspeli{n}.
If~\solspeli{n} is at least as good as~\solspeli{c}, it replaces~\solspeli{c}.
The expected runtime of the \opoea\ for an arbitrary objective function is at \mbox{most~$\scale^{\scale}$}~\cite{DJW2002OTAOTOPOEA}.
Some of the benchmark problems we investigate invoke this boundary.
We apply a slight modification of the \opoea, called the \opoeagz~\cite{CPD2017TAMPARAOEA}:
The standard mutation in each iteration is repeated until at least one bit is flipped~\cite{M1992HGARWMAH}.
No FE is wasted by evaluating a candidate solution identical to the current one.
The probability of this in the \opoea\ is~\mbox{$\left(1-\scale^{-1}\right)^{\scale}$}, which approaches~\mbox{$1/e\approx 0.368$} for~\mbox{$\scale\rightarrow\infty$}.
This small change thus saves more than one third of the FEs while not changing any other characteristic of the algorithm~\cite{CPD2017TAMPARAOEA,LO2018TAOSSA}.
In the following text, expected runtimes for the \opoea\ will therefore be corrected by factor~\opoeaRepCorrection\ to hold for the \opoeagz\ where necessary
\begin{figure
\setlength{\fboxsep}{0pt
\subfloat[\opoeagz]
\includegraphics{figure_1_a
\label{fig:impl:ea
\hfil
\subfloat[\opofeagz]
\includegraphics{figure_1_b
\label{fig:impl:fea
\caption{The simplified pseudo codes of the \opoeagz\ and the \opofeagz, which applies FFA, for minimization problems.
Differences are marked in \textcolor{red}{red}.
Note: In an actual implementation, the algorithms would remember and return the candidate solution with the best encountered \emph{objective value}~\bestSoFarF\ (not fitness).
\label{fig:eaAndFea
\end{figure
In Figure~\ref{fig:eaAndFea}, we put the pseudo code of the \opoeagz\ next to a simplified version of the \opofeagz.
We assume tha
\begin{enumerate
\item the objective function~\ofel\ is subject to minimization, tha
\item its upper bound~\upperBound\ is known, tha
\item all objective values are integers greater or equal to~0, and tha
\item the solution space is~\mbox{$\booleans^{\scale}$}, the bit strings of length~\scale
\end{enumerate
This can be established for many well-known benchmark problems on which the \opoea\ is usually investigated, as well as for many practical optimization problems like \maxsat.
Under these assumptions, only minimal changes to the \opoeagz\ are necessary to introduce FFA:
An array~\ffaH\ of integers of length~\mbox{$\upperBound+1$} is used to hold the frequency of each objective value in~\mbox{$\intFromTo{0}{\upperBound}$}.
Before selecting one of the two candidate solutions with objective values~\ofeli{c} and~\ofeli{n}, the frequencies~\ffaHb{\ofeli{c}} \emph{and}~\ffaHb{\ofeli{n}} of these objective values are increased.
The results of these increments are compared.
Note: Both frequencies are increased, because if \ffaHb{\ofeli{c}} was not incremented, solutions with unique objective values could become traps for the optimization process.
In order to conduct an efficient search under FFA, the set~${\mathbb{Y}}$ of possible objective values for the problem to be solved should not be too big.
FFA must maintain a frequency table~\ffaH, which has the same size as~$\mathbb{Y}$.
Also, FFA attempts to distribute the search effort evenly over all objective values.
In the extreme case where each distinct solution has a different objective value, FFA almost degenerates the search to a random walk.
Most often, the \opoea\ is analyzed as maximization algorithm.
Since the \opofea\ minimizes the objective value frequencies, we also present the \opoea\ for minimization and define the benchmark problems in Section~\ref{sec:experiment} accordingly.
The \opofea\ implementation given in Figure~\ref{fig:impl:fea} can easily be extended towards a \mpleaX.
It can also be modified to handle problems with unknown upper and lower bounds of the objective function (or objective functions that return real numbers but can still be discretized) by implementing~\ffaH\ as hash table~\cite{CLRS2009HT} (see Section~\ref{sec:md5}).
FFA can be introduced into arbitrary metaheuristics
\begin{theorem
\label{theo:invariance
The sequence of candidate \mbox{solutions~$\solspel\in\mathbb{X}$} generated by an optimization process applying FFA is invariant under any \mbox{bijection~$g:\mathbb{Y}\mapsto\mathbb{Z}$} of the objective \mbox{function~$\ofel:\mathbb{X}\mapsto\mathbb{Y}$}, where \mbox{$\mathbb{X}$~is} the solution space, \mbox{$\mathbb{Y}$~is} a finite subset of~$\mathbb{R}$, \mbox{and~$\mathbb{Z}$} is a set of the same size
\begin{IEEEproof
The bijection~$g$ maps each value from~$\mathbb{Y}$ to one value in~$\mathbb{Z}$ and vice versa.
Therefore, if two objective values identify the same (or a different) entry in~\ffaH, so will their bijective transformations.
Under FFA, \emph{only} the entries in~\ffaH\ are modified and compared to make selection decisions
\end{IEEEproof
\end{theorem}
We can also prove this inductively:
Assume that two runs of the \opofeagz\ which minimize \ofel\ and $g\circ\ofel$, respectively, are identical until iteration~$t$:
They have the same random seed, same~$\solspeli{c}$, and $\ffaHb{y}=\ffaHpb{g(y)}\forall y\in\mathbb{Y}$ holds for their respective FFA tables.
Both will sample the same next point~$\solspeli{n}$.
$\ffaHb{\ofelb{\solspeli{c}}}=\ffaHpb{g(\ofelb{\solspeli{c}})}$ and $\ffaHb{\ofelb{\solspeli{n}}}=\ffaHpb{g(\ofelb{\solspeli{n}})}$ will still hold after incrementing the entries.
Hence, both will make the same decision regarding the update of~\solspeli{c} and begin iteration $t+1$ in the same state.
In Sections~\ref{sec:jump}, \ref{sec:trap}, and~\ref{sec:md5}, we provide experimental evidence that this invariance indeed holds
\section{Related Work
\label{sec:relatedWork
FFA was designed as an approach to prevent the premature convergence to a local optimum.
In the context of EAs, it is therefore related to fitness sharing, niching, and clearing~\cite{GR1987GAS,P1996ACPAANMFGA}.
Several such diversity-preserving mechanisms have been plugged into a \mplea{\paramMu}{1} and studied theoretically in~\cite{FOSW2009AODPMFGE} on \twomax, where the original algorithm requires~\bigOof{\scale^\scale}~FEs to find the optimum.
It is found that avoiding fitness or genotype duplicates does not help, whereas with deterministic crowding and sufficiently large~$\paramMu$, the problem can be solved efficiently with high probability.
With fitness sharing and $\paramMu\geq2$, the \mplea{\paramMu}{1} can solve \twomax\ in~\bigOof{\paramMu\scale\log{\scale}}.
Different from the above methods, which only consider the current population, FFA tries to guide the search away from objective values that have been encountered often during the whole course of the optimization process.
In Section~\ref{sec:twomax}, we will show that FFA can help solving \twomax\ efficiently already at~$\paramMu=1$.
Another related idea is Tabu Search (TS)~\cite{GTDW1993AUGTTS}, which improves local search by declaring solutions (or solution traits) which have already been visited as tabu, preventing them from being sampled again.
Like FFA, it utilizes the search history, but usually in form of a list of tabu solutions or solution traits.
Different from FFA, the TS relies on the order of objective values when deciding which solutions to accept.
The Fitness Uniform Selection Scheme (FUSS)~\cite{H2002FUSTPGD} selects solutions in such a way that their corresponding objective values are approximately uniformly distributed within the range of the minimum and maximum objective value in the population.
The Fitness Uniform Deletion Scheme (FUDS)~\cite{HL2006FUO} works similarly, but instead of selecting individuals, it deletes them when slots in the population are required to integrate the offspring.
Both methods need populations, only consider the individuals in the current population, and are only invariant under translation and scaling of the objective function.
The ageing operator in Artificial Immune Systems (AIS) deletes individuals either after they have survived a certain number of iterations, with a certain probability, or both~\cite{COY2020WHAAEAISTOEA,COY2018FAIS,COY2019AISCFAGAFTNNPP}.
Ageing has also been applied in EAs~\cite{GTT1996IAIGA}.
Like FFA, ageing makes solutions less attractive if they remain in the population for a long time.
Different from FFA, the information about these solutions disappears from the optimization process once they ``die.''
Methods which try to balance between solution quality and population diversity are today grouped under the term Quality-Diversity (QD) algorithms~\cite{GLY2019QDTS,CD2018QADOAUMF,PSS2016QDANFFEC}.
Novelty Search (NS)~\cite{LS2011AOETTSFNA} is a QD algorithm.
Instead of an objective function~\ofel, NS uses a (dynamic) novelty metric~$\rho$.
This metric is computed, e.g., as mean behavior difference to the $k$~nearest neighbors in an archive of past solutions.
FFA works on the original objective function and just transforms it to a dynamic fitness measure.
It does not require an archive of solutions but uses a table~\ffaH\ counting the frequency of the objective values.
While NS was aimed to abandon the objective function~\ofel, using it as behavior definition was also tested~\cite{LS2011AOETTSFNA}.
Then, $\rho$~is the mean distance to $k$~neighbors (or all solutions ever found) in the objective space.
Unlike FFA, this uses the assumption that differences between objective values are useful or correlate with diversity.
Novelty Search with Local Competition (NSLC)~\cite{LS2011EADOVCTNSALC} combines the search for finding diverse solutions with a local competition objective rewarding solutions which can outperform those most similar to them.
The \mbox{MAP-Elites} algorithm~\cite{MC2015ISSBME} combines a performance objective~\ofel\ and a user-defined space of features that describe candidate solutions, which is not required by FFA.
\mbox{MAP-Elites} searches for highest-performing solution in each cell of the discretized feature space.
Surprise Search (SS)~\cite{GLY2016SSBOAN} uses the concept of surprise as an alternative to novelty.
A solution is scored by the difference of its observed behavior from the expected behavior.
A history of discovered solution behaviors is maintained and used to predict the behavior of the new solutions.
SS has also been combined with NSLC in a multi-objective fashion~\cite{GLY2019QDTS}.
All of the above algorithms are conceptually different from FFA.
They either are complete optimization methods (NS, QD, TS) or modules for EAs (FUSS/FUDS), while FFA can be plugged into many different optimization algorithms.
Unlike FFA, none of the above methods exhibits an invariance under bijective transformations of the metrics they try to optimize.
From the perspective of invariances, FFA is related to Information-Geometric Optimization (IGO)~\cite{OAAH2017IGOAAUPVIP}.
IGO also replaces the objective function~\ofel\ with an adaptive transformation of it.
This transformation indicates how good or bad an objective value is relative to other observed objective values, i.e., is different from our method which simply compares encounter frequencies.
IGO is invariant under all strictly increasing transformations of~\ofel, whereas FFA creates invariance under all bijective transformations.
IGO is a complete family of optimization methods which can also exhibit invariance under several transformations of the search space.
Since FFA only works on~\ofel, it cannot provide such invariances.
IGO can optimize continuous objective functions, which is not possible with FFA
\section{Experiments
\label{sec:experiment
We now apply the \opoeagz\ and the \opofeagz\ to minimization versions of different classical optimization problems.
We initialize the \opoeagz\ and the \opofeagz\ with the same random seeds for each run, i.e., we always have pairs of runs starting at the same random initial solution and sampling the same first offspring solutions for both algorithms.
The runs are terminated when they discover the optimum.
In some experiments, we additionally limit the computational budget to \mbox{$10^{10}=10'000'000'000$~FEs}.
This should be enough to converge on problems that the algorithms can solve well, as can be seen in Section~\ref{sec:leadingones}.
Leading to several hours to more than a day for a single run on the corresponding problems, this was also the maximum budget we could feasibly allow.
Whenever all runs on an instance succeed to find the optimum, we can compare the mean runtime `\meanRuntime' they need to do so in terms of the consumed~FEs (often called the \emph{first hitting time}).
When some runs fail in the budget-limited settings, we follow the approach from~\cite{HAFR2012RPBBOBES} and use the empirically estimated expected runtime (\ert) instead.
The \ert\ for a problem instance is estimated as the ratio of the sum of all FEs that all the runs consumed until they \emph{either} have discovered a global optimum \emph{or} exhausted their budget, divided by the number of runs that discovered a global optimum~\cite{HAFR2012RPBBOBES}.
The \ert\ is the mean expected runtime under the assumption of independent restarts after failed runs, which then may either succeed (consuming the mean runtime of the successful runs) or fail again (with the observed failure probability, after consuming ${10}^{10}$~FEs).
In order to guarantee the reproducibility of our work, we provide the complete data used in this paper, including the result log files, the scripts used to generate all the figures and tables, as well as the source code of all algorithms and all benchmark problems in~\cite{WWLC2019TEDFTSFFAMOAIUBTOTOF}
\subsection{\onemax~Problems
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_2}
\caption{Illustrations of the \onemax, \twomax, \trap, \jump, and \plateau\ problems for $\scale=32$ and $\jumpWidth=6$.
\label{fig:function_illustrations
\end{figure
\onemax~\cite{M1992HGARWMAH} is a unimodal optimization problem where the goal is to discover a bit string of all ones.
Its minimization version of dimension~\scale\ is defined below and illustrated in Figure~\ref{fig:function_illustrations}
\begin{equation
\onemax(\solspel) = \scale - \countones{\solspel}\textnormal{~where~}\countones{\solspel}=\sum_{i=1}^{\scale} \solspelval{i
\end{equation
It has a black-box complexity of $\bigOmegaOf{\scale/\ln{\scale}}$~\cite{ER1963OTPOIT,DJW2006UALBFRSHIBBO}.
Here, an \opoea\ has an expected runtime of $\bigThetaOf{\scale\ln{\scale}}$~FEs~\cite{M1992HGARWMAH}.
A very exact formula~\cite{HPRTC2018PAOTOPOEA} with our correction factor for the \opoeagz\ is given in Equation~\eqref{eq:opoea:onemax}, where $C_1\approx1.89254$ and $C_2\approx0.59789875$
\begin{equation
\resizebox{0.905\linewidth}{!}{\ensuremath{\opoeaRepCorrected{e\scale\ln{\scale}-C_1\scale+0.5e\ln{\scale}+C_2+\bigOof{(\ln{\scale})/\scale}}}
\label{eq:opoea:onemax
\end{equation
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_3
\caption{The runtime measured for the \opoeagz\ and \opofeagz\ on selected instances of the \onemax\ problem.
\label{fig:onemax_runtime
\end{figure
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_4}\
\caption{9~typical runs of \opofeagz\ on \onemax\ ($\scale=64$)
\label{fig:onemax_progress
\end{figure
We conduct 3333~runs with both the \opoeagz\ and \opofeagz\ on this problem for each~$\scale\in\intFromTo{3}{333}$ and 71~runs for 26~selected larger values of~\scale\ up to~4096, all without budget constraint.
In Figure~\ref{fig:onemax_runtime}, we illustrate the mean runtime to solve the instances with the range of the 15.9\% to the 84.1\% quantiles in the background.\footnote
These quantiles are wider than the inter-quartile range and would represent exactly the range mean-stddev to mean+stddev under a normal distribution
}
In the top-most sub-figure, we illustrate all results for $\scale\in\intFromTo{3}{52}$.
The middle figure is a log-log plot based on the complete data, but with marks only placed at $\scale\in \{2^i, round(2^i/3)\}$ to not clutter the plot.
In both diagrams, we illustrate the results of Equation~\eqref{eq:opoea:onemax} without the \mbox{\bigOof{(\ln{\scale})/\scale}~term}.
They exactly match the results of the \opoeagz.
The mean runtime of the \opofeagz\ seems to be in the scale \mbox{of~$\scale^2\ln{\scale}$} for the investigated range of~\scale.
The illustrated model was obtained using linear regression on the complete set of 1'105'069~runs with the inverse variances of the measured runtimes per distinct \scale~value used as weights.
All regression models in the rest of this article are obtained in the same way.
The curve of the model visually fits to the mean runtimes and the adjusted $R^2$~value of~$0.8$ indicates that it can explain most of the variance in the data.
The observed distribution of the runtime is skewed and the median is lower than the mean on all dimensions.
This is illustrated exemplarily in the histogram for dimension~$\scale=32$ in the lower part of Figure~\ref{fig:onemax_runtime}.
Its shape resembles a log-normal distribution or a sum of parameterized geometric distributions~\cite{D2019ARSHVSD}.
For~$s\leq8$, the histograms look like exponential distributions, caused by the high chance of randomly guessing the optimum.
Figure~\ref{fig:onemax_progress} illustrates nine typical runs of the \opofeagz\ on the \onemax\ problem with~$\scale=64$.
Initially, some of the runs progress towards better solutions, others to worse.
They change the search direction from time to time.
This oscillation is repeated until the global optimum is discovered
\subsection{\leadingones~Problems
\label{sec:leadingones
The \leadingones\ problem~\cite{W1989TGAASPWRBAORTIB,R1997CPOEA} maximizes the length of a leading sequence containing only 1~bits.
Its minimization version of dimension~\scale\ is defined as follows
\begin{equation
\leadingones(\solspel) = \scale - \sum_{i=1}^{\scale} \prod_{j=1}^i \solspelval{j
\end{equation
The problem exhibits epistasis, as the bit at index~2 can only contribute to the objective value if the bit at index~1 has value~1.
The black-box complexity of \leadingones\ is~$\bigThetaOf{\scale\ln{\ln{\scale}}}$~\cite{AADLMW2013TQCOFAHP}.
The \opoea\ has a quadratic expected runtime on \leadingones~\cite{DJW2002OTAOTOPOEA}.
The exact formula~\cite{BDN2010OFAAMRFTLP,S2013ANMFLBOTRTOEA} is presented with our correction factor in Equation~\eqref{eq:opoea:lo}
\begin{equation
\opoeaRepCorrected{0.5\scale^2\left( \left(1-1/\scale\right)^{1-\scale} -1 + 1/\scale\right)
\label{eq:opoea:lo
\end{equation
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_5}
\caption{The runtime measured for the \opoeagz\ and \opofeagz\ on the \leadingones\ problem.
\label{fig:leadingones_runtime
\end{figure
Figure~\ref{fig:leadingones_runtime} has the same structure as Figure~\ref{fig:onemax_runtime} and is based on an experiment with the same parameters, only using the \leadingones\ instead of the \onemax\ problem.
The \opoeagz\ behaves as predicted in Equation~\eqref{eq:opoea:lo}.
The runtime of the \opofeagz\ fits to the illustrated regression model for the investigated range of~\scale\ and can explain almost all of the variance of the data.
Due to the approximately cubic runtime, the mean time to solve \leadingones\ at $\scale=4096$ is $9.9\!\cdot\!10^{9}$~FEs.
The histogram of the observed runtimes for \mbox{dimension~$\scale=32$} in the lower part of Figure~\ref{fig:leadingones_runtime} is slightly skewed
\subsection{\twomax~Problems
\label{sec:twomax
The minimization version of the \twomax~\cite{FQW2018ELDBOAWHTMO,VHGN2002FTTIMEAHSP} problem of dimension~\scale\ can be defined as follows
\begin{equation
\resizebox{0.905\linewidth}{!}{\ensuremath
\twomax(\solspel) = \left\{\!\!\
\begin{array}{l@{~}l
0&\textnormal{if }\countones{\solspel} = \scale\\%
1+\scale-\max\{\countones{\solspel}, \scale-\countones{\solspel}\}&\textnormal{otherwise
\end{array
\right
}
\end{equation
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_6
\caption{The runtime measured for the \opoeagz\ and \opofeagz\ on the \twomax\ problem.
\label{fig:twomax_runtime
\end{figure
The \twomax\ problem introduces deceptiveness in the objective function by having a local and a global optimum.
Since their basins of attraction have the same size, a \opoea\ can solve the problem in \bigThetaOf{\scale\ln{\scale}} steps with probability~0.5 while otherwise needing exponential runtime.
The resulting overall expected runtime is in \bigOmegaOf{\scale^{\scale}}~\cite{FOSW2009AODPMFGE,FQW2018ELDBOAWHTMO}.
On each instance of \twomax\ with $\scale\in\intFromTo{3}{32}$, we conduct 71~runs with \opoeagz\ and 3333~with \opofeagz.
For all experiments from here on except in Section~\ref{sec:jssp}, we use a budget of ${10}^{10}$~FEs.
The \opoeagz\ succeeds in solving the problem only in about half of the runs for~$\scale>10$ within the budget, which was the reason for the \mbox{71-run} limit.
We illustrate its performance in Figure~\ref{fig:twomax_runtime} only for dimensions~$\scale<10$ where it always succeeded.
All runs of \opofeagz\ solved their corresponding instances.
The algorithm exhibits a mean runtime fitting to a model of scale~$\scale^2\ln{\scale}$ for~$\scale\in\intFromTo{3}{32}$, which is a big improvement in comparison to the exponential time needed by the \opoea.
The lower adjusted~$R^2$ and less smooth increase of the runtime with~\scale\ result from the slightly different shapes of the \twomax\ problem for odd and even values of~\scale.
The median runtime is again smaller than the mean.
The histogram of the observed runtimes for $\scale=32$ in the lower part of Figure~\ref{fig:twomax_runtime} again exhibits the familiar skew.
These results are interesting, since avoiding fitness duplicates in a \mplea{\paramMu}{1} does not help to solve the problem efficiently~\cite{FOSW2009AODPMFGE}.
FFA thus does more than this even \mbox{at~$\paramMu=1$}
\subsection{\jump~Problems
\label{sec:jump
The \jump\ functions as defined in~\cite{DJW2002OTAOTOPOEA,FQW2018ELDBOAWHTMO} introduce a deceptive region of width~\jumpWidth\ with very bad objective values right before the global optimum.
The minimization version of the \jump\ function of dimension~\scale\ and jump width~\jumpWidth\ is defined as follows
\footnote
Researchers have formulated different types of \jump\ functions.
The one in~\cite{DDK2015UBBCOJF}, e.g., is similar to our \plateau\ function but differs in the plateau objective value
\begin{equation
\jump(\solspel) = \left\{\!\!\
\begin{array}{l@{~}l
\scale-\countones{\solspel}&\textnormal{if }\left(\countones{\solspel} = \scale\right)\lor\left(\countones{\solspel} \leq \scale-\jumpWidth\right)\\%
\jumpWidth+\countones{\solspel}&\textnormal{otherwise
\end{array
\right
\label{eq:jump
\end{equation
The expected runtime of the \opoea\ on such problems is in \bigThetaOf{\scale^{\jumpWidthRaw}+\scale\ln{\scale}}~\cite{DJW2002OTAOTOPOEA}.
The \jump\ problem is a bijective transformation of the \onemax\ problem.\footnote
For~$\jumpWidth=1$, the \jump\ and \plateau\ problems are \onemax\ problems, which is why we do not perform or illustrate any runs with $\jumpWidth=1$ for either
}
The \opofeagz\ will exhibit the same behavior and runtime requirement on \emph{any} jump problem instance as on a \onemax\ instance of the same dimension~\scale, regardless of the jump width~\jumpWidth
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_7
\caption{The runtime measured for the \opoeagz\ and \opofeagz\ on \jump\ problems with dimension~\scale\ and jump width~$\jumpWidth>1$.
\label{fig:jump_runtime
\end{figure
We conduct experiments with five different jump widths~\jumpWidth, namely
\mbox{$\left\lfloor\ln{\scale}\right\rfloor$},
\mbox{$\left\lfloor\ln{\scale}\right\rfloor+1$},
\mbox{$\left\lfloor\sqrt{\scale}\right\rfloor$},
\mbox{$\left\lfloor\sqrt{\scale}\right\rfloor+1$}, and
\mbox{$\left\lfloor0.5\scale\right\rfloor-1$}.
We illustrate the results in Figure~\ref{fig:jump_runtime} only for those setups where a success rate of 100\% within the $10^{10}$~FEs were achieved in 71~runs.
\opofeagz\ finds the optimum in all runs and all the observed mean runtimes fall on the function fitted to the results on \onemax\ (see Figure~\ref{fig:onemax_runtime}), confirming that the two problems are indeed identical from the perspective of an algorithm using FFA.
As expected, the runtime of the \opoeagz\ steeply increases with the jump width~\jumpWidth\ and it is outperformed by the \opofeagz.
Depending on its configuration, the AIS \mbox{Opt-IA}~\cite{COY2020WHAAEAISTOEA} needs runtimes of at least \bigOof{\scale^2 \ln{\scale}} and \bigOof{\scale^3}~FEs on \onemax\ and \leadingones, respectively.
It seems that our \opofea\ has similar requirements, i.e., compared to the \opoea\ with standard bit mutation, a linear slowdown is incurred on these problems.
However, on the \jump\ problems, \mbox{Opt-IA} needs, again depending on its configuration, at least~\bigOof{\frac{\scale^{\jumpWidthRaw+1}\cdot e^{\jumpWidthRaw}}{\jumpWidthRaw^{\jumpWidthRaw}}}~FEs.
The \mbox{(1+1) Fast-IA}~\cite{COY2018FAIS}, which is at least as fast as the \opoea\ on \onemax\ and \leadingones, needs exponential expected runtime for sufficiently large~\jumpWidth.
The Fast~EA~\cite{DLMN2017FGA} using a heavy-tailed mutation rate, too, needs exponential runtime to solve \jump.
The asymptotic performance of the cGA on \jump\ is not worse than on \onemax\ for logarithmic jump widths~\jumpWidth~\cite{D2019ATRAFTCOJFECCFVANEC}, but it still needs exponential time for larger~\jumpWidth~\cite{D2019AELBFTROTCOJF}.
\opofea, however, performs on \jump\ exactly as on \onemax, regardless of~\jumpWidth
\subsection{\trap\ Function
\label{sec:trap
The \trap\ function~\cite{NB2003AAOTBOSEAOTF,DJW2002OTAOTOPOEA} is very similar to the \onemax\ problem, except that it replaces the worst possible solution there with the global optimum.
Following a path of improving objective values will \emph{always} lead the optimization algorithm away from the global optimum.
The \opoea\ here has an expected runtime of~\bigThetaOf{\scale^{\scale}}~\cite{DJW2002OTAOTOPOEA}.
The minimization version of the \trap\ function can be specified as follows
\begin{equation
\trap(\solspel) = \left\{\!\!\
\begin{array}{l@{~}l
0&\textnormal{if~}\countones{\solspel}=0\\%
\scale-\countones{\solspel}+1&\textnormal{otherwise
\end{array}\right
\end{equation
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_8
\caption{The runtime measured for the \opofeagz\ on the \trap\ problem.
\label{fig:trap_runtime
\end{figure
The \trap\ function is another bijective transformation of the \onemax\ problem.
When we plot the results from 3333~runs of the \opofeagz\ on the \trap\ function in Figure~\ref{fig:trap_runtime}, we find that the results are almost exactly identical to those obtained on \onemax\ and illustrated in Figure~\ref{fig:onemax_runtime}.
The function fitted to the mean runtime on \onemax, again plotted in Figure~\ref{fig:trap_runtime}, passes through the points measured on the \trap\ function
\subsection{\plateau~Problems
The minimization version of the \plateau~\cite{AD2018PRAFP} function of dimension~\scale\ with plateau width~\plateauWidth\ is defined as follows
\begin{equation
\resizebox{0.905\linewidth}{!}{\ensuremath
\plateau(\solspel) = \left\{\!\!\
\begin{array}{l@{~}l
\scale-\countones{\solspel}&\textnormal{if }\left(\countones{\solspel} = \scale\right)\lor\left(\countones{\solspel} \leq \scale-\plateauWidth\right)\\%
\plateauWidth&\textnormal{otherwise
\end{array
\right
}
\end{equation
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_9
\caption{The runtime measured for the \opoeagz\ and \opofeagz\ on \plateau\ problems with dimension~\scale\ and plateau width~$\plateauWidth>1$.
\label{fig:plateau_runtime
\end{figure
The expected runtime of the \opoea\ on such a problem is in~\bigThetaOf{\scale^{\plateauWidthRaw}}~\cite{AD2018PRAFP}.
The \plateau\ problems are no bijective transformation of \onemax.
Instead, they reduce the number of possible objective values \mbox{($|\mathbb{Z}|<|\mathbb{Y}|$)}.
We can expect that the fitness of the solutions on the plateau will get worse quickly under FFA.
We conduct the same experiment as for the \jump\ function with the \plateau\ function and plot the results in the same manner in Figure~\ref{fig:plateau_runtime}.
This time, the \opofeagz\ performs worse than the \opoea.
Interestingly, if we divide the observed mean runtimes of \opofeagz\ by the problem dimension~\scale, we approximately obtain those observed with \opoeagz\ (see the gray marks in Figure~\ref{fig:plateau_runtime}).
This might be a coincidence and more research is necessary
\subsection{Bijection Invariance: Md5 Checksum of Objective Values
\label{sec:md5
We now repeat our experiments with the \opofeagz\ on the \onemax, \twomax, \leadingones, and \trap\ problems \mbox{with~$\scale\in\intFromTo{3}{32}$}, but use a transformation of the objective functions:
Instead of working on the objective values directly, we optimize their \mbox{Md5~checksums}.
We therefore implement~\ffaH\ as a hash table where their encounter frequencies are stored.
The \mbox{Md5~checksum} is a 128~bit message digest published in~\cite{R1992TMMDA}, where \emph{it is conjectured that it is computationally infeasible to produce two messages having the same message digest}.
Although \mbox{Md5~checksums} are not an encryption method, they do allow us to further test the invariance under such ``extreme'' transformations and the idea of implementing~\ffaH\ as hash table without further assumptions.\footnote
It can be assumed that applying \opoea\ to this problem would yield the worst-case complexity and we thus omit doing it.}
We use the same random seeds as in the original runs working on the objective values.
We find that all 3333~runs on all the instances have the same \mbox{(FE, objective value)}-traces as their counterparts, which also follows from Theorem~\ref{theo:invariance}.
Illustrating these results here has no merit, as the figures would be identical to those already shown.
We include the full log files as well as the algorithm implementation in our dataset~\cite{WWLC2019TEDFTSFFAMOAIUBTOTOF}
\subsection{\wmodel~Instances
The \wmodel~\cite{WW2018DFOCOPATTWMBPFST,W2018TWMATBBDOBPIFTBGW,WNSRG2008ATMFMOERANFL} is a benchmark problem which exhibits different difficult fitness landscape features in a tunable fashion
\footnote
There was a mistake in~\cite{WW2018DFOCOPATTWMBPFST}: at line~19 of Algorithm~1, ``start'' should be replaced with ``\wmn''. This was corrected in~\cite{WCLW2019SADSOBIFATMPFBBDOA,W2018RFSSAOTWMFBBOB} and was always correct in the \mbox{\wmodel~implementation~\cite{W2018TWMATBBDOBPIFTBGW}}
}
These include the base size (via parameter~\wmn), neutrality (via parameter~\wmm), epistasis (via parameter~\wmnu), and ruggedness (via parameter~\wmg), from which instances of dimension~\mbox{$\scale=\wmmn$} result.
The \wmodel\ base problem is equivalent to \onemax\ but searches for a string of alternating 0 and 1~bits.
Different transformations are applied to it.
While the ruggedness transformation is a bijective transformation of objective function, the mappings introducing neutrality and epistasis transform the search space itself.
19~diverse \wmodel\ instances have been selected in~\cite{WCLW2019SADSOBIFATMPFBBDOA} based on a large-scale experiment.
No theoretical bounds for the runtimes on these instances are known, but they exhibit different degrees of empirical hardness for different algorithms
\begin{table}[tb
\centerin
\caption
The \ert\ and fraction~\successFrac\ of the~71 \opoeagz\ runs discovering the optimum on the 19~\wmodel\ problem instances selected in~\cite{WCLW2019SADSOBIFATMPFBBDOA}. As all runs of \opofeagz\ reached the optimum, we present its mean and median runtime.
\label{tbl:wmodelresults
\clipbox{0pt}
\resizebox{\linewidth}{!}
\begin{tabular}{@{~}r@{~~}r@{~~}r@{~~}r@{~~}r|rr|rr@{~}
\hlin
\multicolumn{5}{c|}{\wmodel~Instance}&\multicolumn{2}{c|}{\opoeagz}&\multicolumn{2}{c}{\opofeagz}\\%
id&\wmn&\wmm&\wmnu&\wmg&\successFrac&\multicolumn{1}{c|}{\ert}&\multicolumn{1}{c}{\meanRuntime}&\multicolumn{1}{c}{\medianRuntime}\\%
\hlin
1&10&2&6&10&1&5'928&1'090&734\\%
2&10&2&6&18&1&6'605&904&815\\%
3&16&1&5&72&1&9'400&3'646&3'191\\%
4&16&3&9&72&1&864'850&5'856&5'163\\%
5&25&1&23&90&0.66&$5.11\!\cdot\!10^{9}$&3'049&2'218\\%
6&32&1&2&397&0&$+\infty$&1'602&1'355\\%
7&32&4&11&0&0.31&$2.23\!\cdot\!10^{10}$&279'944&238'904\\%
8&32&4&14&0&0.31&$2.23\!\cdot\!10^{10}$&287'286&231'266\\%
9&32&4&8&128&0.75&$3.61\!\cdot\!10^{9}$&219'939&201'524\\%
10&50&1&36&245&0.35&$1.84\!\cdot\!10^{10}$&68'248&60'347\\%
11&50&2&21&256&0.46&$1.23\!\cdot\!10^{10}$&572'874&484'795\\%
12&50&3&16&613&0.24&$3.18\!\cdot\!10^{10}$&639'914&568'120\\%
13&64&2&32&256&0.28&$2.55\!\cdot\!10^{10}$&383'998&359'452\\%
14&64&3&21&16&0.27&$2.74\!\cdot\!10^{10}$&$1.00\!\cdot\!10^{6}$&851'246\\%
15&64&3&21&256&0.17&$4.92\!\cdot\!10^{10}$&$1.28\!\cdot\!10^{6}$&$1.07\!\cdot\!10^{6}$\\%
16&64&3&21&403&0.23&$3.44\!\cdot\!10^{10}$&$1.12\!\cdot\!10^{6}$&884'679\\%
17&64&4&52&2&0.42&$1.37\!\cdot\!10^{10}$&612'610&537'448\\%
18&75&1&60&16&0.27&$2.74\!\cdot\!10^{10}$&225'489&184'933\\%
19&75&2&32&4&0.25&$2.94\!\cdot\!10^{10}$&$1.83\!\cdot\!10^{6}$&$1.61\!\cdot\!10^{6}$\\%
\hlin
\end{tabular
}
\end{table}
We conduct 71~runs for both algorithms on each of these 19~\wmodel\ instances.
In Table~\ref{tbl:wmodelresults}, we presented the fraction~\successFrac\ of runs that found the global optimum and the \ert\ for \opoeagz.
While it can always solve the four easiest instances, its success rate within the $10^{10}$~FEs then drops, which leads to very high \ert~values.
The \opofeagz\ is always faster than \opoeagz\ and all of its runs discovered the global optima of their respective \wmodel\ instances.
In this case, $\meanRuntime=\ert$ and we list it alongside the median runtime~\medianRuntime, which, like on the previously investigated problems, is always smaller than the mean.
Of special interest here is instance~6, which could not be solved by \opoeagz\ at all.
Here, $\scale=\wmmn=32$ and only a ruggedness transformation with $\wmg=397$ is performed, while no additional epistasis ($\wmnu=2$) or neutrality ($\wmm=1$) are introduced in the landscape.
In other words, here, the objective function is equivalent to a (bijective) permutation of the objective values produced by a \onemax\ instance (with a different but equivalent base problem).
This permutation leads to a long deceptive slope in the mid-range of the original objective values and three extremely rugged spikes near the global optimum, i.e., we can expect it to have a hardness similar to the \jump\ or \trap\ functions for the \opoea, which the experiment confirms.
Only for this instance, we conduct 3333 runs with \opofeagz\ and find that the mean~1602 and median~1355 of the runtime are very close to those on the \onemax\ (1620, 1375) and \trap\ functions (1620, 1390), which again confirms the invariance of FFA towards bijective transformations of the objective function
\subsection{\maxsat~Problems
\label{sec:maxsat
The Satisfiability Problem is one of the most prominent problems in artificial intelligence.
An instance is a formula~\mbox{$\maxSatFormula:\booleans^{\scale}\mapsto\booleans$} over~\scale\ Boolean variables.
The variables appear as literals either directly or negated in~\maxSatClauses\ ``{\texttt{or}}'' clauses, which are all combined into one ``{\texttt{and}}''.
Solving a Satisfiability Problem means finding a setting~\solspel\ for the variables so that \maxSatFormulab{\solspel} becomes \texttt{true} (or whether such a setting exists).
This \npHard~\cite{G1979CAIAGTTTONC} decision problem is transformed to an optimization version, the \maxsat\ problem~\cite{HS2005SLSFAA}, where the objective function \ofelb{\solspel}, subject to minimization, computes the number of clauses which are \texttt{false} under~\solspel.
If~\mbox{$\ofelb{\solspel}=0$}, all clauses are \texttt{true}, which solves the Satisfiability Problem.
The worst possible value~\upperBound\ that~\ofel\ can take on is~\maxSatClauses.
The \maxsat\ problem exhibits low epistasis but deceptiveness~\cite{RW1998GABITMD}.
In the so-called phase transition region with~\mbox{$\maxSatClauses/\maxSatVariables\approx4.26$}, the average instance hardness for stochastic local search algorithms is maximal~\cite{HS2000SAORFROS,DNS2017TCAOEAORSkCF,DNS2015IRBFTOPOEOR3CFBOFDC}.
We apply our algorithms as incomplete solvers~\cite{GKSS2008SS} on the ten sets of \emph{satisfiable} uniform random \mbox{3-SAT} instances from SATLib~\cite{HS2000SAORFROS}, which stem from this region.
Here, the number of variables~\scale\ is \mbox{from~$\left\{20\right\}\cup\left\{25i:i\in \intFromTo{2}{10}\right\}$}, where 1000~instances are given for \mbox{$\scale\in\left\{20,50,100\right\}$} and 100 otherwise.
With the \opoeagz, we can only conduct 11~runs for each \mbox{$\scale\in\left\{20,50,75\right\}$} due to the high runtime requirement resulting from many runs failing to solve the problem within \mbox{$10^{10}$~FEs}.
With the \opofeagz, we conduct 11~runs for \mbox{$\scale\in\left\{20,50,100\right\}$} and 110~runs for each dimension other than these, i.e., have \mbox{$110*100=11*1000=11'000$~runs} for each instance dimension~\scale\ in SATLib
\begin{table}[tb
\caption{The fraction~\successFrac\ of successful runs, the \ert, and the mean end objective value~\meanBestF\ for \opoeagz\ and \opoeagz\ on the satisfiable \maxsat\ instances from SATLib.
\label{tbl:maxsat
\centerin
\resizebox{\linewidth}{!}
\begin{tabular}{r|r@{~~}r@{~}r|r@{~~}r@{~}r
\hlin
\multicolumn{1}{c|}{instance}&\multicolumn{3}{c|}{\opoeagz}&\multicolumn{3}{c}{\opofeagz}\\%
\multicolumn{1}{c|}{set}&\multicolumn{1}{c}{\successFrac}&\multicolumn{1}{c}{\ert}&\multicolumn{1}{c|}{\meanBestF}&\multicolumn{1}{c}{\successFrac}&\multicolumn{1}{c}{\ert}&\multicolumn{1}{c}{\meanBestF}\\%
\hlin
\instance{uf20\_*}&0.985&$1.91\!\cdot\!10^{8}$&0.0154&1&3'091&0\\%
\instance{uf50\_*}&0.748&$3.56\!\cdot\!10^{9}$&0.299&1&93'459&0\\%
\instance{uf75\_*}&0.583&$7.41\!\cdot\!10^{9}$&0.528&1&490'166&0\\%
\instance{uf100\_*}&\multicolumn{1}{c}{--}&\multicolumn{1}{c}{--}&\multicolumn{1}{c|}{--}&1&$2.14\!\cdot\!10^{6}$&0\\%
\instance{uf125\_*}&\multicolumn{1}{c}{--}&\multicolumn{1}{c}{--}&\multicolumn{1}{c|}{--}&1&$5.27\!\cdot\!10^{6}$&0\\%
\instance{uf150\_*}&\multicolumn{1}{c}{--}&\multicolumn{1}{c}{--}&\multicolumn{1}{c|}{--}&1&$1.40\!\cdot\!10^{7}$&0\\%
\instance{uf175\_*}&\multicolumn{1}{c}{--}&\multicolumn{1}{c}{--}&\multicolumn{1}{c|}{--}&1&$5.78\!\cdot\!10^{7}$&0\\%
\instance{uf200\_*}&\multicolumn{1}{c}{--}&\multicolumn{1}{c}{--}&\multicolumn{1}{c|}{--}&0.991&$2.44\!\cdot\!10^{8}$&0.00945\\%
\instance{uf225\_*}&\multicolumn{1}{c}{--}&\multicolumn{1}{c}{--}&\multicolumn{1}{c|}{--}&0.994&$2.43\!\cdot\!10^{8}$&0.00555\\%
\instance{uf250\_*}&\multicolumn{1}{c}{--}&\multicolumn{1}{c}{--}&\multicolumn{1}{c|}{--}&0.992&$2.43\!\cdot\!10^{8}$&0.00782\\%
\hlin
\end{tabular
\end{table}
The overall performance of the algorithms aggregated over the instance sets is given in Table~\ref{tbl:maxsat}.
We find that the \opofeagz\ performs much better than the \opoeagz.
While the former can reliably solve instances of all dimensions, the latter already fails in almost half of the runs \mbox{for~$\scale=75$}.
The overall \ert\ of the \opofeagz\ for \mbox{dimension~$\scale=250$} is only about 7\% of the \ert\ that the \opoeagz\ needs over all instances of \mbox{$\scale=50$}.
\begin{figure}[tb
\centerin
\includegraphics[width=0.9999\linewidth]{figure_10
\caption{The ERT-ECDF curves for the SATLib instances: the fraction of instances of a given dimension~\scale\ solved over their empirically determined expected runtime.
\label{fig:maxsat_ert_ecdf
\end{figure}
We now plot the Empirical Cumulative Distribution Function (ECDF~\cite{HAFR2012RPBBOBES}) over the estimated \ert\ in Figure~\ref{fig:maxsat_ert_ecdf}.
Normally, the ECDF shows the fraction of \emph{runs} that could solve their corresponding problem instance over time.
However, we want to illustrate which algorithm can solve which fraction of the \emph{instances} until which (empirically determined expected) time.
For a given dimension~\scale, we therefore compute the \ert\ for each of the corresponding instances based on the conducted runs.
It seems that SATLib contains some instances that the \opoeagz\ can solve quickly, but on many instances it is slow or fails often.
The \ert\ of the instance of dimension~$\scale=250$ hardest for the \opofeagz\ is only~38\% higher than the \ert\ of the \mbox{scale-20} instance hardest for the \opoeagz.
Due to the drop in success rate, the behavior of the \opoeagz\ is already very unstable at dimensions~50 and~75.
This does not happen for the \opofeagz\ at any of the tested dimensions.
In summary, the \opofea\ very significantly outperforms the \opoea\ on a practically-relevant task, which goes beyond the scope of toy problems
\subsection{Job Shop Scheduling Problems (\jssp)
\label{sec:jssp
With the \maxsat, we have investigated an important \npHard\ problem.
While exhibiting interesting features, the \opofeagz\ algorithm we applied is not competitive to the state-of-the-art even two decades ago~\cite{HS2000LSAFSAEE}.
We now want to investigate if FFA can be helpful when the base algorithm is already performing well and we will do so on an entirely different domain.
In a job shop scheduling problem (\jssp)~\cite{LLRKS1993SASAAC} without preemption, there are \jsspMachines~machines and \jsspJobs~jobs.
Each job must be processed by all machines in a job-specific sequence and has, for each machine, a specific processing time.
The goal is to find assignments of jobs to machines that result in an overall shortest makespan, i.e., the schedule which can complete all the jobs the fastest.
The \jssp\ is \npHard~\cite{LLRKS1993SASAAC}.
The objective values are positive integers since the processing times are integers.
We obtain an upper bound~\upperBound\ needed for FFA as the sum of all processing times of all sub-jobs.
We use the \jssp\ as educational example in~\cite{W2019AITOA}, where we discuss all of the following components (except FFA) in great detail.
A solution for the \jssp\ is encoded as permutation with repetition, as integer strings where each of the \jsspJobs~job IDs occurs exactly \jsspMachines~times~\cite{GTK1994SJSSPBGA}.
Such an integer string~\solspel\ is processed from front to end.
When encountering job~$i$, we know to which machine~$j$ it needs to go next based on the job-specific machine sequence and on how often we already saw~$i$ in~\solspel\ before.
We can start it on~$j$ at a time which is the maximum of \emph{1)}~when the previous sub-job assigned to~$j$ will finish and \emph{2)}~when the previous sub-job of~$i$ completes on its corresponding machine.
We develop a memetic algorithm~\cite{NCM2012HOMA} which retains the~\mbox{$\paramMu=16$} best candidate solutions in its population and generates \mbox{$\paramLambda=16$}~new strings in each step via recombination.
Recombination proceeds similar to the solution decoding, but reads unprocessed sub-jobs iteratively from two parent strings (between which it randomly switches) and writes them to an offspring, while marking each processed sub-job in both parents as processed~\cite{W2019AITOA}.
The \paramLambda~new strings each are refined with ten steps of a local search which, in each step, scans the single-swap neighborhood of the string in random order until it finds a makespan-improving move and applies it (or stops if none can be found).
The two algorithms we investigate differ \emph{only} in what they do once this step is completed:
The first, \jsspMA, now applies selection based on the objective values.
In the \jsspFFAMA, on the other hand, the FFA table \ffaH\ is updated by increasing the frequency counter of the corresponding objective value of each of the \mbox{$\paramMu+\paramLambda$} solutions in the joint parent-offspring population.
Selection chooses the $\paramMu$~solutions with the lowest frequency fitness value.
\jsspFFAMA\ still uses the objective function~\ofel\ in the local search and to break ties in FFA. It is therefore not invariant under bijections of~\ofel.
Our goal this time is to achieve the best possible result within five minutes of runtime on an Intel Core \mbox{i7~8700} CPU with \mbox{3.2~GHz} and \mbox{16~GiB} RAM under Java OpenJDK~13 on~\mbox{Ubuntu~19.04}.
This is very different from the previous goals of solving the problems to optimality.
We conduct~$\runs=11$ runs each on 82~well-known \jssp\ instances from the \mbox{OR-Library}~\cite{B1990OLDTPBEM,B1990OLJSS}, namely the sets
\instance{abz*},
\instance{ft*},
\instance{la*},
\instance{orb*},
\instance{swv*}, and
\instance{yn*}, where all processing times are integers
\begin{table
\caption
Best, median, mean, and standard deviation of results from 11~runs of \jsspMA\ and \jsspFFAMA; \tblbettertxt: better value, \tblsolvedtxt: best known solutions (BKS) reached
\label{tbl:jssp:results
\centerin
\clipbox{0pt}
\resizebox{\linewidth}{!}
\begin{tabular}{@{\hspace{0.17em}}l@{~}r@{~}l|rrrr|rrrr@{\hspace{0.17em}}
\hlin
&&&\multicolumn{4}{c|}{\jsspMA}&\multicolumn{4}{c}{\jsspFFAMA}\\%
\multicolumn{1}{l}{inst}&\multicolumn{2}{l|}{BKS+ref}&\multicolumn{1}{c}{best}&\multicolumn{1}{c}{med}&\multicolumn{1}{c}{mean}&\multicolumn{1}{c|}{sd}&\multicolumn{1}{c}{best}&\multicolumn{1}{c}{med}&\multicolumn{1}{c}{mean}&\multicolumn{1}{c}{sd}\\%
\hlin
\instance{abz5}&1234&\cite{AC1991ACSOTJSSP}&1239&1244&1244.1&4.9&\tblsolved{\tblbetter{1234}}&\tblsolved{\tblbetter{1234}}&\tblsolved{\tblbetter{1236.2}}&\tblsolved{2.5}\\%
\instance{abz6}&943&\cite{AC1991ACSOTJSSP}&947&948&949.0&3.9&\tblsolved{\tblbetter{943}}&\tblsolved{\tblbetter{943}}&\tblsolved{\tblbetter{943.4}}&\tblsolved{1.2}\\%
\instance{abz7}&656&\cite{H2002PJSSP}&\tblbetter{679}&693&694.7&10.8&685&\tblbetter{691}&\tblbetter{693.1}&5.5\\%
\instance{abz8}&665&\cite{H2002PJSSP}&698&709&713.2&12.4&\tblbetter{688}&\tblbetter{706}&\tblbetter{705.9}&8.2\\%
\instance{abz9}&678&\cite{ZSRQ2008SNROTSAATTJSSP}&724&737&736.2&6.7&\tblbetter{714}&\tblbetter{727}&\tblbetter{725.6}&6.7\\%
\hlin
\instance{ft06}&55&\cite{CP1989AAFSTJSP}&\tblsolvedReliably{55}&\tblsolvedReliably{55}&\tblsolvedReliably{55.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{55}&\tblsolvedReliably{55}&\tblsolvedReliably{55.0}&\tblsolvedReliably{0.0}\\%
\instance{ft10}&930&\cite{CP1989AAFSTJSP}&937&949&948.2&7.6&\tblsolved{\tblbetter{930}}&\tblsolved{\tblbetter{930}}&\tblsolved{\tblbetter{933.9}}&\tblsolved{6.1}\\%
\instance{ft20}&1165&\cite{CP1989AAFSTJSP}&1173&1178&1177.9&1.8&\tblsolved{\tblbetter{1165}}&\tblsolved{1178}&\tblsolved{\tblbetter{1176.4}}&\tblsolved{4.1}\\%
\hlin
\instance{la01}&666&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{666}&\tblsolvedReliably{666}&\tblsolvedReliably{666.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{666}&\tblsolvedReliably{666}&\tblsolvedReliably{666.0}&\tblsolvedReliably{0.0}\\%
\instance{la02}&655&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{655}&\tblsolvedReliably{655}&\tblsolvedReliably{655.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{655}&\tblsolvedReliably{655}&\tblsolvedReliably{655.0}&\tblsolvedReliably{0.0}\\%
\instance{la03}&597&\cite{AC1991ACSOTJSSP}&\tblsolved{597}&\tblsolved{597}&\tblsolved{599.2}&\tblsolved{4.9}&\tblsolvedReliably{597}&\tblsolvedReliably{597}&\tblsolvedReliably{\tblbetter{597.0}}&\tblsolvedReliably{0.0}\\%
\instance{la04}&590&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{590}&\tblsolvedReliably{590}&\tblsolvedReliably{590.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{590}&\tblsolvedReliably{590}&\tblsolvedReliably{590.0}&\tblsolvedReliably{0.0}\\%
\instance{la05}&593&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{593}&\tblsolvedReliably{593}&\tblsolvedReliably{593.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{593}&\tblsolvedReliably{593}&\tblsolvedReliably{593.0}&\tblsolvedReliably{0.0}\\%
\instance{la06}&926&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{926}&\tblsolvedReliably{926}&\tblsolvedReliably{926.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{926}&\tblsolvedReliably{926}&\tblsolvedReliably{926.0}&\tblsolvedReliably{0.0}\\%
\instance{la07}&890&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{890}&\tblsolvedReliably{890}&\tblsolvedReliably{890.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{890}&\tblsolvedReliably{890}&\tblsolvedReliably{890.0}&\tblsolvedReliably{0.0}\\%
\instance{la08}&863&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{863}&\tblsolvedReliably{863}&\tblsolvedReliably{863.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{863}&\tblsolvedReliably{863}&\tblsolvedReliably{863.0}&\tblsolvedReliably{0.0}\\%
\instance{la09}&951&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{951}&\tblsolvedReliably{951}&\tblsolvedReliably{951.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{951}&\tblsolvedReliably{951}&\tblsolvedReliably{951.0}&\tblsolvedReliably{0.0}\\%
\instance{la10}&958&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{958}&\tblsolvedReliably{958}&\tblsolvedReliably{958.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{958}&\tblsolvedReliably{958}&\tblsolvedReliably{958.0}&\tblsolvedReliably{0.0}\\%
\hlin
\instance{la11}&1222&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1222}&\tblsolvedReliably{1222}&\tblsolvedReliably{1222.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1222}&\tblsolvedReliably{1222}&\tblsolvedReliably{1222.0}&\tblsolvedReliably{0.0}\\%
\instance{la12}&1039&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1039}&\tblsolvedReliably{1039}&\tblsolvedReliably{1039.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1039}&\tblsolvedReliably{1039}&\tblsolvedReliably{1039.0}&\tblsolvedReliably{0.0}\\%
\instance{la13}&1150&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1150}&\tblsolvedReliably{1150}&\tblsolvedReliably{1150.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1150}&\tblsolvedReliably{1150}&\tblsolvedReliably{1150.0}&\tblsolvedReliably{0.0}\\%
\instance{la14}&1292&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1292}&\tblsolvedReliably{1292}&\tblsolvedReliably{1292.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1292}&\tblsolvedReliably{1292}&\tblsolvedReliably{1292.0}&\tblsolvedReliably{0.0}\\%
\instance{la15}&1207&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1207}&\tblsolvedReliably{1207}&\tblsolvedReliably{1207.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1207}&\tblsolvedReliably{1207}&\tblsolvedReliably{1207.0}&\tblsolvedReliably{0.0}\\%
\instance{la16}&945&\cite{AC1991ACSOTJSSP}&946&946&959.4&17.5&\tblsolved{\tblbetter{945}}&\tblsolved{946}&\tblsolved{\tblbetter{945.9}}&\tblsolved{0.3}\\%
\instance{la17}&784&\cite{AC1991ACSOTJSSP}&\tblsolved{784}&\tblsolved{787}&\tblsolved{787.5}&\tblsolved{3.0}&\tblsolvedReliably{784}&\tblsolvedReliably{\tblbetter{784}}&\tblsolvedReliably{\tblbetter{784.0}}&\tblsolvedReliably{0.0}\\%
\instance{la18}&848&\cite{AC1991ACSOTJSSP}&\tblsolved{848}&\tblsolved{848}&\tblsolved{850.0}&\tblsolved{4.5}&\tblsolvedReliably{848}&\tblsolvedReliably{848}&\tblsolvedReliably{\tblbetter{848.0}}&\tblsolvedReliably{0.0}\\%
\instance{la19}&842&\cite{AC1991ACSOTJSSP}&\tblsolved{842}&\tblsolved{852}&\tblsolved{853.4}&\tblsolved{8.7}&\tblsolved{842}&\tblsolved{\tblbetter{842}}&\tblsolved{\tblbetter{842.9}}&\tblsolved{3.0}\\%
\instance{la20}&902&\cite{AC1991ACSOTJSSP}&907&907&907.8&1.8&\tblsolved{\tblbetter{902}}&\tblsolved{907}&\tblsolved{\tblbetter{906.5}}&\tblsolved{1.5}\\%
\hlin
\instance{la21}&1046&\cite{YN1997GAFJSSP}&1056&1068&1067.8&7.5&\tblbetter{1047}&\tblbetter{1053}&\tblbetter{1052.5}&2.7\\%
\instance{la22}&927&\cite{AC1991ACSOTJSSP}&935&941&941.3&8.0&\tblbetter{930}&\tblbetter{935}&\tblbetter{934.5}&1.9\\%
\instance{la23}&1032&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1032}&\tblsolvedReliably{1032}&\tblsolvedReliably{1032.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1032}&\tblsolvedReliably{1032}&\tblsolvedReliably{1032.0}&\tblsolvedReliably{0.0}\\%
\instance{la24}&935&\cite{AC1991ACSOTJSSP}&941&964&960.2&11.7&941&\tblbetter{946}&\tblbetter{945.6}&3.2\\%
\instance{la25}&977&\cite{AC1991ACSOTJSSP}&986&998&1002.3&14.8&\tblbetter{984}&\tblbetter{986}&\tblbetter{986.7}&3.1\\%
\instance{la26}&1218&\cite{AC1991ACSOTJSSP}&\tblsolved{1218}&\tblsolved{1218}&\tblsolved{1222.4}&\tblsolved{11.6}&\tblsolvedReliably{1218}&\tblsolvedReliably{1218}&\tblsolvedReliably{\tblbetter{1218.0}}&\tblsolvedReliably{0.0}\\%
\instance{la27}&1235&\cite{YN1997GAFJSSP}&1252&1269&1268.3&8.3&\tblbetter{1248}&\tblbetter{1264}&\tblbetter{1264.0}&6.7\\%
\instance{la28}&1216&\cite{AC1991ACSOTJSSP}&1225&1232&1238.7&14.5&\tblsolved{\tblbetter{1216}}&\tblsolved{\tblbetter{1225}}&\tblsolved{\tblbetter{1228.1}}&\tblsolved{8.8}\\%
\instance{la29}&1152&\cite{H2002PJSSP}&1199&1222&1224.2&18.9&\tblbetter{1191}&\tblbetter{1219}&\tblbetter{1212.6}&13.0\\%
\instance{la30}&1355&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1355}&\tblsolvedReliably{1355}&\tblsolvedReliably{1355.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1355}&\tblsolvedReliably{1355}&\tblsolvedReliably{1355.0}&\tblsolvedReliably{0.0}\\%
\hlin
\instance{la31}&1784&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1784}&\tblsolvedReliably{1784}&\tblsolvedReliably{1784.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1784}&\tblsolvedReliably{1784}&\tblsolvedReliably{1784.0}&\tblsolvedReliably{0.0}\\%
\instance{la32}&1850&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1850}&\tblsolvedReliably{1850}&\tblsolvedReliably{1850.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1850}&\tblsolvedReliably{1850}&\tblsolvedReliably{1850.0}&\tblsolvedReliably{0.0}\\%
\instance{la33}&1719&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1719}&\tblsolvedReliably{1719}&\tblsolvedReliably{1719.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1719}&\tblsolvedReliably{1719}&\tblsolvedReliably{1719.0}&\tblsolvedReliably{0.0}\\%
\instance{la34}&1721&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1721}&\tblsolvedReliably{1721}&\tblsolvedReliably{1721.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1721}&\tblsolvedReliably{1721}&\tblsolvedReliably{1721.0}&\tblsolvedReliably{0.0}\\%
\instance{la35}&1888&\cite{AC1991ACSOTJSSP}&\tblsolvedReliably{1888}&\tblsolvedReliably{1888}&\tblsolvedReliably{1888.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{1888}&\tblsolvedReliably{1888}&\tblsolvedReliably{1888.0}&\tblsolvedReliably{0.0}\\%
\instance{la36}&1268&\cite{AC1991ACSOTJSSP}&1295&1301&1307.7&15.7&\tblbetter{1281}&\tblbetter{1297}&\tblbetter{1297.5}&9.8\\%
\instance{la37}&1397&\cite{AC1991ACSOTJSSP}&1446&1467&1462.3&13.5&\tblbetter{1432}&\tblbetter{1446}&\tblbetter{1442.5}&6.3\\%
\instance{la38}&1196&\cite{NS1996AFTSAFTJSP}&1251&1263&1262.9&13.5&\tblbetter{1239}&\tblbetter{1240}&\tblbetter{1244.6}&8.7\\%
\instance{la39}&1233&\cite{AC1991ACSOTJSSP}&1251&1256&1267.0&20.5&\tblbetter{1248}&\tblbetter{1250}&\tblbetter{1250.0}&1.2\\%
\instance{la40}&1222&\cite{AC1991ACSOTJSSP}&1241&1264&1262.1&14.0&\tblbetter{1233}&\tblbetter{1247}&\tblbetter{1247.3}&7.7\\%
\hlin
\instance{orb01}&1059&\cite{AC1991ACSOTJSSP}&\tblsolved{\tblbetter{1059}}&\tblsolved{1104}&\tblsolved{1099.2}&\tblsolved{17.5}&1071&\tblbetter{1071}&\tblbetter{1075.4}&5.6\\%
\instance{orb02}&888&\cite{AC1991ACSOTJSSP}&890&919&909.0&14.3&\tblbetter{889}&\tblbetter{889}&\tblbetter{889.0}&0.0\\%
\instance{orb03}&1005&\cite{AC1991ACSOTJSSP}&1026&1058&1060.5&27.8&\tblsolved{\tblbetter{1005}}&\tblsolved{\tblbetter{1022}}&\tblsolved{\tblbetter{1019.5}}&\tblsolved{7.4}\\%
\instance{orb04}&1005&\cite{AC1991ACSOTJSSP}&\tblsolved{1005}&\tblsolved{1028}&\tblsolved{1024.3}&\tblsolved{14.0}&\tblsolved{1005}&\tblsolved{\tblbetter{1011}}&\tblsolved{\tblbetter{1010.0}}&\tblsolved{2.2}\\%
\instance{orb05}&887&\cite{AC1991ACSOTJSSP}&890&905&913.8&18.1&\tblsolved{\tblbetter{887}}&\tblsolved{\tblbetter{890}}&\tblsolved{\tblbetter{890.2}}&\tblsolved{2.4}\\%
\instance{orb06}&1010&\cite{BV1998GLSWSBFJSS}&1013&1031&1031.0&8.6&1013&\tblbetter{1013}&\tblbetter{1017.8}&6.0\\%
\instance{orb07}&397&\cite{H2002PJSSP}&\tblsolved{397}&\tblsolved{397}&\tblsolved{401.7}&\tblsolved{7.4}&\tblsolved{397}&\tblsolved{397}&\tblsolved{\tblbetter{397.1}}&\tblsolved{0.3}\\%
\instance{orb08}&899&\cite{BV1998GLSWSBFJSS}&914&944&941.3&16.5&\tblsolved{\tblbetter{899}}&\tblsolved{\tblbetter{899}}&\tblsolved{\tblbetter{902.0}}&\tblsolved{5.4}\\%
\instance{orb09}&934&\cite{BV1998GLSWSBFJSS}&939&945&947.9&7.2&\tblsolved{\tblbetter{934}}&\tblsolved{\tblbetter{939}}&\tblsolved{\tblbetter{937.9}}&\tblsolved{3.4}\\%
\instance{orb10}&944&\cite{BV1998GLSWSBFJSS}&\tblsolved{944}&\tblsolved{946}&\tblsolved{957.3}&\tblsolved{15.9}&\tblsolvedReliably{944}&\tblsolvedReliably{\tblbetter{944}}&\tblsolvedReliably{\tblbetter{944.0}}&\tblsolvedReliably{0.0}\\%
\hlin
\instance{swv01}&1407&\cite{H2002PJSSP}&1476&1517&1524.1&35.1&\tblbetter{1447}&\tblbetter{1474}&\tblbetter{1483.3}&20.9\\%
\instance{swv02}&1475&\cite{H2002PJSSP}&1550&1585&1582.8&20.9&\tblbetter{1525}&\tblbetter{1548}&\tblbetter{1549.1}&15.5\\%
\instance{swv03}&1398&\cite{H2002PJSSP}&1500&1530&1533.3&28.8&\tblbetter{1489}&\tblbetter{1512}&\tblbetter{1513.1}&15.1\\%
\instance{swv04}&1464&\vphantom{\cite{VLS2015FDSFCBS}}\cite{VLS2015FDSFCBSDER}&1580&1615&1624.1&30.6&\tblbetter{1564}&\tblbetter{1578}&\tblbetter{1586.5}&22.3\\%
\instance{swv05}&1424&\cite{H2002PJSSP}&\tblbetter{1517}&1575&1585.3&49.8&1523&\tblbetter{1554}&\tblbetter{1556.0}&25.8\\%
\instance{swv06}&1671&\cite{VLS2015FDSFCBSDER}&1859&1903&1909.2&44.0&\tblbetter{1824}&\tblbetter{1864}&\tblbetter{1862.7}&27.8\\%
\instance{swv07}&1594&\cite{GR2014AEAGMWABRKGAFJSS}&1766&1814&1816.1&26.4&\tblbetter{1705}&\tblbetter{1755}&\tblbetter{1753.3}&24.5\\%
\instance{swv08}&1752&\cite{VLS2015FDSFCBSDER}&1940&1992&1989.9&38.0&\tblbetter{1930}&\tblbetter{1946}&\tblbetter{1946.4}&13.9\\%
\instance{swv09}&1655&\cite{VLS2015FDSFCBSDER}&1820&1871&1877.9&51.8&\tblbetter{1805}&\tblbetter{1844}&\tblbetter{1844.8}&19.3\\%
\instance{swv10}&1743&\cite{GR2014AEAGMWABRKGAFJSS}&1909&1956&1974.5&44.7&\tblbetter{1904}&\tblbetter{1936}&\tblbetter{1931.4}&15.4\\%
\hlin
\instance{swv11}&2983&\cite{NS2005AATSAFTJSP}&\tblbetter{3439}&\tblbetter{3506}&\tblbetter{3506.1}&52.6&3495&3574&3583.5&62.6\\%
\instance{swv12}&2977&\cite{PLC2015ATSPRATSTJSSP}&\tblbetter{3478}&\tblbetter{3594}&\tblbetter{3594.4}&56.3&3511&3605&3622.1&66.3\\%
\instance{swv13}&3104&\cite{H2002PJSSP}&\tblbetter{3543}&3679&3686.5&81.1&3578&\tblbetter{3664}&\tblbetter{3677.2}&78.6\\%
\instance{swv14}&2968&\cite{H2002PJSSP}&\tblbetter{3358}&3455&\tblbetter{3444.4}&60.4&3369&\tblbetter{3454}&3452.9&69.1\\%
\instance{swv15}&2885&\cite{PLC2015ATSPRATSTJSSP}&3361&\tblbetter{3500}&\tblbetter{3490.6}&89.5&\tblbetter{3356}&3529&3524.1&98.5\\%
\instance{swv16}&2924&\cite{H2002PJSSP}&\tblsolvedReliably{2924}&\tblsolvedReliably{2924}&\tblsolvedReliably{2924.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{2924}&\tblsolvedReliably{2924}&\tblsolvedReliably{2924.0}&\tblsolvedReliably{0.0}\\%
\instance{swv17}&2794&\cite{H2002PJSSP}&\tblsolvedReliably{2794}&\tblsolvedReliably{2794}&\tblsolvedReliably{2794.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{2794}&\tblsolvedReliably{2794}&\tblsolvedReliably{2794.0}&\tblsolvedReliably{0.0}\\%
\instance{swv18}&2852&\cite{H2002PJSSP}&\tblsolvedReliably{2852}&\tblsolvedReliably{2852}&\tblsolvedReliably{2852.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{2852}&\tblsolvedReliably{2852}&\tblsolvedReliably{2852.0}&\tblsolvedReliably{0.0}\\%
\instance{swv19}&2843&\cite{H2002PJSSP}&\tblsolvedReliably{2843}&\tblsolvedReliably{2843}&\tblsolvedReliably{2843.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{2843}&\tblsolvedReliably{2843}&\tblsolvedReliably{2843.0}&\tblsolvedReliably{0.0}\\%
\instance{swv20}&2823&\cite{H2002PJSSP}&\tblsolvedReliably{2823}&\tblsolvedReliably{2823}&\tblsolvedReliably{2823.0}&\tblsolvedReliably{0.0}&\tblsolvedReliably{2823}&\tblsolvedReliably{2823}&\tblsolvedReliably{2823.0}&\tblsolvedReliably{0.0}\\%
\hlin
\instance{yn1}&884&\cite{ZSRQ2008SNROTSAATTJSSP}&921&934&936.2&11.7&\tblbetter{909}&\tblbetter{931}&\tblbetter{931.5}&10.7\\%
\instance{yn2}&904&\cite{GR2014AEAGMWABRKGAFJSS}&953&962&964.0&7.5&\tblbetter{937}&\tblbetter{954}&\tblbetter{952.4}&9.8\\%
\instance{yn3}&892&\cite{NS2005AATSAFTJSP}&929&951&951.9&16.1&\tblbetter{913}&\tblbetter{938}&\tblbetter{938.6}&12.0\\%
\instance{yn4}&968&\cite{H2002PJSSP}&\tblbetter{1022}&1046&1048.9&20.0&1024&\tblbetter{1041}&\tblbetter{1041.9}&9.0\\%
\hlin
\end{tabular
}
\end{table}
From Table~\ref{tbl:jssp:results}, we can find that \jsspMA\ can already discover the best known solution (BKS) on 36~instances at least once and always on~27.
\jsspFFAMA, however, can do so 46 and 32~times, respectively.
\jsspFFAMA\ has better best, median, and mean results 37, 45, and 51~times, respectively, while the same is true for the \jsspMA\ only 8, 3, and 4~times.
In other words, on 93\% of the instances that are not already always solved to optimality by \jsspMA, \jsspFFAMA\ has a better mean result.
The mean (median) result of \jsspFFAMA\ is better than the best result of \jsspMA\ in 17 (13)~instances, while the opposite is never true.
\jsspFFAMA\ has a smaller standard deviation in 48~cases, \jsspMA\ only in~4.
We apply the two-sided Welch's \mbox{t-test} to the results on the 49~instances where the algorithms have different mean results with non-zero standard deviations.
\jsspFFAMA\ performs significantly better than \jsspMA\ on 25 of them at a significance level of~$\alpha=0.01$.
Such high significance is a very strong result at only~11 runs.
The opposite is true only on \instance{swv11}, even if we set~$\alpha=0.1$.
Neither \jsspMA\ nor \jsspFFAMA\ can outperform the state-of-the-art on the JSSP, but they are not very far off, at least if we consider result quality only:
The basic \jsspMA\ obtains better best (mean) results than the GWO proposed in~\cite{JZ2018AOGWOFSCPJSAFJSSC} (2018) in 16 (21) of the 39~instances for which results are provided, while the opposite is never (once) true.
While the HFSAQ~\cite{AKZ2016FSAHWQFSJSSP} (2016) has better mean (best) result quality in 23 (12) of 48~comparable cases, \jsspFFAMA\ scores even in the rest, while having better mean solution quality on 4~instances.
It also 13~times achieves better mean makespans (28 times worse ones) on the 63~common instances compared to the HIMGA~\cite{K2015ANHIMGAFJSSP} (2015), while its best solution is never better.
On instances \instance{swv16} to \instance{swv20}, which can be solved to the BKS by both \jsspMA\ and \jsspFFAMA, budgets of more than 16~min were used in~\cite{H2002PJSSP} to find said BKSes.\footnote
Of course on older hardware, but our Java implementation is not optimized
}
Still, the \jsspFFAMA\ is worse than, e.g., the algorithms in~\cite{CPL2016AHEATSTJSSP}~(2016) and~\cite{VLS2015FDSFCBSDER}~(2015) on every common instance where it does not find the BKS.
In summary, we find that even in a more complicated setup based on an algorithm that already does not perform badly in comparison to recent publications, FFA can lead to a significant performance improvement.
This does not mean that other diversity improvement strategies, e.g., those from Section~\ref{sec:relatedWork}, could not have improved the performance of the \jsspMA\ as well or even better.
Still, together with the results on the \maxsat~problems in Section~\ref{sec:maxsat} and those in our earlier papers on FFA on domains such as Genetic Programming~\cite{WWTY2014EEIAWGP,WWTWDY2014FFA}, this adds evidence to the idea that FFA may not just be of purely academic interest
\section{Conclusions
\label{sec:conclusions
In this paper, we plugged Frequency Fitness Assignment (FFA) into the most basic evolutionary algorithm, the \opoea, and applied the resulting \opofea\ to several problems defined over bit strings of dimension~\scale.
On the one hand, we found that the \opofea\ is slower than the \opoea\ on the \onemax, \leadingones, and \plateau\ functions.
In our experiments with these problems, it seems to increase the mean runtime needed to discover the global optimum by a factor no worse than linear in the number of objective values or in~\scale.
On the other hand, FFA can seemingly decrease the mean runtime on the \trap, \jump, and \twomax\ problems from exponential to the scale \mbox{of~$\scale^2\ln{\scale}$}.
On the \maxsat\ problem and on the \wmodel\ benchmark, the \opofea\ very significantly outperforms the \opoea.
These results are surprising when considering the nature of FFA -- being invariant under bijective transformations of the objective function, i.e., possessing the strongest invariance property known to us.
FFA never compares objective values directly.
An algorithm applying only FFA would exhibit the same performance on the objective function~\ofel\ as on~\mbox{$g\circ\ofel$}, where~$g$ could be an arbitrary encryption method (which we simulate by setting~$g$ to the \mbox{Md5~checksum} routine in Section~\ref{sec:md5}).
This realization is baffling.
Two central assumptions of black-box optimization are that following a trail of improving objective values tends to be a good idea and that ``nice'' optimization problems should exhibit causality, i.e., small changes to a solution should lead to small changes in its objective value.
Under FFA, neither assumption is used.
As a result, properties such as causality, ruggedness, or deceptiveness of a fitness landscape may have little impact on the algorithm performance.
Interestingly, this does seemingly not necessarily come at a high cost in terms of runtime.
Instead of the cost of the invariance, the limitation of the method seems to be that it requires objective functions that can be discretized and do not take on too many different values.
We finally showed that FFA can be combined with ``normal'' optimization and plugged into more complex algorithms.
We inserted it into the selection step of a memetic algorithm whose local search proceeds without FFA and works directly on the objective values.
Here, an FFA variant purely works as population diversity enhancement mechanism and can improve the result quality that the algorithm produces on the \jssp\ within a budget of five minutes.
Notably, while this algorithm does not belong to the state-of-the-art on the JSSP, it seems to be relatively close to it.
Together with our results on the \maxsat\ problem, this means that FFA might even be helpful in cases bordering to practical relevance.
There are several interesting avenues for future work.
First, we want to also plug FFA into other EAs, such as those in~\cite{CPD2017TAMPARAOEA}.
Second, a theoretical analysis of the properties of FFA could be both interesting and challenging, also from the perspective of black-box complexity.
Third, using FFA is the only approach known to us that can solve encrypted optimization problems.
This could open new types of applications in operations research, machine learning, and artificial intelligence.
Fourth, on problems FFA leads to a slowdown. The question whether this slowdown is proportional to the problem dimension or to the number of possible different objective values deserves an investigation.
Finally, it may be possible to adapt ideas from the research on multi-armed bandits to implement an FFA-like approach:
We envisage an Upper Confidence Bound~\cite{ACBF2002FTAOTMBP}-like algorithm, where one solution per encountered objective value is preserved and treated as bandit arm.
Playing an arm would mean to use the solution as input to mutation and the reward could be~1 if the offspring has a new objective value
\section*{Acknowledgments
{\small
We acknowledge support by
the National Natural Science Foundation of China under Grant~61673359,
the Hefei Specially Recruited Foreign Expert Program,
the Youth Project of the Anhui Natural Science Foundation~1908085QF285,
the University Natural Science Research Project of Anhui~KJ2019A0835,
the Key Research Plan of Anhui~201904d07020002,
the Major Project of the Research and Development Fund of Hefei University~19ZR05ZDA,
the Talent Research Fund of Hefei University \mbox{18-19RC26},
and
the \mbox{DIM-RSFI N\textsuperscript{o}~2019-10 C19/1526} project.
The first author especially wants to thank Dr.\ Carola Doerr for the chance to participate in the Dagstuhl Seminar~19431 with the topic \emph{Theory of Randomized Optimization Heuristics}, which gave us the inspiration to investigate FFA on some basic scenarios where it may even be theoretically tractable.
\ifCLASSOPTIONcaptionsof
\newpag
\f
\bibliographystyle{IEEEtran
\small
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,516 |
Sex workers left out of conversation on sex trafficking prevention
· September 10, 2014
Screenshot of message which appears at Myredbook.com
Federal authorities are taking steps to regulate the sex trade under the banner of a nationwide effort to prevent human trafficking. From freezing sex workers' bank accounts to shutting down adult advertising sites, a crackdown on the sex trade is in full swing. A bill making its way through Congress would stricter guidelines for placing "adult service" advertisements. In the process, however, legislators have renewed a debate about the distinction between human trafficking and sex work, and the rights of those involved in both. Taylor Sanders reports.
A bill before the US Senate known as the Stop Advertising Victims of Exploitation – or SAVE – Act, aims to change requirements for placing advertisements for "adult services" online.
Proponents of the bill say it will protect victims of sex trafficking. In June, the FBI shut down a major adult advertising site called Myredbook, which also served as an online resource for sex workers.
"It's important to make the distinction between survival sex and trafficking because we have people on the streets engaged in survival sex," says Meg Muñoz, the director of Abeni – a sex worker advocacy group. She adds the bill – and the Myredbook shutdown – have unintended consequences for many workers just trying to make ends meet. "There is a great deal of argument that X, Y, and Z on the sex work spectrum is illegal so we shouldn't even be having these discussions. But the reality is sex work, historically, has always been around. And historically so has trafficking."
Survival sex is sex work by choice, but under circumstances of economic duress — while sex trafficking, in the U.S., is legally defined as "commercial sex acts induced by force, fraud, coercion or commercial sex acts in which the individual induced to perform commercial sex has not attained 18 years of age." Muñoz says that shutting down adult advertising or putting in new regulations could force some sex workers back into more dangerous situations based on street solicitation.
Online advertising has allowed sex workers to become more independent by permitting them to build their own brand, pre-screen to protect themselves from dangerous clients, and create forums to talk to other sex workers in the area. MyRedBook for example, served as a nation-wide discussion board and social media platform, also providing health information and legal advice. According to Muñoz, this independence debunked the notion that most sex workers are victims of trafficking.
An undercover officer from the Bellingham Police Department, who spoke on condition of anonymity, says police in his unit operate under the assumption that many sex workers are victims of circumstance in need of help: "We go into this investigation with the idea of not making an arrest: trying to get them help, trying to get them out of that circle. Because we understand if we just arrest them it's a misdemeanor crime, their going to be out in 24 hours and they are going to be right back out doing the exact same thing the next day. So our ultimate goal is to try to get them help. And try to get them out of that lifestyle. We understand that obviously a misdemeanor offense for prostitution is going to hurt them in pretty much everything they are trying to do to get back on their feet so that's why we've adopted that mindset."
Some note that not all sex workers are victims – either of trafficking or circumstance. Among them, a sex worker in Seattle, WA who uses the name Savannah Sly. She says there is "a vast spectrum of people who are in the sex industry for a variety of reasons; there are people like me who really love our occupations and then there are a ton of people who really just see it as a job."
Sly says the lack of clear labor protections for those who are in the sex trade voluntarily makes the work more dangerous for all involved. She gives an example of how she once had a pimp and when the group of women she was with tried to fire him, he tried to blackmail them. "If he had not had the leverage of stigma and criminalization he wouldn't not have had that power over us," she said. "We were about to stand up to that anyways, but because the work we were doing was illegal, he had the power to call the police on our group when he saw us advertising elsewhere."
Meg Muñoz says laws that stereotype sex workers as either victims or people in need of salvation miss a key point and leave those most affected out of the conversation. "The sex work community and the sex workers in it not only care incredibly about trafficking — because many of them have experienced that and still have gone back to sex work — but they are our greatest asset when it comes to fighting trafficking. And it is an asset that is untapped, that is being ignored, that is being demonized and is being shut down."
SAVE Act supporters say that cracking down on sex work will protect victims of sex trafficking. But sex worker advocates say measures to protect victims of trafficking could also include more rights for workers in the industry voluntarily.
The SAVE Act – written without any input from sex workers or their advocates – passed the House of Representatives in May and is awaiting consideration by the Senate.
Tags: human traffickingSAVE Actsex workTaylor Sanders
HRW report: I Wanted to Lie Down and Die: Trafficking and Torture of Eritreans in Sudan and Egypt
Canada signs into law a new sex work law to replace one struck down as unconstitutional
Senate lawmakers accuse State Dept. of putting trade policy over human rights
Next story Florida corals face threat of potentially devastating bleaching event
Previous story Kurdish families claim kids kidnapped by PKK | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8 |
Q: Gettext doesnt work what I'm doing wrong? I'm making small plugin to share the post in my word-press
Everything work but I can't get a string to be translatable throw wpml.
Bear with me it's not a question about wpml,
My problem is that I can't figure how to add Get text to my code in the right way.
Here is my code:
function add_social_share_icons($content)
{
$html = "<div class='social-share-wrapper'><div class='share-on'>Share on: </div>";
global $post;
$url = get_permalink($post->ID);
$url = esc_url($url);
if(get_option("social-share-facebook") == 1)
{
$html = $html . "<div class='facebook'><a target='_blank' href='http://www.facebook.com/sharer.php?u=" . $url . "'>Facebook</a></div>";
}
I want to translate this "Share on:" the wpml use the Gettext to detect the string.
Like this <th><?php echo __('Due date:', 'themedomain'); ?></th>
I tried something like this:
$html = "<div class='social-share-wrapper'><div class='share-on'><?php _e('Share on:'); ?></div>";
But it doesn't work!
Can someone help me please?
A: You can't embed a function like that into a string, only simple things can be done that way. Instead, you want to concatenate which can be done in a couple of ways, but this is probably the easiest. Also, instead of _e() which echos the text, I'm using __() to just translate
$html = "<div class='social-share-wrapper'><div class='share-on'>" . __('Share on:') . "</div>";
The second parameter of __() is the text domain, same as _e()
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,552 |
/24 Mar 2020
/Misc
How the Healthcare Technology World is Responding to the Coronavirus Pandemic
It's looking like 2020 will be a deeply challenging year for global health. As the COVID-19 virus (Coronavirus) crisis unfolds, an enormous strain is being placed on the healthcare system – confirmed cases of the virus across America are steeply rising, quarantine measures are complicating the ability of patients to access medical care, and the financial fallout of the crisis has presented never-before-seen supply chain and healthcare delivery challenges.
On top of the more front-and-center challenge of providing immediate medical care, the crisis is putting unprecedented strain on healthcare information and technological infrastructure. It's unclear whether the U.S. healthcare system is prepared to carry the weight, but it's already apparent that significant changes in the ways things like patient data, patient and provider communication, and supply chain processes are managed will be required.
Faced with a truly global pandemic, the role of healthcare IT and innovation has never been more critical. Here are a few ways that the healthcare IT world is – or could soon be – responding to the COVID-19 crisis.
Patient-centered experiences could ease the healthcare burden of COVID-19
Many patients have reported less than optimal experiences trying to get tested for the Coronavirus. Without focusing on the patient perspective, developers could miss opportunities to diminish the effects of the virus. The information patients need may go unseen and the data hospital networks are looking for may go unreported.
For example, one Chicago man reported that when he showed up to the clinic with Coronavirus-like symptoms, he didn't meet the criteria for the test. The hospital deemed self-quarantine unnecessary – only to call him back three days later to get tested for the virus. Not only did that patient lose three days of possible diagnosis, but he could have exposed others to the virus during that time. In this case, sacrificing the patient-centered focus is a mistake that could have very real, and deeply damaging consequences.
Modern hospital networks typically try to cater to as many patients in need as possible, balance logistics and still focus on a patient-centric approach. Giving patients and end-users what they want is an essential step to creating a pleasant user experience in healthcare. Especially during high-stress times like these, healthcare software should cater to an authentic patient journey. That means potentially redesigning basic patient communication systems and experiences, or looking to expand the capabilities of existing digital tools to accommodate the radically different patient expectation during a global pandemic.
For companies creating tools specifically to combat the COVID-19 crisis, every detail of their software matters. The founders of Abi Global Health have responded to the virus by hyper-focusing on the UX effects of their platform. The AI-powered chatbot app helps patients during the "first-mile" of their healthcare journey. Abi effectively reduces physician time per case and in-person visitation; two critical advantages amid the pandemic. CEO and co-founder Kim-Fredrik Schneider explains the significance of UX optimization right now:
"Downloading an app, finding a private space with good lighting and a stable internet connection, getting dressed and ready for a video call when you are feeling unwell; these barriers can be a terrible user experience (UX) for people used to instant service via their mobile devices. A bad UX can lead to inefficient healthcare utilization, which takes a huge financial toll on society; the cost of people going to the doctor when they shouldn't is $50bn, while people not going when they should costs at least $500bn." (Forbes).
Most individuals (and most organizations) have not experienced a public health emergency like this one, so it's difficult to predict how different Americans are reacting. Evidation Health, a healthcare technology company, is using a patient engagement app to survey over 100,000 U.S. citizens about their thoughts on the pandemic. They found that less than 23% of Americans believe the country was prepared for the virus. They also found that between those with health insurance and those without, "[t]he latter group more often said that they would not seek care if they developed coronavirus symptoms (8% versus 21%), and those who would more often said that they would most likely go to an emergency room (19% versus 26%)" (MobihealthNews). These analysis points can help providers track how behaviors may change over time and develop solutions beforehand.
Could supply chain analytics help U.S. healthcare networks be better equipped to handle COVID-19?
Hospital beds per 1,000 people, via the OECD:
South Korea: 12.3
Germany: 8.0
France: 6.0
China: 4.3
Italy: 3.2
United States: 2.8https://t.co/yKavMnNMpm https://t.co/RkhwSTDhb0 pic.twitter.com/kJ57iavncJ
— Christopher Ingraham (@_cingraham) March 11, 2020
As the number of confirmed cases of COVID-19 grows, so does the concern for supplies – and the supply chain providing them. USA Today's conservative guess is "there could be six seriously ill patients for every existing hospital bed" They suggest the gap exists partly because the outbreak hit during the peak of flu season. Hospital beds aren't the only dilemma. Because the virus causes respiratory issues, masks and ventilators are also in high demand.
Some options other countries have turned to were rapidly building more hospitals or, more gravely, limiting care based on age range. One tool American healthcare professionals might want to turn to is data. Effective data analysis can help hospitals with inventory management and cutting costs in healthcare supply chains. one way to fight the pandemic might be to deploy predictive analytics and data-sourcing tools to try and find opportunities for cost-reduction and greater efficiency.
In other words, if we optimize our hospital supply chains now, we may be better equipped to handle large volumes of patients during the peak of COVID-19 diagnoses later.
Establishing supply chain transparency paves the way for responding to unplanned circumstances, and a better understanding of potential demand could allow for medical supplies to be on-site before the need hits. Achieving transparency may include creating a risk index for each supply type based on critical supplier information. Additionally, capacity and distribution optimization could ensure that critical tests aren't needlessly delayed and furthering the effects of an already devastating virus like COVID-19.
In Philadelphia, Penn Medicine's predictive healthcare team has pioneered a tool that helps hospitals prepare for the incoming influx of Coronavirus cases. COVID-19 Hospital Impact Model for Epidemics (CHIME) utilizes SIR modeling, an epidemiological model, to predict the expected count of the COVID-19 diagnoses in a closed population. The software will help hospitals project the number of hospital admissions they might obtain each day and, therefore, prepare their supply chains accordingly. The open-source app was created so that other hospitals can also make use of it by adjusting the perimeters to their specific populations. Penn Medicine hopes CHIME will reduce some of the uncertainty that comes with the Coronavirus, and get hospitals on track to manage the inevitable surge of patients.
Curious what it looks like to bring a digital therapeutics app from concept to market?
Patient data sharing and portability is critical for tracking the spread and impact of COVID-19
The number of confirmed coronavirus cases in the U.S. went from 15 cases on February 15th, to more than 46,000 reported cases across the states as of March 24. But even that number isn't entirely accurate – keeping track of confirmed coronavirus cases in real-time has been a struggle. Infectious disease expert Jeffery Shaman told Buzzfeed News that he wouldn't be surprised if over 50,000 Americans have been infected with the disease at this point. That leaves thousands of Americans being left unrecognized by the U.S. healthcare system. The lack of testing is a huge contributor to that data gap. Is it possible that software could help solve the issue?
Lab researchers of Augusta University in Georgia seem to think so. Dr. Arni S.R. Srinivasa Rao, Director of the Laboratory for Theory and Mathematical Monitoring at Augusta is working with a team of developers to create a free app that helps identify those who are at high risk for contracting COVID-19.
Drs. Arni Rao (foreground) and Jose Vazquez detailing their work on a whiteboard (Health-in-europe.com).
Using a list of symptoms, data about cases in that geographical region, and data from other respondents of the survey, the AI-powered app will use an algorithm to send users a risk assessment. The app will then direct high-risk users to the nearest accredited testing facility. An app like this could help hospitals skip at least some of the screening processes they currently conduct on-site and get likely victims to the hospitals and away from the uninfected.
Healthcare professionals also believe data-mining could more effectively identify hospital outbreaks. According to Lee H. Harrison MD of the University of Pittsburgh, "[r]eal-time data mining could have averted 66% of possible preventable infections if an intervention was implemented within 14 days of identifying the transmission route" (Helio, Feb 20).
Telehealth will be critical for delivering healthcare during the COVID-19 pandemic
During a pandemic, visiting the hospital can carry its own risks. Americans will benefit from sending as few people with unrelated issues to the hospitals as possible. As telemedicine gains popularity in the healthcare industry, it's getting a special amount of attention at the moment.
Doctors are suggesting that telemedicine is critical to flattening the curve. The movement is even getting support from the government. On Tuesday, the Trump administration announced that telemedicine services will be temporarily waived by Medicare, making it easier for healthcare providers to host Medicare-reimbursable office visits through telehealth services. The HHS Office of Inspector General (OIG) is also allowing healthcare providers to "reduce or waive cost-sharing for telehealth visits paid by federal healthcare programs". Private insurers are also looking to cover telehealth practices by waiving costs for virtual visits.
Telemedicine has quickly transitioned from a healthcare "nice-to-have" to a need. G. Cameron Deemer, president of DrFirst, highlights three main roles for telehealth amid the pandemic:
Keeping more hospital beds available by providing in-home care
Protect those with previously existing conditions from a potentially fatal coronavirus diagnosis
Allowing quarantined healthcare providers to see patients through remote channels as they are highly exposed and in limited supply
Telemedicine does not come without its complications, however. While telehealth practices are being gradually adopted nation-wide, most hospitals are not yet equipped to operate virtually. With insurance complications, a lack of hardware, and gaps in knowledge on both the provider and patient-front, telehealth won't always be a simple system to establish.
Nevertheless, organizations are paving the way for virtual visits in their hospitals. Seat Pleasant, a small "smart city" community in Maryland, is responding to the virus by partnering with companies to embark on a $200 million telehealth project. Telemedicine giant Amwell has also reported a 158% increase in app usage nationwide. Telehealth may have a chance to shine amidst an otherwise gloomy time for healthcare.
Deploying emerging technologies in the face of a global pandemic
The immediate and almost unprecedented challenge of the COVID-19 pandemic is also bringing some still-emerging technologies to bear, as organizations seek new and better tools to manage the increased resource demands.
The US Department of Energy is putting its Summit computer, the fastest computer on Earth, up to the task of finding a cure for the virus. Summit computer operators are hopeful that the computer will run simulation results to learn more about which compounds may produce a useful vaccine.
Silicon Valley companies are also looking to employ artificial intelligence to curtail the damage of the virus. Tech giants like Facebook, Google, LinkedIn, Microsoft, Reddit, Twitter, and YouTube have all agreed to help stop the spread of misinformation about the COVID-19 online. Although the technology isn't error-proof yet, the organizations are relying heavily on AI to identify offensive or incorrect information being posted on their platforms. YouTube, for example, is using an automated moderator system to find content that violates their policies, but admits the system still needs work.
It's still too early to understand what the full extent and impact of the COVID-19 pandemic will be, but it's already abundantly clear that U.S. healthcare systems are facing an almost unprecedented challenge. That means that healthcare software developers, healthcare professionals, and the government all face a responsibility to rise to the occasion. By pulling together, leveraging new and emerging technologies, and tackling this unprecedented challenge, the healthcare industry will hopefully find ways to learn from this crisis and ultimately be better prepared for future public health emergencies.
Chief Technology Officer at Monetate Shares Their Product Strategy – Product Hacker – Episode 9
Press Release: Joint Podcast with Point-of-Care Partners
Siara Singleton
TELEHEALTH COULD HELP TACKLE RURAL AMERICA'S BEHAVIORAL HEALTH CRISIS
HOW WILL THE 21ST CENTURY CURES ACT IMPACT THE HEALTHCARE IT SPACE?
INTEROPERABILITY: IMPROVING PATIENT EXPERIENCE
More Healthcare Blogs–>
Learn how we put decision-making power back in patients' hands by building the world's first digital advanced directive.
On Product Hacker, we chat with digital health leaders about engineering healthcare and building patient experiences.
Business Community Design Development Events Fintech Healthcare Jobs Misc People Podcast Product Product Hacker | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,882 |
Mark E. Damon, CPA, MACC
Robert W. Hague, CPA
Jeny L. Grupe, CPA
Susan St. Range, CPA
Rory B. Tosh, CPA
Megan Kurz, CPA
Annie Driver, CPA
Aria Bettinger, CPA
Alyssa Borg, CPA
Daniel A Kosmatka, CPA, PFS
Stewart C. Parmele, CPA
KDP Wealth Management LLC
Record Retention Guidelines
Fed & State Tax Links
Finance Dictionary
ICFiles Login
ICFiles Signup
Check us out on BrokerCheck
Congress at Work
General Business News
Tax and Financial News
What's New in Technology
KDP LLP
CPA Medford OR
Protecting Water, Music and Intellectual Property
America's Water Infrastructure Act of 2018 (S. 3021) – This bill authorizes water pollution control activities as well as conservation and development of water and related resources to facilitate improvements to the rivers and harbors of the United States. The bill was sponsored by Sen. Amy Klobuchar (D-MN) on June 7 and signed into law by the President on Oct. 23.
Orrin G. Hatch-Bob Goodlatte Music Modernization Act (H.R. 1551) – This bill was introduced by Rep. Tom Rice (R-SC) on March 15, 2017 and signed into law by the President on Oct. 11. It represents one of the most significant legislative reforms to U.S. copyright law over the past 20 years by updating licensing agreements to include digital reproduction and distribution. The bill establishes a blanket statutory licensing system on a song-by-song basis administered by a nonprofit mechanical licensing collective. This entity is charged with collecting and distributing royalties, identifying songs and their owners for payment, and maintaining a comprehensive, publicly accessible database for music ownership information for both pre- and post-1972 sound recordings.
Patient Right to Know Drug Prices Act (S. 2554) – Introduced by Sen. Susan Collins (R-ME), this bill is designed to end the practice of insurers and pharmacy benefit managers instituting "gag order" agreements with pharmacists. It has been a practice to prevent pharmacists from informing customers whether it would cost less for a drug if they use their health insurance or pay fully out of pocket. Many times, a consumer would actually save money by paying out of pocket . This legislation formally bans the practice of pharmacy gag clauses at the federal level. The bill was first introduced on March 14 and signed into law on Oct. 10.
Know the Lowest Price Act of 2018 (S. 2553) – Sponsored by Sen. Debbie Stabenow (D-MI), this bill is similar to S. 2554 in that it also prohibits Medicare Part D plans from restricting pharmacies from informing individuals regarding the prices for certain drugs and biologicals. The bill was introduced on March 14 and was signed into law by the President on Oct. 10.
Justice Served Act of 2018 (H.R. 4854) – This legislation increases the capacity of prosecutors to address the backlog of violent crime cases involving suspects identified through DNA evidence. The Department of Justice must allocate a specified percentage of grant funds for such purpose, including at least 5 percent for grants to prosecute cold cases involving violent crime. The bill was introduced by Rep. John Carter (R-TX) on Jan. 19 and enacted on Oct. 9.
Small Business Innovation Protection Act of 2017 (S. 791) – This bill directs the Small Business Administration and the United States Patent and Trademark Office to leverage existing outreach programs to educate more small businesses on intellectual property and domestic/international patent protections. The bill was sponsored by Sen. Gary Peters (D-MI) on March 30, 2017 and signed into law by the President on Oct. 9.
Protecting Religiously Affiliated Institutions Act of 2018 (S. 994) – This bill was introduced by Sen. Orrin Hatch (R-UT) on May 1, 2017 and signed into law by the President on Sept. 28. The bill amends the federal criminal code to broaden the scope of defacing, damaging or destroying religious real property to include threatening such acts. The legislation also establishes criminal penalties of a fine, a prison term of up to five years or both.
Nuclear Energy Innovation Capabilities Act of 2017 (S. 97) – Sponsored by Sen. Michael Crapo (S-ID), this bill enables civilian research and development of advanced nuclear energy technologies by private and public institutions. In addition, the DOE is directed to study the need for a new test reactor to support research and development of advanced reactor systems and, if so, construct such a facility by 2025. The bill was introduced on Jan. 11, 2017 and was signed into law by the President on Sept. 28.
November 1, 2018 Service2Client Congress at Work, Single Blog
Paying the Price for Vice: The Evolving Landscape of Excise Taxes in America
COVID-19 Vaccination Considerations for Employers
Deciding if a Roth IRA Conversion is For You
Prosecution for Use of Performance Enhancement Drugs, Modernizing Government Technology, and Enhancements for Veterans and Their Caregivers
2021 Social Security Tax and Benefit Increases Announced
Dennis William Donnelly - Obituray Oregon Corporate Tax Sick Leave Law Vacation Home
KDP Certified Public Accountants, LLP Remote Technical Support (RTS) is a safe, secure, and easy to use solution for obtaining "on-site" technical support without having to wait for a support specialist to travel to your site. Using a high speed internet connection, RTS is just like having one of our support specialists sitting right at your desk working to resolve your computer problems. Login
How to use RTS
KDP Certified Public Accountants, LLP Client File Transfer (CFT) provides you with the easiest and securest method of transferring your data to us 24 hours a day/7 days a week. Don't waste time burning discs for a single file or struggling to email large files that take forever to load. CFT is a safe, secure, and simple way to transfer to and from KDP. This feature is for CPA materials only. Please contact us for Wealth Management secure transfers.
If you don't already have a client login, click here to get started.
How to use CFT
KDP Certified Public Accountants, LLP
841 O'Hare Parkway, Ste. 200
Email: info@kdpllp.com
© 2019 KDP Certified Public Accountants, LLP All Rights Reserved.
Websites Design by Service2Client.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,156 |
Offensive players help prep for RGIII
The reason Robert Griffin III won the Heisman Trophy and was the second overall pick in this year's draft is his rare combination of talents in both passing and running.
So rare that the Giants have two men on the job simulating the rookie in preparation for the Washington Redskins this week.
Meet David Carr, the arm.
Meet Jerrel Jernigan, the legs.
"How do you do it?" Coughlin said Wednesday, repeating a question about replicating Griffin's speed in practice. "You do the best you can. We have two or three guys that can hopefully do it. David Carr is slick enough to do it, and quick enough to do it, and give us that look. Jerrel Jernigan has given us a few plays over there last week, and the week before."
This is nothing new for the Giants or teams around the league. Just last week, the Giants did the same for backup quarterback and wildcat specialist Colin Kaepernick of San Francisco. Two weeks before that, it was Michael Vick of the Eagles, and a week before that, it was Carolina's Cam Newton. And so on and so forth.
But each week presents a new challenge as the Giants prepare for a quarterback that has thrown for 1,343 yards this season and rushed for another 379, including a 76-yard touchdown last week in the Redskins' win over the Vikings.
"I feel like I'm a little bit faster than him," Carr joked after Wednesday's practice. "I know that he's young, he's got a long road ahead of him, but I've been running for a long time. No, it's fun though. I've said this before, it's fun to kind of simulate those guys. We did it last week with San Francisco, and it worked out pretty good. Defense had a great week. So hopefully we'll do the same thing."
While Carr himself has rushed 302 times for 1,331 yards in his 11-year career, he's only "RGMinus-V" according to his teammates.
"They've used a lot," Carr said of the nicknames thrown his way. "But we'll leave that one out there."
Meanwhile, Jernigan's speed is no joking matter. Neither is his quarterbacking.
The second-year pro was a dual-threat prospect out of Eufaula High School in Alabama before splitting time at wide receiver and under center during his record career at Troy.
"Anytime it was third-and-1, third-and-2, fourth-and-1, I was going in," said Jernigan about playing at Troy, where he saw action at quarterback, H-Back, split end, slot receiver, punt returner and kick returner. "I'm used to it."
Jernigan went on to say it felt good to be back running the option again, even if it was for scouting purposes.
"A lot of them were talking to me today out there, and me, Osi [Umenyiora] and [Justin] Tuck were having fun joking around with it," Jernigan said. "They were like I wasn't that fast to get around the edge on them. I told them I would show them or whatever, but I'm doing pretty good at it."
As for the passing, there's a reason Jernigan was drafted in the third round by the Giants in 2011 as a wide receiver.
"It's no surprise when JJ comes in we're not going to throw the ball," Carr said. "He might have attempted a pass, but I'm not really sure how it worked out." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,738 |
\section{Introduction}
In this work, we study function spaces related to Dirichlet problems for a class
of integrodifferential
operators, which satisfy the maximum principle. We introduce a new function
space, which can be understood as a nonlocal trace space. Let us illustrate our
task with a very
simple problem. Let $\Omega = B_1 \subset \mathbb{R}^d$ be the unit ball and assume
$0 < s < 1$. We ask ourselves the question, for which
functions $g : \mathbb{R}^d \setminus \Omega \to \mathbb{R}$, there is a function $u:\mathbb{R}^d \to
\mathbb{R}$ satisfying
\begin{alignat}{2}
\lim\limits_{\varepsilon \to 0} \int\limits_{\mathbb{R}^d \setminus B_\varepsilon}
\frac{u(x\!+\!h) \!-\! u(x)}{|h|^{d+2s}}\,
\mathrm{d} h &= 0 &&\text{ for } x \in \Omega\,, \label{eq:Dprobeqn}\\
u(x) &= g(x) \quad &&\text{ for } x \in \mathbb{R}^d \setminus
\Omega\,. \label{eq:Dprobdata}
\end{alignat}
Note that \eqref{eq:Dprobeqn} is equivalent to $(-\Delta)^s u = 0$ in $\Omega$.
In order to discuss the possible choices of data $g$, we need to specify the
function space of possible solutions $u$. Moreover, we
have to explain in which sense the above equation is
to be understood. Since the validity of \eqref{eq:Dprobeqn} for some
$x \in
\Omega$ involves values of $u$ on $\mathbb{R}^d \setminus \Omega$, where $u=g$ is
imposed, there is a direct link between the function
space for solutions $u$ and the function space for the data $g$.
\medskip
The set-up of boundary value problems is well understood for differential
operators, i.e., in the limit case $s=1$. However, by considering our
results for $s \to 1^{-}$, we will obtain a new extension result for classical
Sobolev
spaces, cf. \autoref{cor:limit_case-example} and \autoref{cor:limit_case}.
\medskip
Let us explain how to define a variational solution $u$ satisfying
\eqref{eq:Dprobeqn}--\eqref{eq:Dprobdata}, cf. \cite{DRV14, FKV15}. Define two
vector spaces by
\begin{align*}
V^s(\Omega|\R^d) &= \big\{ v \in L^2_{\operatorname{loc}}(\mathbb{R}^d) |
\int\limits_\Omega \int\limits_{\mathbb{R}^d} \frac{\big(v(y) - v(x)
\big)^2}{|x-y|^{d+2s}} \d x \, \d y <
\infty \big\}, \\
H^s_\Omega(\mathbb{R}^d) &= V^s_0(\Omega|\mathbb{R}^d) = \{v \in V^s(\Omega|\mathbb{R}^d)| v= 0 \text{
on } \Omega^c \}\,.
\end{align*}
Let us collect a few basic observations on these spaces.
\begin{enumerate}
\item $V^s(\mathbb{R}^d|\mathbb{R}^d)$ and $V^s(\mathbb{R}^d|\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$ equal the Sobolev-Slobodeckij space
$\dot{H}^s(\mathbb{R}^d)$ and $H^s(\mathbb{R}^d)$, respectively.
\item $H^s_\Omega(\mathbb{R}^d)$ is a Banach space together with the norm
\begin{align*}
\|v\|_{H^s}^2 = \|v\|^2_{L^2} + (1\!-\!s) \int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d}
\frac{\big(v(y) - v(x)
\big)^2}{|x-y|^{d+2s}} \d y \d x
\end{align*}
\end{enumerate}
Let us define the notion of a variational solution.
\begin{defn}[cf. Definition 2.5 in \cite{FKV15}]\label{def:var-sol}
Let $\Omega \subset \mathbb{R}^d$ be open such that $\Omega$ and $\Omega^c$ both have
positive measure. Let $g \in V^s(\Omega|\R^d)$. Then $u \in V^s(\Omega|\R^d)$ is
called a
variational solution to \eqref{eq:Dprobeqn}--\eqref{eq:Dprobdata}, if $u-g \in
H^s_\Omega(\mathbb{R}^d)$ and for every $\varphi \in H^s_\Omega(\mathbb{R}^d)$
\begin{align}
\int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} \frac{\big(u(y) - u(x)
\big) \big(\varphi(y) - \varphi(x) \big) }{|x-y|^{d+2s}} \d y \d x = 0 \,.
\end{align}
\end{defn}
\begin{rem} (i) The above definition implies that solutions $u$ belong to
$L^2(\mathbb{R}^d,\mathrm{d}m)$ with $m(\mathrm{d}x) =
(1+|x|)^{-d-2s} \, \mathrm{d}x$. It would be possible to work under
the assumption $u \in L^1(\mathbb{R}^d, \mathrm{d}m)$, but the presentation would be
less transparent. (ii) In peridynamics, the definition of variational solutions
to nonlocal boundary value problems looks similar, cf. \cite{DuMe14}. However,
it is rather different because of the usage of more restrictive
function spaces. Regularity of $u$ respectively $g$ is required in regions,
which are away from that region, where the nonlocal equation is considered. The
above definition avoids such an assumption.
\end{rem}
With the above
definition at hand, we are now in the position to explain the main question
addressed in this
article. In order to apply \autoref{def:var-sol} one needs to prescribe the
data
function $g$ in the vector space $V^s(\Omega|\R^d)$, i.e. in particular one needs
to
prescribe all values of $g$ in $\mathbb{R}^d$. This leads to the following question:
\medskip
\begin{tabbing}
{\bf Question:} \= For which Banach space of functions $g: \Omega^c \to \mathbb{R}$ \\
\> (a) is there an extension operator $g \mapsto \ext(g) \in V^s(\Omega|\R^d)$, and \\
\> (b) is there a trace operator from $V^s(\Omega|\R^d)$ into this
space?
\end{tabbing}
\medskip
Extension and trace theorems are well known in the study of classical
local Dirichlet problems. Thus, for the case of Sobolev spaces of integer
order, these questions are classical and answers were given long time
ago , cf.
\cite{Slo58} for an early work and \cite{AdFo03} for a general exposition. For
a large class of domains $\Omega$, functions in $H^{1/2}(\partial \Omega)$
can be extended to elements of $H^1(\Omega)$ and these themselves have a trace
in $H^{1/2}(\partial \Omega)$. A side result of our research on nonlocal
quantities is that instead of $H^1(\Omega)$ one could also consider the
much larger space of all $L^2(\Omega)$-functions $v$ with
\begin{align}
\int_{\Omega} \int_{\Omega}
\frac{|v(x)-v(y)|^2}{(|x-y|+\delta_x+\delta_y)^{d+2}}\,dy\,dx < \infty \,,
\end{align}
where $\delta_z = \operatorname{dist}(z, \partial \Omega)$
for $z \in \mathbb{R}^d$.
\medskip
Trace and extension results have been established for various function spaces
including Sobolev spaces with fractional order of differentiability.
To our best knowledge, extensions from the complement of a domain to the whole
space have not been dealt with so far. One reason for this might be that
Dirichlet problems with prescribed data on the complement have not yet been
studied intensively.
\medskip
Let us formulate our main result, which answers the aforementioned question. We allow the domain $\Omega$ to have a rather rough
boundary, but we stress the fact that our results are new even for domains $\Omega$ with a smooth boundary. See \autoref{sec:prelims} for the definition of inner respectively exterior thickness of domains. Note that any bounded Lipschitz domain has these properties. The inner radius of an open set $D\subset \mathbb{R}^d$ is defined as
$\inr(D)=\frac12 \sup_{B\subset D} \diam(B)$, where the supremum is taken
over all
balls $B\subset D$.
\begin{thm}\label{thm:answer}
Assume $0 < s < 1$. Let $\Omega\subset \mathbb{R}^d$ be open, interior thick and
exterior thick such
that $\partial \Omega$ has Lebesgue measure zero, and $\inr(\Omega)<\infty$ or
$\inr(\Omega^c)=\infty$. Then the following is true:
\begin{enumerate}
\item[(a)]
If $f\in L^p_{loc}(\mathbb{R}^d)$, $1 \leq p < \infty$, satisfies
\begin{equation}\label{e:space}
\int_{\Omega} \int_{\mathbb{R}^d} \frac{|f(x)-f(y)|^p}{|x-y|^{d+sp}}\,dy\,dx <
\infty,
\end{equation}
then
\begin{equation}\label{e:spacecomp}
\int_{\Omega^c} \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\,dx < \infty \,.
\end{equation}
\item[(b)] There exists a linear extension operator $\ext$, which maps
$L^p_{loc}(\Omega^c)$, $1 \leq p < \infty$, to measurable functions defined on
$\mathbb{R}^d$ such that
\begin{align}
(1-s) \int_{\Omega} \int_{\mathbb{R}^d}
\frac{|\ext(f)(x)-\ext(f)(y)|^p}{|x-y|^{d+sp}}\,dy\,dx \asymp \int_{\Omega^c}
\int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\, dx \,,
\end{align}
with constants that depend only on $\inf s$, $p$, $d$ and $\Omega$.
\end{enumerate}
\end{thm}
\autoref{thm:answer} follows directly from \autoref{thm:norm} and
\autoref{thm:extension}.
\medskip
Considering the limit $s \to 1^{-}$, \autoref{thm:answer} implies a new
extension-type result for classical Sobolev spaces. We formulate this
observation in the special case $\Omega = B_1 \subset \mathbb{R}^d$ and refer to
\autoref{cor:limit_case} for the general case and to \autoref{rem:Du} for some
related result.
\begin{cor}\label{cor:limit_case-example}
Given $1 < p < \infty$, there is a constant $c = c(d,p) \geq 1$ such that
\[
\int \limits_{B_1} |\nabla \ext(f)|^p \leq c \int\limits_{B_2\setminus B_1}\!
\int\limits_{B_2\setminus B_1}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+p}}~dx\,dy
\]
for every $f \in L^p(B_2\setminus B_1)$ such that the right-hand side is finite.
\end{cor}
\autoref{cor:limit_case-example} is a special case of \autoref{cor:limit_case}.
\medskip
The article is organized a follows. In \autoref{sec:setup_results} we present
the setup of our work together with the main results, \autoref{thm:norm} and
\autoref{thm:extension}. \autoref{sec:prelims}
provides basic properties of the function spaces under consideration. In
\autoref{sec:proof1} we present the proof of \autoref{thm:norm}. The proof of
\autoref{thm:extension} is given in \autoref{sec:proof2}.
\section{Setup and detailed results}\label{sec:setup_results}
Throughout the whole paper we assume that
$\Omega\subset\mathbb{R}^d$ is an open set with the property that both, $\Omega$ and
$\Omega^c = \mathbb{R}^d \setminus \Omega$, have positive Lebesgue measure. For our
main result, we will assume some very mild additional assumption. We will use
the symbol $g \lesssim h$ to denote that the inequality $g \leq c
h$ holds with a positive constant $c$ that is independent of $g$ and
$h$. We adopt the convention that $0^a=\infty$ for $a<0$, in particular,
$\frac{1}{0}=\infty$. We assume $0<p<\infty$ and $0<s\leq 1$.
\medskip
In short, our main result answers the question from the previous section. It
roughly says that the vector space of all functions $g \in
L^2_{loc}(\Omega^c)$ with
\begin{align}
\int_{\Omega^c} \int_{\Omega^c}
\frac{|g(x)-g(y)|^2}{(|x-y|+\delta_x+\delta_y)^{d+2s}}\,dy\,dx < \infty
\end{align}
has the desired properties, see \autoref{thm:answer}. A special feature of our
result is that the limit
case $s=1$ can be included. Thus we obtain a new extension result for
$W^{1,2}(\Omega)$-functions, see below for details.
Let us now explain the set-up in detail. For $f\in L^p(\mathbb{R}^d)$, define
\begin{align}
|f|_{W^{s,p}(\Omega|\Omega^c)}^p &:= \int_\Omega \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{|x-y|^{d+sp}}\,dy\,dx, \label{eq:WspOm-seminorm}\\
\|f\|_{W^{s,p}(\Omega|\Omega^c)}^p &:= \|f\|_{L^p(\mathbb{R}^d)}^p + |f|_{W^{s,p}(\Omega|\Omega^c)}^p,
\label{eq:WspOm-norm}
\end{align}
and let $W^{s,p}(\Omega|\Omega^c) = \{ f\in L^p(\mathbb{R}^d): \|f\|_{W^{s,p}(\Omega|\Omega^c)} < \infty \}$. If $f
\in L^p(\mathbb{R}^d)$, then $f \in W^{s,p}(\Omega|\Omega^c)$, if $f$ satisfies some regularity
condition across the boundary $\partial \Omega$, whereas the behavior of $f$
far from $\partial \Omega$ is not considered.
\begin{example*}
Consider a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ and $f \in
L^p(\mathbb{R}^d)$, $1 \leq p < \infty$, given by
$f = \mathbbm{1}_\Omega$. Then the function $f$ belongs to
$W^{s,p}(\Omega|\Omega^c)$ if and only if $s < \frac{1}{p}$.
\end{example*}
Recall that the inner radius of an open set $D\subset \mathbb{R}^d$ is defined as
$\inr(D)=\frac12 \sup_{B\subset D} \diam(B)$, where the supremum is taken
over all
balls $B\subset D$. For $x\in \mathbb{R}^d$, set $\delta_x = \dist(x,\Omega)$.
For $0<\delta,\varepsilon \leq \infty$ set
\[
\Omega^{\rm int}_\delta = \{x\in\Omega : \dist(x,\Omega^c) \leq \delta\}, \qquad
\Omega^{\rm ext}_\varepsilon = \{x\in\Omega^c : \dist(x,\Omega) < \varepsilon \}.
\]
Note that $\Omega^{\rm int}_{\inr(\Omega)} = \Omega$.
For a function $g$
let
\[
|g|_{A,B}^{s,p} := \int_A \int_{B}
\frac{|g(x)-g(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\,dx.
\]
\medskip
The following result introduces a useful (semi)norm that is equivalent to
$\|f\|_{W^{s,p}(\Omega|\Omega^c)}$ respectively $|f|_{W^{s,p}(\Omega|\Omega^c)}$. For the definition of interior thick
domains we refer the reader to \autoref{subsec:whitney}, here let us only mention
that bounded Lipschitz domains are interior thick.
\begin{thm}\label{thm:norm}
Let $0<p<\infty$ and $0<s\leq 1$.
Suppose that $\Omega\subset \mathbb{R}^d$ is an open interior thick set.
Then there exists a constant $c=c(p,\Omega)$ not depending on $s$, such that
\begin{equation}\label{eq:comp-semi}
c^{-1} |f|_{W^{s,p}(\Omega|\Omega^c)}^p \leq
\int_{ \Omega \cup \Omega^{\rm ext}_{\inr(\Omega)} } \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\,dx
\leq \frac{c}{s} |f|_{W^{s,p}(\Omega|\Omega^c)}^p
\end{equation}
for every $f\in L^p(\mathbb{R}^d)$. The following norms
\[
\left( \| \cdot \|_{L^p(\Omega, (1+|x|)^{-d-sp}\,dx)}^p + |\cdot|_{W^{s,p}(\Omega|\Omega^c)}^p
\right)^{1/p},
\]
\[
\left( \| \cdot \|_{L^p(\mathbb{R}^d, (1+|x|)^{-d-sp}\,dx)}^p + |\cdot|_{W^{s,p}(\Omega|\Omega^c)}^p
\right)^{1/p} \,,
\]
and
\[
\left( \|f\|_{L^p(\mathbb{R}^d, (1+|x|)^{-d-sp}\,dx)}^p +
\int_{ \mathbb{R}^d } \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\,dx \right)^{1/p}
\]
are comparable with constants depending only on $p$, $\Omega$ and the lower
bound for $s$.
\end{thm}
\begin{rem}
Note that, for the case $s \to 1^{-}$, the different $s$-dependence on the two
sides in \eqref{eq:comp-semi} is not important.
\end{rem}
\begin{example} Define $f: \mathbb{R}^d \to \mathbb{R}$ by $f(x) =
\sqrt{|x|-1}$ for $1 < |x| < 2$ and $f(x) = 0$ elsewhere. Assume $\Omega = B_1
\subset \mathbb{R}^d$ as in \autoref{cor:limit_case-example}. Then both expressions,
\begin{align*}
\int_{ \Omega \cup \Omega^{\rm ext}_{\inr(\Omega)} } \int_{\Omega^c}
\frac{|f(x)-f(y)|^2}{(|x-y|+\delta_x+\delta_y)^{d+2s}}\,dy\,dx
\quad \text{ and } \quad |f|_{W^{s,2} (\Omega|\Omega^c)}^2
\end{align*}
diverge for $s \to 1^{-}$. As observed in \cite[Sec. 2.2.4]{PV-thesis}, the
expression $(1-s) |f|_{W^{s,2}(\Omega|\Omega^c)}$ remains bounded.
\end{example}
\medskip
The following theorem contains our main result.
\begin{thm}\label{thm:extension}
Let $\Omega\subset \mathbb{R}^d$ be an open set which is exterior thick and such that
$\partial \Omega$ has Lebesgue measure zero,
and $\inr(\Omega)<\infty$ or $\inr(\Omega^c)=\infty$. Then there exists
a~linear operator $\ext$ which maps $L^1_{loc}(\Omega^c)$ to
the space of measurable functions on $\mathbb{R}^d$ with the following properties.
\begin{enumerate}
\item[(a)]
For all $f\in L^1_{loc}(\Omega^c)$, $\ext(f)|_{\Omega^c} = f$ and
$\ext(f)|_\Omega \in C^\infty(\Omega)$.
Furthermore, if $z_0\in \partial \Omega$ and the limit
$ g=\lim_{\Omega^c\ni x\to z_0} f(x)$
exists, then also the limit $\lim_{\Omega\ni x\to z_0} \ext(f)(x)$ exists and
equals $g$.
\item[(b)]
Let $1\leq p<\infty$.
There exists a~constant $c=c(\Omega,p)$ such that the following inequalities
hold
for all $f\in L^1_{loc}(\Omega^c)$ and $0<\delta\leq \varepsilon \leq \infty$
\begin{align}
|\ext(f)|_{\Omega^{\rm int}_\delta, \Omega^{\rm ext}_\varepsilon}^{s,p}
&\leq \frac{c}{s} |f|_{\Omega^{\rm ext}_\delta,\Omega^{\rm ext}_\varepsilon}^{s,p}, \quad 0<s\leq 1, \label{e:int-ext}\\
|\ext(f)|_{\Omega^{\rm int}_\delta, \Omega^{\rm int}_\delta}^{s,p}
&\leq \frac{c}{s(1-s)} |f|_{\Omega^{\rm ext}_\delta, \Omega^{\rm ext}_\delta}^{s,p}, \quad 0<s<1. \label{e:int-int}
\end{align}
In particular,
\begin{align}
|\ext(f)|_{\Omega, \Omega^c}^{s,p}
&\leq \frac{c}{s} |f|_{\Omega^c,\Omega^c}^{s,p}, \quad 0<s\leq 1, \label{e:simpl1}\\
|\ext(f)|_{\mathbb{R}^d, \mathbb{R}^d}^{s,p}
&\leq \frac{c}{s(1-s)} |f|_{\Omega^c, \Omega^c}^{s,p}, \quad 0<s<1. \label{e:simpl2}
\end{align}
\item[(c)]
Let $1\leq p < \infty, \beta\in \mathbb{R}$ or $p = \infty, \beta =0$. There
exists a~constant
$c=c(\Omega,\beta, p)$ such that the following inequality holds
for all $f\in L^1_{loc}(\Omega^c)$
\[
\| \ext(f)\|_{L^p(\Omega, (1+|x|)^\beta dx)} \leq c \| f \|_{L^p(\Omega^{\rm ext}_{\inr(\Omega)},
(1+|x|)^\beta dx)}.
\]
\end{enumerate}
\end{thm}
From \autoref{thm:norm} and \autoref{thm:extension}, the answer
to the question posed earlier immediately follows, cf. \autoref{thm:answer}.
\begin{cor}\label{cor:limit_case}
Let $\Omega$ be a bounded Lipschitz-domain and $1<p<\infty$. Then there
exists a~constant $c=c(\Omega,p)$ such that \begin{equation}\label{e:extW1p}
|\ext(f)|_{W^{1,p}(\Omega)} \leq c
|f|_{\Omega^{\rm ext}_{\inr(\Omega)},\Omega^{\rm ext}_{\inr(\Omega)}}^{1,p}, \quad f\in
L^1_{loc}(\Omega^c),
\end{equation}
where we take $|\ext(f)|_{W^{1,p}(\Omega)} = \|\nabla \ext(f)\|_{L^p(\Omega)}$,
if $\ext(f)\in W^{1,p}(\Omega)$, and $|\ext(f)|_{W^{1,p}(\Omega)}=\infty$
otherwise.
\end{cor}
\begin{proof}
We put $\delta=\varepsilon=\inr(\Omega)$ in \eqref{e:int-int},
multiply its both sides by $(1-s)$ and take $s\to
1^{-}$. If $|f|_{\Omega^{\rm ext}_{\inr(\Omega)}, {\Omega^{\rm ext}_{\inr(\Omega)}}}^{s,p}<\infty$ for
some~$s$,
then $f\in L^p(\Omega)$ and inequality \eqref{e:extW1p} follows from
\cite[Theorem 2]{BBM}. In the other case inequality \eqref{e:extW1p} is trivial.
\end{proof}
\begin{rem}\label{rem:Du}
In the case $p=2$, a result related to \autoref{cor:limit_case} has
recently
been established in \cite{DuTi16}. For the trace map $T$, the authors prove an
estimate of the form
\begin{align}
\|Tu\|_{H^{1/2}(\partial \Omega)} \leq C \|u\|_{\mathcal{S}(\Omega)} \,,
\end{align}
where
\begin{align*}
\|u\|^2_{\mathcal{S}(\Omega)} = \|u\|^2_{L^2(\Omega)} + \int\limits_{\Omega}
\int\limits_{\Omega \cap B(x,\delta(x))} \frac{\big(u(y) -
u(x)\big)^2}{\delta(x)^{d+2}} \mathbbm{1}_{B_1}(|y-x|) dy \, dx \,,
\end{align*}
and $\delta$ denotes the distance function with respect to $\partial \Omega$.
The authors of \cite{DuTi16} are interested in models from
peridynamics. It is interesting that
our approach to nonlocal function spaces, in the limit case $s \to 1^{-}$, leads
to
a similar nonlocal trace theorem as their approach. Note that \cite{DuTi16}
does not contain extension results like \autoref{thm:extension}.
\end{rem}
\section{Preliminary results}\label{sec:prelims}
In this section, we prove basic properties of the function spaces $W^{s,p}(\Omega|\Omega^c)$ and
collect several result on inner thick respectively exterior thick domains.
\subsection{Basic properties \texorpdfstring{of $W^{s,p}(\Omega|\Omega^c)$}{}}
Recall the definitions from \eqref{eq:WspOm-seminorm} and \eqref{eq:WspOm-norm}.
\begin{prop}
Let $\Omega\subset \mathbb{R}^d$ be an open set, $0<s<1$ and $p\geq 1$.
Then the space $W^{s,p}(\Omega|\Omega^c)$ equipped with the norm $\|\cdot \|_{W^{s,p}(\Omega|\Omega^c)}$ is
a~Banach space.
\end{prop}
\begin{proof}
The proof is straightforward. Let $(f_n)$ be a~Cauchy sequence in $(W^{s,p}(\Omega|\Omega^c),
\|\cdot \|_{W^{s,p}(\Omega|\Omega^c)} )$.
Then $(f_n)$ is a~Cauchy sequence in $L^p(\mathbb{R}^d)$, hence there exists a
function $f \in
L^p(\mathbb{R}^d)$ such that $f_n\to f$ in $L^p(\mathbb{R}^d)$.
Let $f_{k_n}$ be a~subsequence convergent a.e. to $f$.
By the Fatou lemma
\begin{align*}
|f_{k_n}-f|_{W^{s,p}(\Omega|\Omega^c)}^p &=
\int_\Omega \int_{\Omega^c} \liminf_{l \to\infty}
\frac{|(f_{k_n}-f_l)(x)-(f_{k_n}-f_l)(y)|^p}{|x-y|^{d+sp}}\,dy\,dx \\
&\leq \liminf_{l \to\infty} \int_\Omega \int_{\Omega^c}
\frac{|(f_{k_n}-f_l)(x)-(f_{k_n}-f_l)(y)|^p}{|x-y|^{d+sp}}\,dy\,dx \to 0
\text{ as } n\to \infty\,.
\end{align*}
From the above calculation and triangle inequality we deduce that $f\in
W^{s,p}(\Omega|\Omega^c)$.
Since $(f_n)$ is a~Cauchy sequence in $(W^{s,p}(\Omega|\Omega^c), \|\cdot
\|_{W^{s,p}(\Omega|\Omega^c)} )$ and its subsequence converges to $f$, the whole
sequence converges to $f$.
\end{proof}
\begin{rem}
For $p\in (0,1)$ the space $W^{s,p}(\Omega|\Omega^c)$ equipped with a~metric
$\rho(f,g):=\|f-g\|_{W^{s,p}(\Omega|\Omega^c)}^p$ is complete. The proof is
basically the same as above.
\end{rem}
\begin{prop}
If a~measurable function $f:\mathbb{R}^d\to\mathbb{R}$ satisfies $|f|_{W^{s,p}(\Omega|\Omega^c)} < \infty$, then
$f\in L^p(\mathbb{R}^d, (1+|x|)^{-d-sp}\,dx)$.
Furthermore, the norms
\[
\left( \| \cdot \|_{L^p(\Omega, (1+|x|)^{-d-sp}\,dx)}^p + |\cdot|_{W^{s,p}(\Omega|\Omega^c)}^p
\right)^{1/p},
\]
\[
\left( \| \cdot \|_{L^p(\mathbb{R}^d, (1+|x|)^{-d-sp}\,dx)}^p + |\cdot|_{W^{s,p}(\Omega|\Omega^c)}^p
\right)^{1/p}
\]
are comparable.
\end{prop}
\begin{proof}
Let $R>1$ be large enough so that $B(0,R)$ intersects both $\Omega$ and
$\interior \Omega^c$. For a~given $f$ as in the proposition, let $n \in {\mathbb{N}}$ be
such
that for $E_n=\{x\in \mathbb{R}^d: |f(x)|\leq n\}$ the intersections $F_n = E_n \cap
\Omega\cap B(0,R)$ and $G_n = E_n \cap \Omega^c\cap B(0,R)$ are of positive
Lebesgue measure. Note that
\[
|w-z| \leq R+|z| \leq R(1+|z|), \qquad \text{for $w\in B(0,R)$ and $z\in
\mathbb{R}^d$.}
\]
Therefore
\begin{align*}
2|f|_{W^{s,p}(\Omega|\Omega^c)}
&\geq
\int_{F_n} \int_{\Omega^c\setminus E_{2n}}
\frac{|f(x)-f(y)|^p}{|x-y|^{d+sp}}\,dy\,dx
+ \int_{\Omega \setminus E_{2n}} \int_{G_n}
\frac{|f(x)-f(y)|^p}{|x-y|^{d+sp}}\,dy\,dx \\
&\geq \frac{2^{-p}}{R^{d+sp}} \left(
\int_{F_n} \int_{\Omega^c\setminus E_{2n}} \frac{|f(y)|^p}{(1+|y|)^{d+sp}}
\,dy
\,dx + \int_{\Omega \setminus E_{2n}} \int_{G_n}
\frac{|f(x)|^p}{(1+|x|)^{d+sp}} \,dy\,dx \right) \\
&\geq
\frac{2^{-p}(|{F_n}|\wedge |{G_n}|)}{R^{d+sp}}
\int_{\mathbb{R}^d\setminus E_{2n}} \frac{|f(x)|^p}{(1+|x|)^{d+sp}} \,dx.
\end{align*}
Choose $n \in {\mathbb{N}}$ sufficiently large so that $|{F_n}|\wedge |{G_n}|$ is
positive. Since obviously $\int_{E_{2n}} |f(x)|^p (1+|x|)^{-d-sp} \,dx <
\infty$, we
conclude that $f\in L^p(\mathbb{R}^d, (1+|x|)^{-d-sp}\,dx)$.
Comparability of the first two norms follows from the following inequalities
\begin{align*}
\| f \|_{L^p(\Omega, (1+|x|)^{-d-sp}\,dx)}^p + |f|_{W^{s,p}(\Omega|\Omega^c)}^p
&
\gtrsim
\int_{\Omega\cap B(0,R)} \left( |f(x)|^p + \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{(1+|y|)^{d+sp}} \,dy \right) \,dx \\
&\gtrsim
\int_{\Omega\cap B(0,R)} \int_{\Omega^c}
\frac{|f(x)|^p+|f(x)-f(y)|^p}{(1+|y|)^{d+sp}} \,dy \,dx \\
&\gtrsim
\int_{\Omega^c} \frac{|f(y)|^p}{(1+|y|)^{d+sp}} \,dy
\end{align*}
with constants depending only on $\Omega$, $R$, $d$, $s$, $p$.
\end{proof}
\subsection{Whitney decomposition, thickness and plumpness}\label{subsec:whitney}
We recall several geometric notions needed in the
sequel. They allow us to present our main results for rather general
domains $\Omega \subset \mathbb{R}^d$. Note that, however, \autoref{thm:norm} and
\autoref{thm:extension} are new even for domains with a smooth boundary.
\medskip
For a nonempty open set $D\subset \mathbb{R}^d$, $D\neq \mathbb{R}^d$, we fix
a Whitney decomposition $\mathcal{W}(D)$ \cite[VI.1]{MR0290095} and write
$\mathcal{W}_m(D)$
for the family of Whitney cubes with side length $2^{-m}$, $m\in\mathbb{Z}$.
If $Q\in\mathcal{W}(D)$, then
\begin{equation}\label{dist_est}
\diam(Q)\le \dist(Q,\partial D)\le 4\diam(Q)\,.
\end{equation}
For any cube $Q$, its side length is denoted by $\ell(Q)$ and its center by
$x_Q$.
By $Q^*$ we denote a cube with the same center as $Q$, but side length
$\ell(Q^*)=(1+\varepsilon)\ell(Q)$, where $0<\varepsilon<1/4$ is fixed once for
all.
Such cubes have the property that
\[
{\mathbf{1}}_D \leq \sum_{Q\in \mathcal{W}(D)} {\mathbf{1}}_{Q^*} \leq M {\mathbf{1}}_D
\]
with some constant $M$ depending only on $d$.
The next two definitions are slightly modified versions of \cite[Definition
3.1]{Triebel2008}.
Our definitions and \cite[Definition 3.1]{Triebel2008} coincide if $D$ or
$D^c$ has finite inner radius.
In the case when $\inr(D)=\inr(D^c)=\infty$, if the domain $D$ is
$I$-thick in the sense of \autoref{def:Ithick}, then it is also
$I$-thick
in the sense of \cite[Definition 3.1]{Triebel2008}.
\begin{defn}\label{def:Ithick}
An open set $D\subset \mathbb{R}^d$ is called \emph{$I$-thick} (\emph{interior
thick}), if
for every $M>0$ there exists a constant $C$ such that for every cube $Q\in
\mathcal{W}(\mathbb{R}^d\setminus \overline{D})$ with $\diam Q < M \inr(D)$ there exists
a~\emph{reflected} cube $\widetilde{Q} \in \mathcal{W}(D)$ satisfying
\begin{equation}\label{e:Ithick}
C^{-1} \diam(Q) \leq \diam(\widetilde{Q}) \leq C \diam(Q) \quad\text{and}\quad
\dist(\widetilde{Q}, Q) \leq C \dist(Q,\partial D).
\end{equation}
\end{defn}
\begin{defn}\label{def:Ethick}
An open set $D\subset \mathbb{R}^d$ is called \emph{$E$-thick} (\emph{exterior
thick}), if
for every $M>0$ there exists a constant $C$ such that for every cube $Q\in
\mathcal{W}(D)$ with $\diam Q < M \inr(D^c)$ there exists
a~\emph{reflected} cube $\widetilde{Q} \in \mathcal{W}(\mathbb{R}^d\setminus\overline{D})$
satisfying
\begin{equation}\label{e:Ethick}
C^{-1} \diam(Q) \leq \diam(\widetilde{Q}) \leq C \diam(Q) \quad\text{and}\quad
\dist(\widetilde{Q}, Q) \leq C \dist(Q,\partial D).
\end{equation}
\end{defn}
\begin{rem}
The definitions of $I$- and $E$-thickness do not depend on the choice of the
families of Whitney cubes $\mathcal{W}(D)$ and $\mathcal{W}(\mathbb{R}^d\setminus \overline{D})$.
\end{rem}
\begin{rem}\label{rem:lambda}
Let $\lambda>0$ be fixed.
In \autoref{def:Ithick} we may additionally assume that the reflected
cubes satisfy
\begin{equation}\label{e:Ithicklambda}
\diam \widetilde{Q} \leq \lambda \diam Q.
\end{equation}
Indeed, if the opposite inequality holds, then
\[
\dist(\widetilde{Q}, \partial D) \geq \diam(\widetilde{Q}) > \lambda \diam Q,
\]
so in the ball
$B(x_{\widetilde{Q}}, 5\diam \widetilde{Q})$ there exists a~point $z\in D$
with
$\dist(z,\partial D) = \lambda \diam Q$.
Take $Q'$ to be a~cube from $\mathcal{W}(D)$ containing $z$. Then
\[
\frac{\lambda}{5} \diam Q \leq \frac15 \dist(z,\partial D) \leq \diam Q'\leq
\dist(z,\partial D) = \lambda \diam Q,
\]
so $\diam Q$ and $\diam Q'$ are comparable. Moreover, since $z\in
B(x_{\widetilde{Q}}, 5\diam \widetilde{Q}) \cap Q'$,
we obtain
\[
\dist(Q', \widetilde{Q}) \leq \dist(z,x_{\widetilde{Q}}) \leq
5\diam\widetilde{Q},
\]
therefore,
\[
\dist(Q', Q) \leq \diam Q' + \dist(Q', \widetilde{Q}) + \diam\widetilde{Q} +
\dist(\widetilde{Q}, Q) \lesssim \dist(Q,\partial D)
\]
with a constant depending only on $\lambda$ and $C$.
Consequently,
\eqref{e:Ithick} holds also for $Q'$ in place of $\widetilde{Q}$ (perhaps with
an
enlarged $C$).
Hence by redefining reflected cubes both \eqref{e:Ithick} and
\eqref{e:Ithicklambda} hold.
A similar remark applies to \autoref{def:Ethick}.
\end{rem}
\begin{rem}\label{rem:Ithickmod}
In \autoref{def:Ithick} we may additionally assume that the reflected
cubes satisfy
\begin{equation}\label{e:Ithickmod}
\widetilde{Q}\subset \{x\in D: \dist(x,\partial D) < \inr(D^c) \}.
\end{equation}
Indeed, by taking $\lambda \leq\frac15$ in \autoref{rem:lambda} and perhaps
redefining reflected cubes, we obtain
$\dist(w,\partial D) \leq 5\diam \widetilde{Q} \leq \diam Q < \inr(D^c)$
for
$w\in \widetilde{Q}$, as desired.
A similar remark applies to \autoref{def:Ethick}.
\end{rem}
\begin{rem}\label{r:overlap}
Let $D$ be exterior thick.
The family of all reflected cubes in the sense of the definition above, i.e.,
$\mathcal{F}:=\{\widetilde{Q} : Q\in
\mathcal{W}(D),\ \diam Q < M \inr(D^c)\}$ has the bounded overlap property, i.e.,
there exists a~constant $N$ such that
\[
\sum_{Q\in \mathcal{W}(D)} {\mathbf{1}}_{\widetilde{Q}} \leq N{\mathbf{1}}_{\mathbb{R}^d\setminus
\overline{D}}.
\]
This estimate holds true because the size of $\widetilde{Q}$ and its distance
to
$Q$ are comparable to the
size of $Q$. An analogous property holds for interior thick sets $D$.
\end{rem}
From \cite[Proposition 3.6]{Triebel2008} it follows that if $D$ is a bounded
$(\epsilon,\delta)$-domain \cite[Definition 3.1(i)]{Triebel2008}, then $D$ is
$I$-thick and $\partial D$ has Lebesgue measure zero. Bounded Lipschitz
domains are both $I$-thick and $E$-thick \cite[Proposition 3.8]{Triebel2008}.
We will show that the assumption that $D$ is an $(\epsilon,\delta)$-domain
may
be replaced by a~weaker one. To this end we need the following definition.
\begin{defn}\cite{MR927080, mv}
A set $A\subset \mathbb{R}^d$ is {\em $\kappa$-plump}
with $\kappa\in (0,1)$ (or simply \emph{plump}) if, for each $0<r<
\mathrm{diam}(A)$ and each $x\in \bar{A}$, there
is $z\in \overline{B(x,r)}$ such that
$B(z,\kappa r)\subset A$.
\end{defn}
\begin{lem}\label{l.plump}
If $D\subset \mathbb{R}^d$ is plump, then it is also $I$-thick and $\partial D$ has
$d$-dimensional Lebesgue measure zero.
\end{lem}
\begin{proof}
Let us note that if $D$ is plump, then its boundary $\partial D$ is
\emph{porous},
i.e., there exists a~constant $\alpha$
with the following property: for every $x\in \mathbb{R}^d$ and $0<r\leq 1$, there
exists
$y\in B(x,r)$
such that $B(y,\alpha r) \subset B(x,r) \setminus \partial D$.
Therefore $\partial D$ has Lebesgue measure zero, see e.g. \cite{Luukkainen}.
Let $M>1$.
For each cube $Q\in \mathcal{W}(\interior D^c)$ such that $\diam Q < M \inr(D)$ we
will associate a~\emph{reflected} cube $\widetilde{Q} \in \mathcal{W}(D)$ in the
following way. Let $y_{Q}\in \partial D$ be a~fixed point
satisfying $|x_{Q}-y_{Q}| = \dist(x_{Q}, \partial D)$. We consider a~ball
$B(y_{Q}, \frac{\diam Q}{M})$.
By plumpness condition, there exist a~ball $B\subset B(y_{Q}, \frac{\diam
Q}{M})
\cap D$ of radius
$\kappa \frac{\diam Q}{M}$. Let $z$ be its center; as $\widetilde{Q}$ we fix any
of
the Whitney cubes from $\mathcal{W}(D)$ containing $z$.
Let $z$ be a~point as above. Then
\[
\kappa\frac{\diam Q}{M} \leq \dist(z, \partial D) \leq \frac{\diam Q}{M},
\]
and hence by properties of Whitney cubes
\[
\frac{\kappa}{5M} \diam Q \leq \diam \widetilde{Q} \leq \frac{\diam Q}{M}.
\]
Furthermore, for $x\in Q$ and $w\in \widetilde{Q}$
\begin{align}
|x-w|
&\leq |x-x_{Q}| + |x_{Q}-y_{Q}| + |y_{Q}-z| + |z-w| \leq (1+5+1+1)\diam Q \\
&\leq 8\dist(Q, \partial D). \label{eq:q1Q}
\end{align}
To summarize, the five numbers $\diam(Q)$, $\diam(\widetilde{Q})$,
$\dist(Q,\partial
D)$, $\dist(\widetilde{Q},\partial D)$, $\dist(Q,\widetilde{Q})$ are
comparable
with constants depending only on $\kappa$ and $M$.
\end{proof}
\cite[Remark 3.7]{Triebel2008} provides an example of an interior
thick set
$\Omega$ such that $|\partial \Omega|>0$.
It follows from \autoref{l.plump} that such $\Omega$ is not plump. This
example is however not completely satisfactory in our case, since in our
results we assume that $|\partial \Omega|=0$. Therefore we provide another
example.
\begin{example*}
Consider annuli $A_n = \{x\in \mathbb{R}^2: 2^{-n-1} \leq |x| < 2^{-n}\}$ and let
$a_n=2^{-n-1}/n$, where $n=1,2,\ldots$.
Let $O_n\subset A_n$ be a~maximal set such that balls centered at points from
$O_n$ with radii $a_n$ are pairwise disjoint and contained in $A_n$. Clearly
$O_n\neq \emptyset$. Set
\[
\Omega = \bigcup_{n=1}^\infty \bigcup_{x\in O_n} B(x, \frac{a_n}{2}).
\]
It is easy to observe that $\Omega$ is both
interior and exterior thick. However, the largest ball that is
contained in $B(0,2^{-n})$, has a radius smaller than $3 a_n/2$. Since
$(3 a_n/2)/(2^{-n}) = 3/(4n)\to 0$, the set $\Omega$
is not plump. Moreover, $|\partial\Omega|=0$.
\end{example*}
\section{Proof \texorpdfstring{of \autoref{thm:norm}}{}}\label{sec:proof1}
Let $f \in L^p(\mathbb{R}^d)$, $1 \leq p < \infty$. Recall the definition $\delta_z = \operatorname{dist}(z, \partial \Omega)$
for $z \in \mathbb{R}^d$. The first inequality in
\eqref{eq:comp-semi}
follows from the fact that $|x-y| + \delta_x + \delta_y \leq 3|x-y|$ for $x\in
\Omega$ and $y\in \Omega^c$. This implies
\begin{align*}
3^{-d-p} \int_{ \Omega} \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{|x-y|^{d+sp}}\,dy\,dx \leq
\int_{ \Omega \cup \Omega^{\rm ext}_{\inr(\Omega)} } \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\,dx \,.
\end{align*}
This estimate implies the desired inequality.
The remainder of this section is devoted to the proof of the
second inequality. We observe that
\begin{align*}
& \int_{ \{x\in \mathbb{R}^d: \delta_x < \inr(\Omega)\} } \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\,dx\\
&\leq
|f|_{ W^{s,p}(\Omega|\Omega^c)}^p
+ \int_{ \{x\in \Omega^c: \delta_x < \inr(\Omega) \} } \int_{\Omega^c}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\,dx
=: |f|_{ W^{s,p}(\Omega|\Omega^c)}^p + I.
\end{align*}
We note that if $x\in \interior \Omega^c$ satisfies $\delta_x<\inr(\Omega)$,
then a Whitney cube $Q \in \mathcal{W}(\interior \Omega^c)$
containing $x$ satisfies $\diam Q \leq \dist(Q,\partial \Omega) <
\inr(\Omega)$.
Moreover, since $\partial \Omega$ has Lebesgue measure zero, we obtain
\begin{equation}\label{eq:doublesum}
I \leq \sum_{Q_1 \in \mathcal{W}^b}
\sum_{Q_2 \in \mathcal{W}(\interior \Omega^c) }
\int_{Q_1} \int_{Q_2}
\frac{|f(x)-f(y)|^p}{(|x-y|+\delta_x+\delta_y)^{d+sp}}\,dy\,dx,
\end{equation}
where
\[
\mathcal{W}^b = \{ Q_1 \in \mathcal{W}(\interior \Omega^c): \diam Q_1 < \inr(\Omega) \}.
\]
We take $M=1$ in the \autoref{def:Ithick} so that $\widetilde{Q}$ exists for
all cubes $Q\in \mathcal{W}^b$,
and let $C$ be the corresponding constant.
Let $Q_1 \in \mathcal{W}^b$ and $Q_2 \in \mathcal{W}(\interior \Omega^c)$. For $y\in Q_2$ and
$w\in \widetilde{Q}_1$
\begin{align}
|y-w| &\leq \dist(y,Q_1) + \diam Q_1 + \dist(Q_1,\tilde{Q_1}) + \diam
\tilde{Q_1} \nonumber\\
&\leq \dist(y,Q_1) + (5C+1)\diam Q_1. \label{eq:q2Q}
\end{align}
Recall that for any cube $Q$, its side length is denoted by $\ell(Q)$ and its center by
$x_Q$. For $x\in Q_1$ and $y\in Q_2$ we denote
\[
w=w(x,y) = x_{\widetilde{Q}_1} + \left(\frac{x-x_{Q_1}}{2\ell(Q_1)} +
\frac{y-x_{Q_2}}{2\ell(Q_2)} \right) \ell(\widetilde{Q}_1)
\]
and observe that $w\in \widetilde{Q}_1$.
We come back to estimating the double integral in \eqref{eq:doublesum}
\begin{align*}
\int_{Q_1} \int_{Q_2} & \frac{|f(x)-f(y)|^p}{ (|x-y| + \delta_x +
\delta_y)^{d+sp}}\,dy\,dx\\
&\leq
(2^{p-1}\vee 1)
\int_{Q_1} \int_{Q_2} \frac{|f(x)-f(w(x,y))|^p}{ (\dist(Q_1,Q_2) + \dist(Q_1,
\partial\Omega))^{d+sp}}\,dy\,dx \\
&\quad +
(2^{p-1}\vee 1)
\int_{Q_1} \int_{Q_2} \frac{|f(w(x,y))-f(y))|^p}{ (\dist(y,Q_1) +\dist(Q_1,
\partial\Omega))^{d+sp}}\,dy\,dx \\
&=:(2^{p-1}\vee 1) (I_1(Q_1, Q_2) +I_2(Q_1, Q_2)).
\end{align*}
To estimate $I_1(Q_1, Q_2)$, we change the variable $y$ to $w=w(x,y)$ in the
integral and
obtain
\begin{align*}
I_1(Q_1, Q_2) &\leq \frac{2^d \ell(Q_2)^d}{\ell(\widetilde{Q}_1)^d} \int_{Q_1}
\int_{\widetilde{Q}_1}
\frac{|f(x)-f(w))|^p}{ (\dist(Q_1,Q_2) + \dist(Q_1,
\partial\Omega))^{d+sp}}\,dw\,dx \\
&\lesssim
\int_{Q_1} \int_{\widetilde{Q}_1} \frac{|f(x)-f(w))|^p}{
|x-w|^{d+sp}}\,dw\,dx
\cdot
\frac{\int_{Q_2} \left(1+\frac{\dist(Q_1,Q_2)}{\dist(Q_1,
\partial\Omega)}\right)^{-d-sp} \,dy}{\ell(\widetilde{Q}_1)^d}
\end{align*}
In the last passage we have used \eqref{eq:q1Q} with $D:= \Omega$, $Q:= Q_1$ and the inequality $s\leq 1$
(although any upper bound for $s$ would suffice). We obtain
\begin{align*}
&\sum_{Q_1 \in \mathcal{W}^b } \sum_{Q_2\in \mathcal{W}(\interior \Omega^c)}
\!\!\!\!\!\! I_1(Q_1, Q_2)\\
&\lesssim
\sum_{Q_1 \in \mathcal{W}^b}
\int_{Q_1} \int_{\widetilde{Q}_1} \frac{|f(x)-f(w))|^p}{
|x-w|^{d+sp}}\,dw\,dx
\cdot
\sum_{Q_2} \frac{\int_{Q_2} \left(1+\frac{\dist(Q_1,Q_2)}{\dist(Q_1,
\partial\Omega)}\right)^{-d-sp} \,dy}{\ell(\widetilde{Q}_1)^d}.
\end{align*}
By properties of Whitney cubes,
\begin{align*}
\sum_{Q_2}& \frac{\int_{Q_2} \left(1+\frac{\dist(Q_1,Q_2)}{\dist(Q_1,
\partial\Omega)}\right)^{-d-sp} \,dy}{\ell(\widetilde{Q}_1)^d}\\
&\leq
\ell(\widetilde{Q}_1)^{-d} c(d) \int_{\mathbb{R}^d} \left(1+
\frac{|y-x_{Q_1}|}{\dist(Q_1,
\partial\Omega)}\right)^{-d-sp}\,dy \\
&=\Big( \frac{\dist(Q_1, \partial\Omega)}{\ell(\widetilde{Q}_1)} \Big)^d c(d)
\int_{\mathbb{R}^d} (1+|z|)^{-d-sp}\,dz
\leq \frac{c(d,C)}{s},
\end{align*}
where the constant $c(d,C)$ depends only on $d$ and $C$, but not on the cube
$Q_1$. Thus by \autoref{r:overlap}
\begin{equation}\label{eq:sum-norm}
\sum_{Q_1 \in \mathcal{W}^b } \sum_{Q_2\in \mathcal{W}(\interior \Omega^c)}
\!\!\!\!\!\! I_1(Q_1, Q_2)
\leq \frac{c(d,p,C)}{s} |f|_{ W^{s,p}(\Omega|\Omega^c)}^p.
\end{equation}
We are left with estimating $I_2(Q_1, Q_2)$. We interchange the order of
integration and
change the variable $x$ to $w=w(x,y)$.
By \eqref{eq:q2Q}, this gives us
\begin{align*}
I_2(Q_1, Q_2) &\leq \frac{2^d \ell(Q_1)^d}{\ell(\widetilde{Q}_1)^d} \int_{Q_2}
\int_{\widetilde{Q}_1}
\frac{|f(w)-f(y))|^p}{ (\dist(y,Q_1) + \dist(Q_1,
\partial\Omega))^{d+sp}}\,dw\,dy \\
&\leq c(d,p,C) \int_{Q_2} \int_{\widetilde{Q}_1}
\frac{|f(w)-f(y))|^p}{ |w-y|^{d+sp}}\,dw\,dy.
\end{align*}
By \autoref{r:overlap}, we get an estimate of the form \eqref{eq:sum-norm}
for $I_2(Q_1, Q_2)$ instead of
$I_1(Q_1, Q_2)$. The proof is complete.
Note that the constant in \autoref{thm:norm} depends on $\Omega$ only
through $d$ and the constant $C$ from \autoref{def:Ithick} taken for
$M=1$.
\qed
\section{Proof \texorpdfstring{ of \autoref{thm:extension}}{}}\label{sec:proof2}
We may assume that $\Omega\neq \emptyset$.
We fix $M=1$ if $\inr(\Omega^c)=\infty$ and $M=\frac{2\sqrt{d}
\inr(\Omega)}{\inr(\Omega^c)}$ if $\inr(\Omega^c)<\infty$,
and we fix $\lambda = 1/125$.
We take reflected cubes and the constant $C$ as in \autoref{def:Ithick} and
\autoref{rem:lambda}
for these particular choices of $M$ and $\lambda$.
By \autoref{rem:Ithickmod},
the reflected cubes satisfy $\eqref{e:Ithickmod}$ with $D=\Omega^c$.
\subsection{Definition of the extension.}
Let $Q_0=[0,1]^d$. We fix a~function $\psi_0\in C_c^\infty(Q_0^*)$ such that
$\psi_0=1$ on $Q_0$ and $0\leq \psi_0 \leq 1$. We shift and rescale this
function to other cubes, i.e., we let
\[
\psi_Q(x)= \psi_0\left(\frac{x-x_{Q}}{\ell(Q)} + x_{Q_0}\right), \quad x\in
\mathbb{R}^d.
\]
Recall from \autoref{subsec:whitney} that for $Q\in\mathcal{W}(D)$ we have $\diam(Q)\le \dist(Q,\partial D)\le 4\diam(Q)$. For any cube $Q$, its side length is denoted by $\ell(Q)$ and its center by $x_Q$. By $Q^*$ we denote a cube with the same center as $Q$, but side length
$\ell(Q^*)=(1+\varepsilon)\ell(Q)$, where $0<\varepsilon<1/4$ is fixed as above.
We consider the following family of functions
\[
\phi_Q(x) =\frac{\psi_Q(x)}{\sum_{R\in \mathcal{W}(\Omega)} \psi_R(x)},\quad x\in \mathbb{R}^d.
\]
Thus $\phi_Q\geq 0$ and $\sum_{Q\in \mathcal{W}(\Omega)} \phi_Q = {\mathbf{1}}_\Omega$. Let $f
\in L^1_{loc}(\Omega^c)$. Let
\[
a_Q = \frac{1}{|\widetilde{Q}|} \int_{\widetilde{Q}} f(x)\,dx, \quad Q\in
\mathcal{W}(\Omega)\,.
\]
Note that the reflected cube $\widetilde{Q}$ is well defined thanks to the
choice
of
$M$.
We extend a given function $f \in L^1_{loc}(\Omega^c)$ from $\Omega^c$ to
$\mathbb{R}^d$
by defining $\ext(f)$ as follows:
\begin{align*}
\ext(f)(x) =
\begin{cases}
\sum_{Q\in \mathcal{W}(\Omega)} a_Q \phi_Q(x) \quad &\text{ if } x \in \Omega \,, \\
f(x) &\text{ if } x \in \Omega^c \,.
\end{cases}
\end{align*}
Let ${\mathcal{N}_\Omega}(Q) = \{R\in \mathcal{W}(\Omega): R\cap Q^* \neq\emptyset \}$ be the
collection
of Whitney cubes intersecting $Q$. Observe that for $x\in Q_1\in \mathcal{W}(\Omega)$
and any $t \in \mathbb{R}$
\begin{equation}\label{e:fdecomp}
\ext(f)(x) = \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)} a_Q \phi_Q(x) = t + \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)}
(a_Q-t) \phi_Q(x).
\end{equation}
\subsection{A remark on reflected cubes}\label{subs:reflected}
Let
\[
\mathcal{W}^{<\delta}(\Omega) = \{ Q:\mathcal{W}(\Omega) : \dist(Q,\partial \Omega) < \delta\}.
\]
Let us note that if $Q_1 \in \mathcal{W}^{<\delta}(\Omega)$, $Q_2\in {\mathcal{N}_\Omega}(Q_1)$ and
$Q_3\in{\mathcal{N}_\Omega}(Q_2)$,
then
\[
\diam Q_3 \leq 5\diam Q_2 \leq 25\diam Q_1.
\]
Therefore, if $z\in \widetilde{Q}_3$, then
\[
\dist(z,\partial \Omega) \leq 5\diam \widetilde{Q}_3 \leq 5\lambda \diam Q_3
\leq
125\lambda \diam Q_1 \leq 125\lambda \dist(Q_1, \partial\Omega) < \delta,
\]
that is, $\widetilde{Q}_3\subset \Omega^{\rm ext}_\delta$. In particular,
$\widetilde{Q}_1, \widetilde{Q}_2 \subset \Omega^{\rm ext}_\delta$, because we may take
$Q_3=Q_2$ or $Q_3=Q_1$.
\subsection{\texorpdfstring{An estimate of $|a_{Q_1}-a_{Q}|^p$}{First
estimate}.}
We claim that, for $Q_1,Q \in \mathcal{W}(\Omega)$,
\begin{align}\label{e:aQ-est}
|a_{Q_1}-a_{Q}|^p \lesssim \frac{(\dist(Q_1,Q)+\ell(Q_1)+\ell(Q))^{d+sp}}{|Q_1|
|Q|}
\,|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p}\,.
\end{align}
Indeed,
\begin{align*}
|a_{Q_1}-a_{Q}| &=
\frac{1}{|\widetilde{Q}_1| |\widetilde{Q}|}
\left|
|\widetilde{Q}| \int_{\widetilde{Q}_1} f(y) \,dy -
|\widetilde{Q}_1| \int_{\widetilde{Q}} f(x) \,dx \right| \\
&\leq
\frac{1}{|\widetilde{Q}_1| |\widetilde{Q}|}
\int_{\widetilde{Q}_1} \int_{\widetilde{Q}} |f(y)-f(x)| \,dy \, dx \,.
\end{align*}
From $|Q_j| \lesssim |\widetilde{Q}_j|$, and Jensen inequality we deduce
\begin{align*}
|a_{Q_1}-a_{Q}|^p &\lesssim
\frac{1}{|Q_1||Q|}
\int_{\widetilde{Q}_1} \int_{\widetilde{Q}} |f(y)-f(x)|^p \,dy \, dx,
\end{align*}
and the claim follows.
\subsection{\texorpdfstring{An estimate of $|\phi_Q|_{Q_1,Q_2}^{s,p}$ for
$s\leq 1$ and arbitrary cubes $Q, Q_1$, $Q_2$.}{Second estimate}}
It is easy to check that $|\nabla \phi_Q| \lesssim \ell(Q)^{-1}$. Therefore
$|\phi_Q(x) - \phi_Q(y)| \lesssim \ell(Q)^{-1}|x-y| \wedge 1$ for all $x,y$.
As a result, we obtain
\begin{align}
|\phi_Q|_{Q_1,Q_2}^{s,p} &\lesssim \int_{Q_1} \int_{Q_2}
\frac{\ell(Q)^{-p}|x-y|^p \wedge 1}{|x-y|^{d+sp}}\,dy\,dx \nonumber\\
& \lesssim |Q_1| |Q_2| \Big(\frac{\ell(Q_1)^{p-sp-d} \ell(Q)^{-p}}{1-s}
\wedge
\frac{\ell(Q_2)^{p-sp-d} \ell(Q)^{-p}}{1-s} \wedge \dist(Q_1,Q_2)^{-d-sp}
\Big)\,.
\label{e:phiQ-est}
\end{align}
We note that the above inequality for $s=1$ is nontrivial only if
$\dist(Q_1,Q_2)>0$.
\subsection{Proof of part (b)\texorpdfstring{, formula \eqref{e:int-int}}{}}
It holds
\begin{align}\label{e:f-omega-omega}
|\ext(f)|_{\Omega^{\rm int}_\delta, \Omega^{\rm int}_\delta}^{s,p} &\leq
\sum_{Q_1\in \mathcal{W}^{<\delta}(\Omega)} \sum_{Q_2\in \mathcal{W}^{<\delta}(\Omega)}
|\ext(f)|_{Q_1,Q_2}^{s,p}.
\end{align}
For $Q_1$, $Q_2\in \mathcal{W}^{<\delta}(\Omega)$, we use \eqref{e:fdecomp} twice with
$t=a_{Q_1}$ and obtain
\begin{align*}
|\ext(f)|_{Q_1,Q_2}^{s,p}
&\lesssim
\sum_{Q\in {\mathcal{N}_\Omega}(Q_1)\cup{\mathcal{N}_\Omega}(Q_2) } |a_Q-a_{Q_1}|^p |\phi_Q|_{Q_1,Q_2}^{s,p}
\, .
\end{align*}
If additionally $Q_2 \in {\mathcal{N}_\Omega}(Q_1)$, then for $Q\in {\mathcal{N}_\Omega}(Q_1)\cup{\mathcal{N}_\Omega}(Q_2)$
\begin{align*}
|a_Q-a_{Q_1}|^p |\phi_Q|_{Q_1,Q_2}^{s,p} &\lesssim
\frac{(\dist(Q_1,Q)+\ell(Q_1)+\ell(Q))^{d+sp}}{|Q_1| |Q|}
\,|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p}
\cdot \frac{|Q_1| |Q_2| \ell(Q_1)^{-sp-d}}{1-s} \\
&\lesssim \frac{\,|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p}}{1-s}\,.
\end{align*}
Otherwise, if $Q_2\in \mathcal{W}^{<\delta}(\Omega) \setminus {\mathcal{N}_\Omega}(Q_1)$, then
$\dist(Q_1,Q_2)>0$ and consequently $\ell(Q_1)$, $\ell(Q_2) \lesssim
\dist(Q_1,Q_2)$. Then
for $Q\in {\mathcal{N}_\Omega}(Q_1)$
\begin{align*}
|a_Q-a_{Q_1}|^p |\phi_Q|_{Q_1,Q_2}^{s,p} &\lesssim
\frac{(\ell(Q_1)+\ell(Q))^{d+sp}}{|Q_1| |Q|}
\,|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p} \cdot \frac{|Q_1|
|Q_2|}{\dist(Q_1,Q_2)^{d+sp}}\\
&\lesssim \frac{\ell(Q_1)^{sp} |Q_2|\, |f|_{\widetilde{Q}_1,
\widetilde{Q}}^{s,p}}{\dist(Q_1,Q_2)^{d+sp}}.
\end{align*}
On the other hand, for $Q\in {\mathcal{N}_\Omega}(Q_2)$
\begin{align*}
|a_Q-a_{Q_1}|^p |\phi_Q|_{Q_1,Q_2}^{s,p} &\lesssim
\frac{(\dist(Q_1,Q)+\ell(Q_1)+\ell(Q))^{d+sp}}{|Q_1| |Q|}
\,|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p} \cdot \frac{|Q_1|
|Q_2|}{\dist(Q_1,Q_2)^{d+sp}}\\
&\lesssim |f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p}.
\end{align*}
Let $\mathcal{W}^{<\delta}_{\mathcal{N}}(\Omega) = \{ Q\in \mathcal{W}(\Omega) : {\mathcal{N}_\Omega}(Q) \cap
\mathcal{W}^{<\delta}(\Omega) \neq\emptyset \}$.
Combining the above inequalities yields
\begin{align}
|\ext(f)|_{\Omega^{\rm int}_\delta, \Omega^{\rm int}_\delta}^{s,p}
&\lesssim
\sum_{Q_1\in \mathcal{W}^{<\delta}(\Omega)} \bigg( \sum_{Q_2\in {\mathcal{N}_\Omega}(Q_1)}
\sum_{Q\in
{\mathcal{N}_\Omega}(Q_1)\cup{\mathcal{N}_\Omega}(Q_2) } \frac{\,|f|_{\widetilde{Q}_1,
\widetilde{Q}}^{s,p}}{1-s}
\nonumber\\
&\qquad\qquad\qquad+ \sum_{Q_2\in \mathcal{W}^{<\delta}(\Omega) \setminus {\mathcal{N}_\Omega}(Q_1)}
\sum_{Q\in
{\mathcal{N}_\Omega}(Q_1)} \frac{\ell(Q_1)^{sp} |Q_2|\, |f|_{\widetilde{Q}_1,
\widetilde{Q}}^{s,p}}{\dist(Q_1,Q_2)^{d+sp}}\nonumber\\
&\qquad\qquad\qquad+ \sum_{Q_2\in \mathcal{W}^{<\delta}(\Omega) \setminus {\mathcal{N}_\Omega}(Q_1)}
\sum_{Q\in
{\mathcal{N}_\Omega}(Q_2)} |f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p} \bigg) \nonumber\\
&\lesssim
\sum_{Q_1\in \mathcal{W}^{<\delta}(\Omega)} \bigg( \sum_{Q: {\mathcal{N}_\Omega}(Q)\cap{\mathcal{N}_\Omega}(Q_1)
\neq\emptyset
} \frac{\,|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p}}{1-s} \nonumber\\
&\qquad\qquad\qquad+ \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)}
\, |f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p} \int_{\Omega\setminus \bigcup
{\mathcal{N}_\Omega}(Q_1)}
\frac{\ell(Q_1)^{sp} \,dx}{\dist(x,Q_1)^{d+sp}}\nonumber\\
&\qquad\qquad\qquad+ \sum_{Q_2\in \mathcal{W}^{<\delta}_{\mathcal{N}}(\Omega) \setminus
\{Q_1\}}
|f|_{\widetilde{Q}_1, \widetilde{Q}_2}^{s,p} \bigg) \nonumber\\
&\lesssim
\frac{ |f|_{\Omega^{\rm ext}_\delta, \Omega^{\rm ext}_\delta}^{s,p}}{s(1-s)}. \label{e:f-o-o}
\end{align}
The fact that in the last expression there are sets $\Omega^{\rm ext}_\delta$ follows from
a~remark
in~\autoref{subs:reflected}.
\subsection{Proof of part (b)\texorpdfstring{, formula \eqref{e:int-ext}}{}}
It holds
\begin{align}\label{e:aim}
|\ext(f)|_{\Omega^{\rm int}_\delta, \Omega^{\rm ext}_\varepsilon}^{s,p} &\leq
\sum_{Q_1\in \mathcal{W}^{<\delta}(\Omega)} \sum_{Q_2\in \mathcal{W}^{<\varepsilon}(\interior
\Omega^c)}
|\ext(f)|_{Q_1,Q_2 \cap \Omega^{\rm ext}_\varepsilon}^{s,p}.
\end{align}
Now let $Q_1\in \mathcal{W}^{<\delta}(\Omega)$ and $Q_2\in
\mathcal{W}^{<\varepsilon}(\interior \Omega^c)$. We again
use
\eqref{e:fdecomp} with $t=a_{Q_1}$,
\begin{align*}
|\ext(f)|_{Q_1,Q_2 \cap \Omega^{\rm ext}_\varepsilon}^{s,p}
& =
\int_{Q_1} \int_{Q_2 \cap \Omega^{\rm ext}_\varepsilon} \frac{\big| \sum_{Q\in
{\mathcal{N}_\Omega}(Q_1)} (a_Q -
a_{Q_1})\phi_Q(x) + a_{Q_1}- f(y)\big|^p}{(|x-y| +\delta_x+\delta_y)^{d+sp}}
\,dy\,dx\\
&\lesssim
\sum_{Q\in {\mathcal{N}_\Omega}(Q_1)} \int_{Q_1} \int_{Q_2} \frac{\big| (a_Q -
a_{Q_1})(\phi_Q(x)-\phi_Q(y)) \big|^p}{(|x-y| +\delta_x+\delta_y)^{d+sp}}
\,dy\,dx \\
&\qquad+ \dist(Q_1,Q_2)^{-d-sp} \int_{Q_1} \int_{Q_2 \cap \Omega^{\rm ext}_\varepsilon}
\Big| a_{Q_1}-
f(y)\Big|^p\,dy\,dx =: A+ \frac{B}{\dist(Q_1,Q_2)^{d+sp}}.
\end{align*}
The first term above is estimated using \eqref{e:aQ-est} and
\eqref{e:phiQ-est},
\begin{align*}
A &\lesssim \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)}
\frac{(\dist(Q_1,Q)+\ell(Q_1)+\ell(Q))^{d+sp}}{|Q_1| |Q|}
\,|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p}\,
\frac{|Q_1||Q_2|}{\dist(Q_1,Q_2)^{d+sp}}
\\
&\lesssim \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)}
|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p}\, \frac{ \ell(Q_1)^{sp}
|Q_2|}{\dist(Q_1,Q_2)^{d+sp}}.
\end{align*}
For the second term,
\begin{align*}
B &= |Q_1| \int_{Q_2 \cap \Omega^{\rm ext}_\varepsilon} \left| \frac{1}{|\widetilde{Q}_1|}
\int_{\widetilde{Q}_1}
(f(x)-f(y)) \,dx \right|^p \,dy \lesssim \int_{Q_2 \cap \Omega^{\rm ext}_\varepsilon}
\int_{\widetilde{Q}_1}
|f(x)-f(y)|^p \,dx \,dy \\
&\lesssim \dist(Q_1,Q_2)^{d+sp}
\int_{Q_2 \cap \Omega^{\rm ext}_\varepsilon} \int_{\widetilde{Q}_1}
\frac{|f(x)-f(y)|^p}{(|x-y|
+\delta_x+\delta_y)^{d+sp}} \,dx \,dy = \dist(Q_1,Q_2)^{d+sp} |f|_{Q_2 \cap
\Omega^{\rm ext}_\varepsilon,
\widetilde{Q}_1}^{s,p}.
\end{align*}
Inequalities obtained for $A$ and $B$ together with \eqref{e:aim} yield
\begin{align}
|\ext(f)|_{\Omega^{\rm int}_\delta, \Omega^{\rm ext}_\varepsilon}^{s,p}
&\lesssim
\sum_{Q_1\in \mathcal{W}^{<\delta}(\Omega)}
\bigg(
\sum_{Q\in {\mathcal{N}_\Omega}(Q_1)}
|f|_{\widetilde{Q}_1, \widetilde{Q}}^{s,p} \int_{\Omega^c}\frac{\ell(Q_1)^{sp}
\,dx}{\dist(x,Q_1)^{d+sp}} +
\sum_{Q_2\in \mathcal{W}^{<\varepsilon}(\interior \Omega^c)}
|f|_{Q_2\cap \Omega^{\rm ext}_\varepsilon, \widetilde{Q}_1}^{s,p} \bigg) \nonumber\\
&\lesssim \frac{1}{s} |f|_{\Omega^{\rm ext}_\delta, \Omega^{\rm ext}_\varepsilon}^{s,p},
\label{e:f-o-oc}
\end{align}
since by~\autoref{subs:reflected},
the cubes $\widetilde{Q}_1$ and $\widetilde{Q}$ above are contained in
$\Omega^{\rm ext}_\delta \subset \Omega^{\rm ext}_\varepsilon$.
\subsection{Proof of part (b)\texorpdfstring{, formulas \eqref{e:simpl1} and
\eqref{e:simpl2}}{}}
Formula \eqref{e:simpl1} follows directly by taking $\delta=\inr \Omega$ and
$\varepsilon=\infty$ in \eqref{e:int-ext} and enlarging the right hand side;
alternatively, one may also apply \eqref{e:int-ext} to
$\delta=\varepsilon=\infty$.
To prove \eqref{e:simpl2} we proceed as follows,
\begin{align*}
|\ext(f)|_{\mathbb{R}^d, \mathbb{R}^d}^{s,p} &=
|\ext(f)|_{\Omega, \Omega}^{s,p} + 2|\ext(f)|_{\Omega, \Omega^c}^{s,p} +
|f|_{\Omega^c, \Omega^c}^{s,p}
\leq \frac{c}{s(1-s)} |f|_{\Omega^c, \Omega^c}^{s,p},
\end{align*}
by \eqref{e:int-ext} and \eqref{e:int-int} applied to
$\delta=\varepsilon=\infty$.
We note that the constants depend on $\Omega$ only through $d$, $M$ and $C$.
\subsection{Proof of part (a).}
The smoothness of $\ext(f)$ on $\Omega$ follows directly from the definition.
The proof of the second part is omitted as it is straightforward, it is based
on the fact that if the cubes $Q\in \mathcal{W}(\Omega)$ approach $z\in \partial
\Omega$,
then so do the reflected cubes $\widetilde{Q}$.
\qed
\subsection{Proof of part (c).}
If $p=\infty$, then
\begin{align*}
\|\ext(f)\|_{L^\infty(\Omega)}
&=
\sup_{Q_1 \in \mathcal{W}(\Omega)} \, \sup_{x\in Q_1} \left| \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)}
a_Q \phi_Q(x) \right| \\
&\leq \sup_{Q_1 \in \mathcal{W}(\Omega)} \# {\mathcal{N}_\Omega}(Q_1) \cdot
\|f\|_{L^\infty(\Omega^{\rm ext}_{\inr(\Omega)})}
\lesssim \|f\|_{L^\infty(\Omega^c)}.
\end{align*}
Now let $p<\infty$.
We first observe that
\begin{equation}\label{eq:weightcomparison}
|x|+1 \asymp |\tilde{x}|+1 \qquad \textrm{whenever $x\in Q\in \mathcal{W}(\Omega)$
and
$\tilde{x}\in \widetilde{Q} \in \mathcal{W}(\interior \Omega^c)$,}
\end{equation}
with constants dependent only on the domain $\Omega$. Indeed, let
$R=\dist(0,\partial \Omega)$, then
\[
|\tilde{x}| \leq |\tilde{x}-x| + |x| \lesssim \diam Q + |x|,
\]
and
\[
\diam Q \leq \dist(Q,\partial \Omega) \leq |x-0| + \dist(0,\partial \Omega) =
|x|+R,
\]
so $|\tilde{x}|+1 \lesssim |x|+1$, as claimed. The proof of the opposite
estimate is similar and omitted.
Let $\omega(x)=(1+|x|)^\beta$. Since the numbers $\# {\mathcal{N}_\Omega}(Q_1)$ for $Q_1 \in
\mathcal{W}(\Omega)$ are bounded from above by a~constant depending only on the domain
$\Omega$, we obtain
\begin{align}
\|f\|_{L^p(\Omega,\,\omega(x)dx)}^p
&= \sum_{Q_1 \in \mathcal{W}(\Omega)} \, \int_{Q_1} \left| \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)} a_Q
\phi_Q(x) \right|^p \omega(x)\,dx\nonumber\\
&\lesssim
\sum_{Q_1 \in \mathcal{W}(\Omega)} \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)} |a_Q|^p \int_{Q_1}
|\phi_Q(x)|^p \omega(x)\,dx \label{eq:pestimate}
\end{align}
By Jensen inequality, comparability of the sizes of cubes $Q$, $\widetilde{Q}$
and
$Q_1$ as in the sum above, and \eqref{eq:weightcomparison} we can estimate each
summand as follows
\begin{align*}
|a_Q|^p \int_{Q_1} |\phi_Q(x)|^p \omega(x)\,dx
&\leq \frac{1}{|\widetilde{Q}|} \int_{\widetilde{Q}} |f(x)|^p\,dx \cdot |Q_1|
\cdot
\sup_Q \omega
\lesssim \int_{\widetilde{Q}} |f(x)|^p \omega(x)\,dx.
\end{align*}
Using boundedness of $\# {\mathcal{N}_\Omega}(Q_1)$, and \autoref{r:overlap} we obtain from
the estimate \eqref{eq:pestimate} the following estimate:
\begin{align*}
\|f\|_{L^p(\Omega,\,\omega(x)dx)}^p
&\lesssim
\sum_{Q_1 \in \mathcal{W}(\Omega)} \sum_{Q\in {\mathcal{N}_\Omega}(Q_1)} \int_{\widetilde{Q}}
|f(x)|^p
\omega(x)\,dx
\lesssim
\sum_{Q \in \mathcal{W}(\Omega)} \int_{\widetilde{Q}} |f(x)|^p \omega(x)\,dx \\
&\lesssim
\|f\|_{L^p(\Omega^{\rm ext}_{\inr(\Omega)},\,\omega(x)dx)}^p.
\end{align*}
This completes the proof of part (c) and thus the proof of
\autoref{thm:extension}.
\qed
\def$'${$'$}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,537 |
Q: Xamarin iOS monotouch dialog - labels overlapping text input When you create a dialog similar to the one below
And then navigate to a new view controller by tapping on a radiobutton group (in my example industry type), then return back your labels are endinf up overlapping the text. Tested on simulator and Apple iPad mini 2.
Has anyone found a way to fix this without creating a custom class ?
A: The only way was to create a new class and explicitly specify constraints by overriding GetCell method:
public class BaseEntryElement:EntryElement
{
.....
public override UITableViewCell GetCell (UITableView tv)
{
var c= base.GetCell (tv);
c.ContentView.SubviewsDoNotTranslateAutoresizingMaskIntoConstraints ();
c.ContentView.AddConstraints (
c.ContentView.Subviews[0].WithSameCenterY(c.ContentView),
c.ContentView.Subviews[0].AtLeftOf(c.ContentView,BaseEntryElement.offset),
c.ContentView.Subviews[0].AtTopOf(c.ContentView),
c.ContentView.Subviews[1].ToRightOf(c.ContentView.Subviews[0],40),
c.ContentView.Subviews[1].WithSameCenterY(c.ContentView.Subviews[0])
);
return c;
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,222 |
Q: Create a column indicating the presence of a value in a set of columns I have a table looks like this:
table1 <- data.table(ID = 1:10,
col1 = c("x1", "x2", "x3", NA, NA, NA, "x7", "x8", "x9", "x10"),
col2 = c("x1", "x2", "x3", "x4", NA, NA, "x7", NA, "x9", "x10"),
col3 = c("x1", NA, NA, "x4", "x5", "x6", NA, "x8", NA, "x10"))
ID col1 col2 col3
1: 1 x1 x1 x1
2: 2 x2 x2 <NA>
3: 3 x3 x3 <NA>
4: 4 <NA> x4 x4
5: 5 <NA> <NA> x5
6: 6 <NA> <NA> x6
7: 7 x7 x7 <NA>
8: 8 x8 <NA> x8
9: 9 x9 x9 <NA>
10: 10 x10 x10 x10
I need to create column, that indicates combinations value of presence in columns col1, col2 and col3 by their colnames.
Expected output is provided below:
ID features
1: 1 col1:col2:col3
2: 2 col1:col2
3: 3 col1:col2
4: 4 col2:col3
5: 5 col3
6: 6 col3
7: 7 col1:col2
8: 8 col1:col3
9: 9 col1:col2
10: 10 col1:col2:col3
The features column reflects in which combination the corresponding ID is present in table1. How can i do it?
A: Since you're using data.table, you can do:
table1[, .(features = paste(names(table1)[-1][!is.na(.SD)], collapse = ":")),
ID]
## ID features
## 1: 1 col1:col2:col3
## 2: 2 col1:col2
## 3: 3 col1:col2
## 4: 4 col2:col3
## 5: 5 col3
## 6: 6 col3
## 7: 7 col1:col2
## 8: 8 col1:col3
## 9: 9 col1:col2
## 10: 10 col1:col2:col3
Or you can melt the data and then aggregate using paste:
melt(table1, "ID", na.rm = TRUE)[, .(features = paste(variable, collapse = ":")), ID]
A: In base R, it can be used apply():
#Code
table1$Var <- apply(table1[,-1],1,function(x) paste0(names(x)[!is.na(x)],collapse = ':'))
Output:
table1
ID col1 col2 col3 Var
1: 1 x1 x1 x1 col1:col2:col3
2: 2 x2 x2 <NA> col1:col2
3: 3 x3 x3 <NA> col1:col2
4: 4 <NA> x4 x4 col2:col3
5: 5 <NA> <NA> x5 col3
6: 6 <NA> <NA> x6 col3
7: 7 x7 x7 <NA> col1:col2
8: 8 x8 <NA> x8 col1:col3
9: 9 x9 x9 <NA> col1:col2
10: 10 x10 x10 x10 col1:col2:col3
Or using tidyverse functions:
library(dplyr)
library(tidyr)
#Code
new <- table1 %>% pivot_longer(-ID) %>%
group_by(ID) %>%
summarise(val=paste0(name[!is.na(value)],collapse = ":"))
Output:
# A tibble: 10 x 2
ID val
<int> <chr>
1 1 col1:col2:col3
2 2 col1:col2
3 3 col1:col2
4 4 col2:col3
5 5 col3
6 6 col3
7 7 col1:col2
8 8 col1:col3
9 9 col1:col2
10 10 col1:col2:col3
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,816 |
"""
.. _tutorial-deploy-model-on-rasp:
Deploy the Pretrained Model on Raspberry Pi
===========================================
**Author**: `Ziheng Jiang <https://ziheng.org/>`_, \
`Hiroyuki Makino <https://makihiro.github.io/>`_
This is an example of using Relay to compile a ResNet model and deploy
it on Raspberry Pi.
"""
import tvm
from tvm import te
import tvm.relay as relay
from tvm import rpc
from tvm.contrib import utils, graph_executor as runtime
from tvm.contrib.download import download_testdata
######################################################################
# .. _build-tvm-runtime-on-device:
#
# Build TVM Runtime on Device
# ---------------------------
#
# The first step is to build the TVM runtime on the remote device.
#
# .. note::
#
# All instructions in both this section and next section should be
# executed on the target device, e.g. Raspberry Pi. And we assume it
# has Linux running.
#
# Since we do compilation on local machine, the remote device is only used
# for running the generated code. We only need to build tvm runtime on
# the remote device.
#
# .. code-block:: bash
#
# git clone --recursive https://github.com/apache/tvm tvm
# cd tvm
# mkdir build
# cp cmake/config.cmake build
# cd build
# cmake ..
# make runtime -j4
#
# After building runtime successfully, we need to set environment varibles
# in :code:`~/.bashrc` file. We can edit :code:`~/.bashrc`
# using :code:`vi ~/.bashrc` and add the line below (Assuming your TVM
# directory is in :code:`~/tvm`):
#
# .. code-block:: bash
#
# export PYTHONPATH=$PYTHONPATH:~/tvm/python
#
# To update the environment variables, execute :code:`source ~/.bashrc`.
######################################################################
# Set Up RPC Server on Device
# ---------------------------
# To start an RPC server, run the following command on your remote device
# (Which is Raspberry Pi in our example).
#
# .. code-block:: bash
#
# python -m tvm.exec.rpc_server --host 0.0.0.0 --port=9090
#
# If you see the line below, it means the RPC server started
# successfully on your device.
#
# .. code-block:: bash
#
# INFO:root:RPCServer: bind to 0.0.0.0:9090
#
######################################################################
# Prepare the Pre-trained Model
# -----------------------------
# Back to the host machine, which should have a full TVM installed (with LLVM).
#
# We will use pre-trained model from
# `MXNet Gluon model zoo <https://mxnet.apache.org/api/python/gluon/model_zoo.html>`_.
# You can found more details about this part at tutorial :ref:`tutorial-from-mxnet`.
from mxnet.gluon.model_zoo.vision import get_model
from PIL import Image
import numpy as np
# one line to get the model
block = get_model("resnet18_v1", pretrained=True)
######################################################################
# In order to test our model, here we download an image of cat and
# transform its format.
img_url = "https://github.com/dmlc/mxnet.js/blob/main/data/cat.png?raw=true"
img_name = "cat.png"
img_path = download_testdata(img_url, img_name, module="data")
image = Image.open(img_path).resize((224, 224))
def transform_image(image):
image = np.array(image) - np.array([123.0, 117.0, 104.0])
image /= np.array([58.395, 57.12, 57.375])
image = image.transpose((2, 0, 1))
image = image[np.newaxis, :]
return image
x = transform_image(image)
######################################################################
# synset is used to transform the label from number of ImageNet class to
# the word human can understand.
synset_url = "".join(
[
"https://gist.githubusercontent.com/zhreshold/",
"4d0b62f3d01426887599d4f7ede23ee5/raw/",
"596b27d23537e5a1b5751d2b0481ef172f58b539/",
"imagenet1000_clsid_to_human.txt",
]
)
synset_name = "imagenet1000_clsid_to_human.txt"
synset_path = download_testdata(synset_url, synset_name, module="data")
with open(synset_path) as f:
synset = eval(f.read())
######################################################################
# Now we would like to port the Gluon model to a portable computational graph.
# It's as easy as several lines.
# We support MXNet static graph(symbol) and HybridBlock in mxnet.gluon
shape_dict = {"data": x.shape}
mod, params = relay.frontend.from_mxnet(block, shape_dict)
# we want a probability so add a softmax operator
func = mod["main"]
func = relay.Function(func.params, relay.nn.softmax(func.body), None, func.type_params, func.attrs)
######################################################################
# Here are some basic data workload configurations.
batch_size = 1
num_classes = 1000
image_shape = (3, 224, 224)
data_shape = (batch_size,) + image_shape
######################################################################
# Compile The Graph
# -----------------
# To compile the graph, we call the :py:func:`relay.build` function
# with the graph configuration and parameters. However, You cannot to
# deploy a x86 program on a device with ARM instruction set. It means
# Relay also needs to know the compilation option of target device,
# apart from arguments :code:`net` and :code:`params` to specify the
# deep learning workload. Actually, the option matters, different option
# will lead to very different performance.
######################################################################
# If we run the example on our x86 server for demonstration, we can simply
# set it as :code:`llvm`. If running it on the Raspberry Pi, we need to
# specify its instruction set. Set :code:`local_demo` to False if you want
# to run this tutorial with a real device.
local_demo = True
if local_demo:
target = tvm.target.Target("llvm")
else:
target = tvm.target.arm_cpu("rasp3b")
# The above line is a simple form of
# target = tvm.target.Target('llvm -device=arm_cpu -model=bcm2837 -mtriple=armv7l-linux-gnueabihf -mattr=+neon')
with tvm.transform.PassContext(opt_level=3):
lib = relay.build(func, target, params=params)
# After `relay.build`, you will get three return values: graph,
# library and the new parameter, since we do some optimization that will
# change the parameters but keep the result of model as the same.
# Save the library at local temporary directory.
tmp = utils.tempdir()
lib_fname = tmp.relpath("net.tar")
lib.export_library(lib_fname)
######################################################################
# Deploy the Model Remotely by RPC
# --------------------------------
# With RPC, you can deploy the model remotely from your host machine
# to the remote device.
# obtain an RPC session from remote device.
if local_demo:
remote = rpc.LocalSession()
else:
# The following is my environment, change this to the IP address of your target device
host = "10.77.1.162"
port = 9090
remote = rpc.connect(host, port)
# upload the library to remote device and load it
remote.upload(lib_fname)
rlib = remote.load_module("net.tar")
# create the remote runtime module
dev = remote.cpu(0)
module = runtime.GraphModule(rlib["default"](dev))
# set input data
module.set_input("data", tvm.nd.array(x.astype("float32")))
# run
module.run()
# get output
out = module.get_output(0)
# get top1 result
top1 = np.argmax(out.numpy())
print("TVM prediction top-1: {}".format(synset[top1]))
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,306 |
\section{Introduction}
\label{intro}
\begin{figure}[tbp]
\includegraphics[width=0.95\columnwidth, clip]{ph_diag_z2_large.eps}
\caption{Sketch of the phase diagram of the 3D ${\mathbb Z}_2$ gauge
Higgs model (\ref{HiggsH}). The dashed line is the self-dual line,
cf. Eq.~(\ref{selfdual}), the thick line corresponds to first-order
transitions on the self-dual line, extending for a finite interval.
The two lines labelled ``${\mathbb Z}_2$" are related by duality,
cf. Eq.~(\ref{dualitymap}), and correspond to Ising-like continuous
transitions. They end at $J = J_{\rm Is} \approx 0.22165$,
$\kappa=\infty$ and at $J =0$, $\kappa = \kappa_c \approx 0.76141$.
The three lines are conjectured to meet at a multicritical point
(MCP) on the self-dual line, at $[\kappa^\star\approx
0.7525,J^\star\approx 0.2258]$. We argue in the paper that the
multicritical behavior belongs to the $XY$ universality class. The
other endpoint of the first-order transition line should give rise
to a critical endpoint (CEP). }
\label{phadia}
\end{figure}
The three-dimensional (3D) ${\mathbb Z}_2$ gauge Higgs model is one of
the simplest gauge theories with matter fields, that shows a
nontrivial phase diagram characterized by the presence of a
topological phase, see, e.g.,
Refs.~\onlinecite{Wegner-71,BDI-74,BDI-75,FS-79,Kogut-79,JSJ-80,HL-91
GGRT-03,Kitaev-03,VDS-09,TKPS-10,GHMS-11,DKOSV-11,WDP-12
Fradkin-book,Sachdev-19,SSN-21,Grady-21,HSAFG-21}. The model can
also be related to the quantum two-dimensional toric model in the
presence of external {\em magnetic} fields, by a quantum-to-classical
mapping~\cite{Wegner-71,Kitaev-03,TKPS-10}, and to a statistical
ensemble of membranes~\cite{HL-91,GHMS-11}.
A notable feature of the model~\cite{Wegner-71,FS-79,Kogut-79} is the
existence of a duality transformation, which relates the free energy
at different points of the phase
diagram~\cite{Wegner-71,BDI-75,FS-79,Kogut-79}. A particular line in
the phase diagram, which will play an important role in the following,
is the self-dual line which is left invariant by the duality
transformation. In Fig.~\ref{phadia} we sketch the phase diagram of
the model, in the space of the Hamiltonian parameters [they are
defined in Eq.~(\ref{HiggsH})]. It presents a topologically ordered
deconfined phase, delimited by two continuous Ising transition lines
that are related by duality. In the context of two-dimensional
quantum systems, such a topological ordered phase is realized in
${\mathbb Z}_2$ spin
liquids~\cite{RC-89,Kivelson-89,RS-91,Wen-91,SF-00,MSF-01}, which is
the phase of matter realized by the toric code~\cite{Kitaev-03}.
Moreover, the 3D ${\mathbb Z}_2$ gauge Higgs model presents a
first-order transition line running along the self-dual line, for a
limited range of the Hamiltonian
parameters~\cite{TKPS-10,GGRT-03,JSJ-80}.
The available numerical results are consistent with the existence of a
multicritical point (MCP), where the first-order transition line and
the two continuous Ising transition lines meet, see, e.g.,
Refs.~\onlinecite{TKPS-10,SSN-21}. Assuming the existence of the MCP, an
interesting question concerns the nature of the multicritical
behavior. This issue has been recently investigated in
Ref.~\onlinecite{SSN-21}, which reported apparently puzzling results,
leading to estimates of the critical exponents that are substantially
consistent with those of the $XY$ universality class. This may suggest
that the multicritical behavior at the MCP is controlled by the 3D
$XY$ fixed point, with an effective enlargement of the symmetry of the
multicritical modes to the continuous O(2) group. This scenario was
considered unlikely in Ref.~\onlinecite{SSN-21}, because of the unclear
relationship between the multicritical $XY$ behavior and the mutual
statistics of the condensing
quasiparticles~\cite{TKPS-10,VDS-09,GM-12,Burnell-18} along the two
distinct Ising transition lines meeting at the MCP. These mutual
statistics do not affect critical exponents on the Ising lines,
because only one of the two excitations is massless on them,
but both excitations must become massless at the MCP. Therefore, it is
not clear how their competition can give rise to the effective
enlargement of the symmetry at the MCP, as required by the $XY$
universality class.
In this paper we investigate the multicritical behavior at the MCP.
We argue that the multicritical behavior is controlled by the stable
$XY$ fixed point of the 3D multicritical Landau-Ginzburg-Wilson (LGW)
field theory with two competing scalar fields associated with the
${\mathbb Z}_2$ transition lines meeting at the
MCP~\cite{LF-72,FN-74,NKF-74,PV-02,CPV-03}. Duality properties play a
crucial role for the realization of the multicritical $XY$ scenario,
which implies an effective enlargement of the symmetry of the
multicritical modes, to the continuous symmetry group O(2). To
provide further support to this scenario, we also report some
numerical finite-size scaling (FSS) analyses of data from Monte Carlo
(MC) simulations.
The paper is organized as follows. In Sec.~\ref{model} we present the
3D lattice ${\mathbb Z}_2$ gauge Higgs model, and summarize the known
features of its phase diagram. In Sec.~\ref{multicr} we discuss the
multicritical theory appropriate for the MCP, and apply the
multicritical LGW field theory to predict a multicritical $XY$
behavior. In Sec.~\ref{numres} we report some numerical results
supporting the multicritical $XY$ scenario, obtained by FSS analyses
of MC simulations. Finally in Sec.~\ref{conclu} we draw our
conclusions.
\section{The ${\mathbb Z}_2$ gauge Higgs model}
\label{model}
\subsection{Hamiltonian and duality transformations}
\label{modelH}
We consider a lattice gauge model with ${\mathbb Z}_2$ gauge
invariance defined on a cubic 3D lattice with periodic boundary
conditions. The fundamental variables are Ising spins $s_{\bm x}=\pm
1$ defined on the lattice sites and Ising spins $\sigma_{{\bm
x},\mu}=\pm 1$ defined on the bonds ($\sigma_{{\bm x},\mu}$ is
associated with the bond starting from site ${\bm x}$ in the $\mu$
direction, $\mu=1,2,3$). The model is defined by the lattice
Hamiltonian~\cite{Wegner-71,FS-79,Kogut-79}
\begin{eqnarray}
&& H = - J \sum_{{\bm x},\mu}
s_{\bm x} \, \sigma_{{\bm x},\mu} \,
s_{{\bm x}+\hat{\mu}}
- \kappa \sum_{{\bm x},\mu>\nu} \Pi_{{\bm
x},\mu\nu}\,,
\label{HiggsH}\\
&&\Pi_{{\bm x},\mu\nu}=
\sigma_{{\bm
x},\mu} \,\sigma_{{\bm x}+\hat{\mu},\nu} \,\sigma_{{\bm
x}+\hat{\nu},\mu} \,\sigma_{{\bm x},\nu}\,.
\label{plaquette}
\end{eqnarray}
The corresponding partition function and free-energy density are
\begin{equation}
Z = \sum_{\{s,\sigma\}} e^{-\beta H(J,\kappa)}\,, \qquad
F(J,\kappa) = - {T\over L^d} \ln Z\,,
\label{partfuncmodel}
\end{equation}
where $\beta=1/T$ is the inverse temperature, and $L^d$ is the volume
of the system. This paper only consider three-dimensional systems, and
therefore $d=3$. However, when arguments are independent of the space
dimension, we keep $d$ generic. In the following, energies are
measured in units of $T$, which is equivalent to fix $\beta=1$ in
Eq.~\eqref{partfuncmodel}.
The model can be simplified by considering the so-called unitary
gauge. Indeed, the site variables $s_{\bm x}$ can be eliminated by
redefining $\sigma_{{\bm x},\mu}$ as
\begin{equation}
s_{\bm x} \, \sigma_{{\bm x},\mu} \, s_{{\bm x}+\hat{\mu}}\,\to\,
\sigma_{{\bm x},\mu} \,.
\label{gaugeun}
\end{equation}
Correspondingly, the partition function can be written as
\begin{eqnarray}
&& Z = \sum_{\{\sigma\}} e^{-H_{\rm ug}(J,\kappa)}\,,\label{zhiggs}\\
&& H_{\rm ug} = - J \sum_{{\bm x},\mu} \sigma_{{\bm x},\mu} - \kappa
\sum_{{\bm x},\mu>\nu} \Pi_{{\bm x},\mu\nu}\,.
\label{HiggsHug}
\end{eqnarray}
An important property of the 3D lattice ${\mathbb Z}_2$ gauge Higgs
model is the existence of a duality mapping~\cite{BDI-75} between the
Hamiltonian parameters, that leaves the partition function unchanged,
modulo a regular function of the parameters. If
\begin{eqnarray}
\left(J^\prime, \kappa^\prime\right)=
\left( -{1\over 2} {\rm ln}\,{\rm
tanh}\,\kappa\,, -{1\over 2} {\rm ln}\,{\rm tanh}\, J \right)\,,
\label{dualitymap}
\end{eqnarray}
we have \cite{BDI-75}
\begin{equation}
F(J^\prime, \kappa^\prime) = F(J, \kappa) - {3\over 2}
\ln[\sinh(2J)\sinh(2\kappa)]\,.
\label{dualityZ}
\end{equation}
One can also define a self-dual line,
\begin{equation}
D(J,\kappa) = J - J^\prime =
J + {1\over 2} {\rm ln}\,{\rm tanh}\,\kappa = 0\,,
\label{selfdual}
\end{equation}
where the duality transformation maps the model into itself, i.e.
$J^\prime = J$ and $\kappa^\prime = \kappa$. Note that $D(J,\kappa)$
is odd under the duality mapping $(J,\kappa) \to
(J^\prime,\kappa^\prime)$, i.e., $D(J,\kappa) = -
D(J^\prime,\kappa^\prime)$.
\subsection{The phase diagram}
\label{phasdiagr}
Some features of the phase diagram are well established, see, e.g.,
Refs.~\onlinecite{FS-79,TKPS-10,SSN-21}. A sketch of the phase diagram is
shown in Fig.~\ref{phadia}. For $\kappa\to\infty$ an Ising transition
occurs at~\cite{FXL-18} $J_{\rm Is} = 0.221654626(5)$. By duality, in
the pure ${\mathbb Z}_2$ gauge model a transition occurs in the
corresponding point, $J=0$ and
\begin{equation}
\kappa_c = -{1\over 2} {\rm ln}\,{\rm tanh}\,J_{\rm Is} =
0.761413292(11)\,.
\label{z2gaugecr}
\end{equation}
Two Ising-like continuous transition lines, related by the duality
transformation (\ref{dualitymap}), start from these
points~\cite{FS-79} and intersect along the self-dual
line~\cite{SSN-21}. Moreover, some numerical
studies~\cite{TKPS-10,GGRT-03} have provided evidence of first-order
transitions along the self-dual line, in the relatively small interval
\begin{equation}
0.688 \lesssim \kappa \lesssim 0.753\,,\qquad
0.258\gtrsim J \gtrsim 0.226\,.
\label{foint}
\end{equation}
Since the first-order transition line is limited to an interval along
the self-dual line, there are only two phases, separated by the two
continuous transition lines, see Fig.~\ref{phadia}. For small $J$ and
large $\kappa$ there is a topological deconfined phase. The remaining
part of the phase diagram corresponds to a single phase that extends
from the disordered small-$J,\kappa$ region to the whole large-$J$
region. In particular, no phase transition occurs along the line
$\kappa=0$, where the model (\ref{HiggsHug}) becomes trivial.
A natural conjecture is that the first-order and the two continuous
Ising transition lines meet at the same point located along the
self-dual line, giving rise to a multicritical point (MCP). Numerical
results~\cite{SSN-21,TKPS-10} are consistent with this conjecture. In
particular, Ref.~\onlinecite{SSN-21} reported evidence of a critical
transition point along the self-dual line---we identify it with the
MCP---with critical parameters $\kappa^\star \approx 0.7526$ and
$J^\star\approx 0.2257$. The corresponding critical exponents are
close to, and substantially consistent with, those associated with the
$XY$ universality class~\cite{PV-02,CPV-03,CHPV-06,HV-11}. In spite
of these results, Ref.~\onlinecite{SSN-21} considered an $XY$ multicritical
behavior unlikely. In the paper, we rediscuss the issue, and give
additional theoretical and numerical arguments that support the
hypothesis that the MCP belongs to the $XY$ universality class.
We finally note that the first-order transition line starting from the
MCP ends at $J\approx 0.258$ and $\kappa\approx 0.688$. We expect
this endpoint to correspond to a continuous transition, likely
belonging to the Ising universality class.
\section{Multicritical behavior}
\label{multicr}
As discussed above, the phase diagram of the lattice ${\mathbb Z}_2$
gauge Higgs model shows a MCP, where a first-order and two continuous
transition lines meet (this MCP is usually called
bicritical~\cite{LF-72,FN-74,NKF-74}). In the following, we first
discuss the expected behavior of the model close to the MCP, on the
basis of the renormalization-group (RG) theory. Then, we discuss a
a LGW field theory characterized by two interacting local real scalar
fields~\cite{LF-72,FN-74,NKF-74,CPV-03}, which may describe the
multicritical behavior.
\begin{figure}[tbp]
\includegraphics[width=0.95\columnwidth, clip]{ph_diag_z2_small.eps}
\caption{Sketch of the phase diagram close to the MCP. We report the
first-order transition line (thick line), the self-dual line (dashed
line), the two continuous transition lines (continuous lines), and
the line (dotted line) where $u_2 = 0$. The line $u_1 = 0$ coincides
with the self-dual line. }
\label{phadiabis}
\end{figure}
\subsection{Multicritical scaling theory}
\label{sec3A}
At a MCP, the singular part of the free-energy density
can be written as
\begin{equation}
F_{\rm sing}(J,\kappa,L) = L^{-d} {\cal F}(\{u_i L^{y_i}\})\,,
\label{freeen}
\end{equation}
where $u_i$ are the nonlinear scaling fields and the RG exponents
$y_i$ are ordered so that
\begin{equation}
y_1>y_2>y_3>y_4 > ...\,,
\label{defyi}
\end{equation}
In the present model, we expect two relevant RG
perturbations. Therefore, $y_1$ and $y_2$ are positive, and the
corresponding scaling fields $u_1$ and $u_2$ vanish at the MCP. The
exponents $y_i$ with $i\ge 3$ are instead negative and control the
corrections to the multicritical behavior. All the scaling fields
$u_i$ are analytic functions of the model parameters $J$ and
$\kappa$. In the infinite-volume limit and neglecting subleading
corrections, we can rewrite the singular part of the free energy
density as
\begin{eqnarray}
&&F_{\rm sing}(J,\kappa) = |u_2|^{d/y_2} {\cal F}_\pm (X)\,,
\label{freeen2}\\
&& X \equiv
u_1 |u_2|^{-\phi}\,, \qquad \phi\equiv {y_1/y_2}>1\,,
\label{phidef}
\end{eqnarray}
where the functions ${\cal F}_\pm(X)$ apply to the parameter regions
in which $\pm u_2 > 0$, and $\phi$ is the so-called crossover exponent
associated with the MCP. Close to the MCP, the transition lines
follow the equation
\begin{equation}
X = u_1 |u_2|^{-\phi} = {\rm const}\,,
\label{traju1u2}
\end{equation}
with a different constant for each transition line. Since $\phi > 1$,
they are tangent to the line $u_1 = 0$.
The duality mapping (\ref{dualitymap}), and in particular
Eq.~(\ref{dualityZ}), implies the relation
\begin{equation}
F_{\rm sing}(J^\prime,\kappa^\prime) = F_{\rm sing}(J,\kappa)\,.
\label{fsingdua}
\end{equation}
Then, if we set
\begin{equation}
u^\prime_1 =
u_1(J^\prime,\kappa^\prime)\,,\quad u^\prime_2 =
u_2(J^\prime,\kappa^\prime)\,,
\label{u1u2p}
\end{equation}
using Eq.~(\ref{freeen2}) we obtain the equality
\begin{equation}
|u_2|^{d/y_2}{\cal F}_\pm (u_1 |u_2|^{-\phi}) =
|u_2^\prime|^{d/y_2}{\cal F}_\pm (u_1^\prime |u_2^\prime|^{-\phi})\,.
\label{eqSF-freeenergy}
\end{equation}
Since along the self-dual line $u_1=u_1^\prime=0$, this relation can only be
satisfied if $|u_2| = |u_2^\prime|$. If we then expand the scaling
function ${\cal F}_\pm(X)$ in powers of $X$, Eq.~(\ref{eqSF-freeenergy})
implies $u_1^m = (u_1^\prime)^m$ for all values of $m$ such that the
derivative
\begin{equation}
{\cal F}_{m} = \left. {\partial^{m} {\cal F}(X)\over \partial X^m}
\right \vert_{X=0}
\label{deffm}
\end{equation}
is nonvanishing. This condition can be satisfied only if $u_1$ changes
at most by a sign under duality. As we shall argue below, $u_1$ is odd
under duality, i.e., $u_1^\prime = -u_1$. In this case, we should
additionally require ${\cal F}_m = 0$ for any odd $m$: the functions
${\cal F}_\pm(X)$ are even in $X$.
To show that $u_1$ is odd, note that, as discussed in
Sec.~\ref{phasdiagr}, the first-order transition line runs along the
self-dual line (\ref{selfdual}) ending at the MCP, located at
\begin{equation}
J = J^\star,\quad \kappa = \kappa^\star =
-{1\over 2} {\rm ln}\,{\rm tanh}\,J^\star\,.
\label{tcpco}
\end{equation}
This transition line is expected to coincide~\cite{LF-72,FN-74,NKF-74}
with the line $u_1 = 0$, close to the MCP. Since the self-dual line is
given by $D(J,\kappa) = 0$, we can make the identification
\begin{eqnarray}
u_1 = D(J,\kappa)\,,
\label{nonlinu1}
\end{eqnarray}
close to the MCP. As noted before, $D(J,\kappa)$ is odd under duality.
The scaling field $u_2$ is then necessarily even under duality and is
therefore given by
\begin{equation}
u_2(J,k) = - J + J^\star +
{1\over 2} \ln {\tanh \kappa^{\phantom{\star}} \over \tanh\kappa^\star}\,.
\label{u2def}
\end{equation}
The scaling fields can be straightforwardly linearized obtaining
\begin{eqnarray}
&&u_1 \approx \Delta J + c\, \Delta \kappa\,, \qquad
u_2 \approx - \Delta J + c \,\Delta \kappa\,,
\label{ulinear}
\end{eqnarray}
where
\begin{eqnarray}
&&\Delta J = J-J^\star\,,\qquad\Delta \kappa = \kappa-\kappa^\star\,,
\nonumber\\
&&c = \sinh(2J^\star) = {1\over \sinh(2\kappa^\star)} \approx 0.467\,.
\label{defcconstant}
\end{eqnarray}
In terms of $u_1$ and $u_2$, close to the MCP the first-order
transition line corresponds to $ X = 0$, $u_2<0$. The two continuous
transition lines are defined by $X = \pm k$ with $u_2 > 0$.
Using the above results we can also predict how the latent heat
$\Delta_h$ vanishes along the first-order transition line when
approaching the MCP. A straightforward scaling
argument~\cite{CPPV-04} gives
\begin{equation}
\Delta_t \sim |u_2|^\theta\,,\qquad \theta = {d-y_1\over y_2}\,.
\label{latheat}
\end{equation}
Note that this scaling behavior is the same as that of the
magnetization $M$ at the Ising transition, with the correspondence
$y_1=y_h$ and $y_2=1/\nu$: $M$ indeed vanishes as $M\sim
|T-T_c|^\beta$ with $\beta = (d-y_h)\nu$, see, e.g.,
Ref.~\onlinecite{PV-02}.
\subsection{Scaling of the energy cumulants }
\label{sec3B}
Due to the fact that we are considering a lattice gauge theory, and
therefore that we cannot easily access the order parameters associated
with the phase transitions, we focus on the multicritical behavior of
the energy operators. We define
\begin{eqnarray}
H_J &=& \sum_{x\,\mu} \sigma_{x,\mu}\,, \qquad
H_\kappa = \sum_{x\,\mu>\nu} \Pi_{x,\mu\nu}\,,\label{hamjk}\\
H &=& -J H_J - \kappa H_\kappa\,.\nonumber
\end{eqnarray}
We consider the cumulants
\begin{equation}
C_{nm} = - L^d {\partial^{n+m} \over \partial J^n \partial
\kappa^m} F(J,\kappa,L)\,,
\label{cumulant-def}
\end{equation}
where $F$ is the free-energy density. For $n+m\le 3$, $C_{nm} =
M_{nm}$, where $M_{nm}$ are the central moments defined by
\begin{equation}
M_{nm} = \langle (H_J - E_J)^n (H_\kappa - E_\kappa)^m \rangle\,,
\label{nmomdef}
\end{equation}
with $E_J = \langle H_J \rangle$ and $E_\kappa = \langle H_\kappa
\rangle$. For $n+m\ge 4$, central moments and cumulants differ. For
instance, $C_{40} = M_{40} - 3 M_{20}^2$.
Using the cumulants $C_{mn}$ we can easily construct the cumulants
$K_n$ of the total energy $H$, defined by the derivatives of $\ln Z$
with respect to $\beta$, see Eq.~\eqref{partfuncmodel}. For example,
we have
\begin{eqnarray}
K_2 &=& J^2 C_{20} + 2 J \kappa C_{11} + \kappa^2 C_{02}\,,\label{kncnm}\\
K_3 &=& - \left( J^3 C_{30} + 3 J^2 \kappa C_{21} + 3 J \kappa^2 C_{12} +
\kappa^3 C_{03}\right)\,,\nonumber
\end{eqnarray}
etc... Note that the specific heat is given by
$C_V=K_2/V$.
In the following, we consider periodic boundary conditions, which
preserve the duality property in finite-size systems. Using
Eq.~(\ref{dualityZ}), and taking the appropriate derivatives with
respect to $J$ and $\kappa$, we can obtain an infinite series of exact
relations among the expectation values $E_J,\,E_\kappa$ and the
cumulants $C_{mn}$ at $(J,\kappa)$ and at the corresponding
duality-transformed couplings $(J^\prime, \kappa^\prime)$,
cf. Eq.~(\ref{dualitymap}). Along the self-dual line where
$(J,\kappa)=(J^\prime, \kappa^\prime)$, they turn into an infinite
series of exact relations among expectation values of cumulants
computed on the self-dual line. The lowest-order cumulants satisfy
the relations
\begin{eqnarray}
&& E_{\kappa}+\sinh(2J)\,E_J-3\cosh(2J)L^3=0\,,
\label{exadualrel1}\\
&& \sinh^2(2J)\,C_{20}-C_{02}-2\cosh(2J)\,E_{\kappa}+6L^3=0\,.\qquad
\label{exadualrel2}
\end{eqnarray}
Relations for higher-order cumulants are more cumbersome. Neglecting
the regular terms arising from the second term of the r.h.s. of
Eq.~(\ref{dualitymap}), third-order cumulants satisfy the relations
\begin{eqnarray}
&& C_{12} + \sinh(2J)\,C_{21} + 2 \cosh(2J) \,C_{11} \approx
0\,, \label{thirdcurel} \\
&& C_{03} + \sinh^3(2J) \,C_{30}+ 6 \cosh(2J)\,C_{02} +\nonumber\\
&&\quad + 2 [3 + \cosh(4J)]E_\kappa \approx 0 \,.
\nonumber
\end{eqnarray}
The scaling behavior of the cumulants $C_{nm}$ can be derived by
differentiating the asymptotic scaling relation
\begin{equation}
F_{\rm sing}(J,\kappa,L) \approx L^{-d} f(x_1, x_2)\,, \qquad x_i =
u_i L^{y_i}\,,
\label{scalF-finiteV}
\end{equation}
where we only keep the relevant RG contributions. Note that the
duality relation (\ref{dualityZ}) for the free energy, and the duality
properties of $u_1$ and $u_2$, imply that
\begin{equation}
f(-x_1,x_2) = f(x_1,x_2)\,.
\label{SF-funf}
\end{equation}
Introducing the derivatives
\begin{equation}
f_{n,m}(x_1,x_2) = {\partial^{n+m} f(x_1,x_2) \over \partial x_1^n
\partial x_2^m}\,,
\label{calcnmdef}
\end{equation}
the leading critical contribution is generally given by
\begin{eqnarray}
C_{nm}(J,\kappa,L) \approx u_{1,J}^n u_{1,\kappa}^m L^{(n+m)y_1}
f_{n+m,0}(x_1,x_2) \,,
\label{genscalingCnm}
\end{eqnarray}
where $u_{1,J}$ and $u_{1,\kappa}$ are the derivatives of $u_1$ with
respect to $J$ and $\kappa$.
The cumulants of the total energy are expected to develop an
analogous scaling behavior, i.e.
\begin{eqnarray}
K_n(J,\kappa,L) \approx L^{n y_1} {\cal K}_{n}(x_1,x_2) \,.
\label{genscalingK}
\end{eqnarray}
Along the self-dual line $u_1=0$ the duality symmetry leads to some
cancellations, as a consequence of Eq.~(\ref{SF-funf}). For $n+m$
even, the leading terms of the cumulants $C_{nm}$ are given by
\begin{eqnarray}
C_{nm}(J,\kappa,L) &\approx& u_{1,J}^n u_{1,\kappa}^m L^{(n+m)y_1}
f_{n+m,0}(0,x_2) \nonumber \\
&\approx& c^m L^{(n+m) y_1} f_{n+m,0}(0,x_2) \,.
\label{scaling-even}
\end{eqnarray}
Note that Eq.~(\ref{scaling-even}) is consistent with the exact
relations derived from duality, such as Eq.~(\ref{exadualrel2}). For
$n+m$ odd, the relation (\ref{SF-funf}) implies that
\begin{equation}
f_{n+m,0}(0,x_2)= 0\,.
\label{cmnx10}
\end{equation}
Therefore, the leading scaling behavior is obtained by differentiating
once with respect to $u_2$. Thus, for odd $n+m$ we obtain
\begin{eqnarray}
C_{nm} &\approx& L^{(n+m-1) y_1 + y_2} f_{n+m-1,1}(0,x_2) \times
\label{scaling-odd}\\
&&\;\;\times (n\ u_{1,J}^{n-1} u_{1,\kappa}^{m} u_{2,J} +
m\ u_{1,J}^n u_{1,\kappa}^{m-1} u_{2,\kappa}) \nonumber \\ &\approx& (m -
n) c^m L^{(n+m-1) y_1 + y_2} f_{n+m-1,1}(0,x_2) \,,
\nonumber
\end{eqnarray}
where $u_{2,J}$ and $u_{2,\kappa}$ are the derivatives of $u_2$ with
respect to $J$ and $\kappa$, respectively. Using these asymptotic
behaviors along the self-dual line and the relations (\ref{kncnm}), we
can also derive the corresponding asymptotic FSS of the cumulants
$K_n$ of the total energy, which behave as
\begin{eqnarray}
K_n & \approx & L^{ny_1}\, \widetilde{\cal K}_n(x_2) \quad {\rm for}
\;\;{\rm even}\;\; n\,,
\label{Hcumulants-scaling-even}\\
K_n &\approx& L^{(n-1)y_1+y_2} \, \widetilde{\cal K}_n(x_2)
\quad {\rm for} \;\;{\rm odd}\;\; n\,.
\label{Hcumulants-scaling-odd}
\end{eqnarray}
It is also useful to consider combinations whose cumulants have
definite properties under duality transformations. We
define
\begin{eqnarray}
A &=& H_J - \sinh(2\kappa)\,H_\kappa\,, \label{defA}\\
S &=& H_J + \sinh(2\kappa)\,H_\kappa\,.\label{defS}
\end{eqnarray}
Since
\begin{eqnarray}
{\partial u_1\over \partial J} + \sinh(2\kappa)
{\partial u_1 \over \partial \kappa}
&=& 0\,, \\
{\partial u_2\over \partial J} - \sinh(2\kappa)
{\partial u_2 \over \partial \kappa} &=& 0 \,,\nonumber
\end{eqnarray}
one can easily check that the cumulants $A_{n}$ of the operator $A$,
defined in Eq.~(\ref{defA}), do not receive contributions associated
with the scaling field $u_1$. Therefore, they generally scale as
\begin{eqnarray}
A_{n} \approx L^{n y_2} {\cal A}_n(x_1, x_2) \,,
\quad {\cal A}_n =
(-2)^n f_{0n}(x_1,x_2) \,.
\label{HnAsca}
\end{eqnarray}
The cumulants $S_{n}$ of the operator $S$ behave as
\begin{eqnarray}
S_{n} \approx L^{n y_1} {\cal S}_n(x_1,x_2) \,, \quad
{\cal S}_n =
2^n f_{n0}(x_1,x_2) \,.
\label{HnSsca}
\end{eqnarray}
Along the self-dual line, however, this diverging behavior is not
observed for $n$ odd, since $f_{n0}(0,x_2)=0$, thus $S_n$ is expected
to diverge as $L^{(n-1) y_1}$.
We finally note that the above scaling equations assume that the
leading contribution is due to the singular part of the free
energy. However, contributions due to the regular free-energy term, of
order $L^d$, may provide the leading contribution for the lowest-order
cumulants, depending on the values of the RG exponents $y_1$ and
$y_2$.
\subsection{Multicritical field theory}
\label{sec3C}
The results of Sections \ref{sec3A} and \ref{sec3B} only rely on the
existence of a duality transformation and make no assumption on the
nature of the MCP. To go further and make more quantitative
predictions, it is crucial to understand the nature of the order
parameters. Along the finite-$J$ transition line that ends at $\kappa
= \infty$, the order parameter is expected to be a local function of
the $s_x$ fields, which should correspond to the Ising
magnetization. Of course, because of gauge invariance, any rigorous
definition requires the introduction of an appropriate gauge fixing,
which however would not change any gauge-invariant correlation
function (in Ref.~\onlinecite{BN-87} this approach has been used to obtain
rigorous results for the phase behavior of the U(1) Abelian-Higgs
model). The order parameter for the ${\mathbb Z}_2$ gauge theory is
instead expected to be nonlocal and indeed the transition has a
topological nature. Apparently, this observation seems to indicate
that one cannot use standard symmetry arguments to understand the
critical behavior at the MCP, as they assume that the order parameters
are coarse-grained local functions of the microscopic fields.
We wish now to argue that, at the MCP (and only there), because of
duality, we can assume that both order parameters are local. Strictly
speaking, duality is only a mapping of the Hamiltonian parameters, but
here we will enlarge its role and assume that duality provides a
mapping for all RG operators. Essentially, let us assume that we are
working in the infinite-dimensional space of Hamiltonians on which the
RG transformations act~\cite{Wegner}. If we start from a ${\mathbb
Z}_2$ gauge Hamiltonian, under RG transformations, we will generate
a flow towards a ${\mathbb Z}_2$ gauge-invariant fixed point, while
starting from the usual Ising model, we will observe a flow towards
the Wilson-Fisher ${\mathbb Z}_2$ fixed point. The existence of an
exact microscopic relation between the ${\mathbb Z}_2$ gauge model and
the Ising model allows us to conjecture that the two fixed points are
equivalent, with the same set of RG dimensions and operators. In other
words, there is a mapping (we call it duality) between all RG
operators at the different fixed points. It is then plausible that
this duality transformation maps the local order parameter of the
Ising model to the nonlocal order parameter of the gauge model. The
mapping changes the Hamiltonian parameters, except on the self-dual
line, and therefore at the MCP. Here, the mapping would imply the
equivalence of the local and of the nonlocal order parameters for the
same model. Therefore, it seems reasonable to describe the
multicritical behavior in terms of two local quantities. We thus
consider two different scalar fields $\varphi_1({\bm x})$ and
$\varphi_2({\bm x})$, associated with the two transition lines.
To derive a Lagrangian for the effective model, we note that the
theory should be invariant under a change of sign of both fields, so
that only even powers of each field are allowed. Under these
conditions the LGW Hamiltonian is~\cite{LF-72,FN-74,NKF-74}
\begin{eqnarray}
{\cal H} &=&
\frac{1}{2} \Bigl[ ( \partial_\mu \varphi_1)^2 + (
\partial_\mu \varphi_2)^2\Bigr] +
\frac{1}{2} \Bigl( r_1 \varphi_1^2 + r_2 \varphi_2^2 \Bigr)
\nonumber\\
&&+
\frac{1}{4!} \Bigl[ v_1 \varphi_1^4 + v_2 \varphi_2^4 +
2 w\, \varphi_1^2\varphi_2^2 \Bigr] \,.
\label{bicrHH}
\end{eqnarray}
This model has been studied at length. In the mean-field
approximation~\cite{LF-72,FN-74,NKF-74}, the field theory
(\ref{bicrHH}) admits a bicritical point analogous to the one
appearing in Fig.~\ref{phadiabis}. Moreover, if the transition is
continuous, it should belong to the $XY$ universality class
\cite{LF-72,FN-74,NKF-74,PV-02,CPV-03} thereby leading to an effective
enlargement of the symmetry from ${\mathbb Z}_2\oplus{\mathbb Z}_2$ to
O(2).
Several field-theoretical and numerical works have determined the
exponents $y_i$ entering the multicritical scaling ansatz
(\ref{freeen}), see, e.g., Refs.~\onlinecite{CPV-03,HV-11}. As shown in
Ref.~\onlinecite{CPV-03}, the leading exponents correspond to the RG
dimensions at the isotropic $XY$ fixed point of quadratic and quartic
perturbations that belong to different representations of O(2)
group. The leading RG exponent $y_1$ is associated with the quadratic
spin-two perturbation. The corresponding RG dimension
is~\cite{CPV-03,HV-11}
\begin{eqnarray}
y_1 = 1.7639(11)\,.
\label{y1XY}
\end{eqnarray}
The second largest exponent is associated with the spin-zero
quadratic operator, and is directly related to the
correlation-length critical exponent at standard $XY$ transitions:
\begin{equation}
y_2 = {1\over \nu_{xy}} = 1.4888(2)\,,
\label{y2XY}
\end{equation}
where we used the estimate $\nu_{xy}=0.6717(1)$ (see, e.g.,
Refs.~\onlinecite{GZ-98,PV-02,CHPV-06,KP-17,Hasenbusch-19,CLLPSSV-20} for
theoretical results by various methods). Using the above results, we
can estimate the crossover exponent,
\begin{equation}
\phi = y_1/y_2 = 1.1848(8)\,.
\label{phiXY}
\end{equation}
Scaling corrections at the multicritical $XY$ point are controlled by
the negative RG dimensions $y_i$. The most relevant ones are
\cite{CPV-03,CPV-00,HV-11,Hasenbusch-19}
\begin{eqnarray}
y_3 &=& -0.108(6)\,, \label{y3XY} \\
y_4 &=& -0.624(10)\,, \\
y_5 &=& -0.789(4)\,, \label{y5est}
\end{eqnarray}
which are related to the spin-4, spin-2, and spin-zero quartic
perturbations, respectively. Note that, at standard $XY$ transitions,
corrections decay as $L^{-\omega}$ with $\omega = - y_5 \approx 0.79$. At
the MCP, scaling corrections decay much slower, as $L^{y_3}\approx
L^{-0.108}$, which may complicate the analysis of the universal
multicritical $XY$ behavior. Moreover, corrections with any integer
combination of the subleading exponents are also expected, and thus
corrections $L^{ny_3}$ with $n=2,3,\ldots$ should also appear.
In the LGW approach the analogue of the duality mapping is the
exchange of the two fields ($\varphi_1\to\varphi_2$,
$\varphi_2\to\varphi_1$). The RG operators associated with the scaling
fields $u_i$ must have definite properties under these
transformations. The leading operator of RG dimension $y_1$ is odd
under the field exchange. This implies that $u_1$ is odd under the
simultaneous exchange of $r_1$, $r_2$ and of $v_1$, $v_2$. In the
${\mathbb Z}_2$ gauge Higgs model this implies that the scaling field
$u_1$ is odd under duality, in agreement with the arguments presented
in Sec.~\ref{sec3A}. Analogously, we predict $u_2$ to be even, as
already discussed before. We can also predict the transformation
properties of the irrelevant scaling fields: $u_3$ and $u_5$ are even
functions under duality, while $u_4$ is odd. In particular, there are
no corrections with exponent $y_4$ on the self-dual line.
\section{Numerical results}
\label{numres}
In this section we report some numerical results supporting our
hypothesis of an emerging $XY$ multicriticality at the MCP, as
discussed in the previous sections. For this purpose, we present FSS
analyses of MC simulations of the unitary-gauge model
(\ref{HiggsHug}), using a standard Metropolis upgrading of the
discrete spin link variables~\cite{Metropolis:1953am}. We consider
cubic systems of size $L$ with periodic boundary conditions.
We perform simulations along the self-dual line $u_1=0$ and along the
line $u_2 = 0$, measuring the energy cumulants defined in
Sec.~\ref{sec3B}. We verify the predicted FSS behavior, using the RG
exponents $y_1 = 1.7639(11)$ and $y_2 = 1.4888(2)$ of the $XY$
universality class. We should remark that the observation of the
asymptotic scaling behaviors predicted by the multicritical $XY$
scenario is made difficult by the existence of several sources of
slowly decaying scaling corrections. The leading ones decay very
slowly, as $L^{n y_3} \approx L^{- 0.108 n}$ with $n=1,2,\ldots$.
Then, we should consider terms decaying as $L^{-(y_1-y_2)} \approx
L^{-0.28}$ [they are absent on the self-dual line because of
Eq.~(\ref{SF-funf})], as $L^{-2(y_1-y_2)} \approx L^{-0.55}$,
$L^{-y_4}\approx L^{-0.62}$ (they are absent along the self-dual
line), $L^{-y_5} \approx L^{-0.79}$. For $m+n=2$ also the regular
background plays a role, giving rise to corrections of order
$L^{3-2y_1} \approx L^{-0.53}$.
\begin{figure}[tbp]
\includegraphics[width=0.95\columnwidth, clip]{u1zero_u2_H3.eps}
\caption{Scaling behavior of the third cumulant $K_3$ of the
Hamiltonian along the $u_1=0$ line as a function of $u_2 L^{y_2}$.
We use the $XY$ exponents $y_1 = 1.7639$, $y_2=1.4888$ and
$\kappa^{\star} = 0.7525$.}
\label{h3u10}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.95\columnwidth, clip]{u1zero_u2_H3A.eps}
\includegraphics[width=0.95\columnwidth, clip]{u1zero_u2_H4A.eps}
\caption{Scaling behavior of cumulants $A_{3}$ (top) and $A_{4}$ (bottom)
along the $u_1=0$ line as a function of $u_2 L^{y_2}$. We use the
$XY$ exponents $y_1 = 1.7639$, $y_2=1.4888$ and $\kappa^{\star} =
0.7525$. }
\label{HAu10}
\end{figure}
Along the self-dual line the scaling field $u_1$ vanishes. Thus,
according to the RG analysis reported in Sec.~\ref{sec3B}, we expect
that the asymptotic scaling behavior of the energy cumulants depends
on the FSS variable $x_2=u_2 L^{y_2}$. Along the self-dual line the
numerical FSS analyses of the energy cumulants $K_n\,,A_n\,,S_n$ are
consistent with the predictions of the multicritical theory, see
Eqs.~(\ref{Hcumulants-scaling-even}) and
(\ref{Hcumulants-scaling-odd}) for the total energy, once the $XY$
values of the RG exponents reported in Eqs.~(\ref{y1XY}) and
(\ref{y2XY}) are used. The most accurate estimate of the MCP point is
obtained by biased analyses of the third cumulant $A_{3}\sim L^{3y_2}$
of the operator $A$, see Eq.~(\ref{defA}), along the self-dual line,
using the $XY$ values for the exponents. Fitting the data to
Eq.~(\ref{HnAsca}), we obtain
\begin{equation}
\kappa^\star = 0.7525(1)\,,\quad J^\star = 0.22578(5)\,.
\label{mcpco}
\end{equation}
This estimate of the MCP is consistent with the results reported in
Ref.~\onlinecite{SSN-21}. The analysis of the other cumulants gives
consistent results.
The accuracy of the description in terms of the multicritical $XY$
predictions is demonstrated by the scaling plots of the data of the
cumulants using the $XY$ exponents and the estimates (\ref{mcpco}).
In Fig.~\ref{h3u10} we show data for the third cumulant $K_3$ of the
Hamiltonian, which is expected to scale with the power law $L^{2y_1 +
y_2}$, cf. Eq.~(\ref{Hcumulants-scaling-odd}). We observe a
reasonably good scaling: scaling corrections are hardly visible within
the statistical errors. Note that, according to the multicritical $XY$
scenario, one expects slowly decaying corrections with exponent
$|y_3|\approx 0.11$, cf. Eq.~(\ref{y3XY}). We do not observe them
here. In our range of values of $L$, $L^{y_3}$ varies only slightly,
and thus it is conceivable that they do not affect the divergent
behavior of the observables, but only the accuracy of the scaling
functions. In Fig.~\ref{HAu10} we report the scaling plots of $A_{3}$
and $A_{4}$. Data are again in good agreement with the theoretical
predictions for their asymptotic scaling behavior,
cf. Eqs.~(\ref{HnAsca}). We do not report the second cumulant
$A_{2}$, whose singular part should scale as $L^{2 y_2}$. Since $2 y_2
\approx 2.9775 < 3$, its behavior is dominated by the regular
contribution, that scales as the volume $L^3$.
\begin{figure}[t]
\includegraphics[width=0.95\columnwidth, clip]{u2zero_u1_H3.eps}
\includegraphics[width=0.95\columnwidth, clip]{u2zero_u1_Cv.eps}
\caption{ Scaling plot of the second cumulant $K_2$ (bottom)
and of the third cumulant $K_3$ (top) of the Hamiltonian
along the $u_2=0$ line, using the $XY$ exponent $y_1=1.7639$ and
$\kappa^{\star}=0.7525$. Data confirm the
scaling prediction, Eq.~(\ref{knscau20}) .
Notice that the error bars of $K_3$ for
$u_1 L^{y_1}\gtrsim 1$ may be underestimated. In this
region of the phase diagram we observe an increasing
inefficiency of the MC algorithm.}
\label{h3u20}
\end{figure}
Beside checking the consistency of the numerical data
with the multicritical $XY$ scenario, we can also perform unbiased
fits, to determine $y_1$ and $y_2$.
If we fit the third and fourth cumulant of the Hamiltonian
(they should scale as $K_3\sim L^{2 y_1 + y_2}$ and $K_4\sim L^{4
y_1}$, respectively) we obtain $2 y_1 + y_2 = 5.0(1)$ and $4 y_1 =
7.0(1)$, which are consistent with the predictions $2 y_1 + y_2
\approx 5.02$ and $4 y_1 \approx 7.06$. The exponent $y_2$ can also
be estimated from $A_{n}$. We obtain $y_2 = 1.495(10)$ and $y_2 =
1.48(2)$ from $A_{3}$ and $A_{4}$, respectively. The agreement with
the conjectured $XY$ values is quite good.
We also performed simulations along the $u_2=0$ line, see
Eq.~(\ref{u2def}), using the estimate $\kappa^{\star}=0.7525$ obtained
from the FSS analyses along the self-dual $u_1=0$ line. Along the
$u_2=0$ line, the asymptotic FSS of the cumulants of the total energy
is that given in Eq.~(\ref{genscalingK}), i.e.
\begin{equation}
K_n \approx L^{ny_1} {\cal K}_n(x_1,0)\,.
\label{knscau20}
\end{equation}
Note that, for $n$ odd, consistency with the FSS behavior along the
self-dual line, see Eq.~(\ref{Hcumulants-scaling-odd}), requires
${\cal K}_n(0,0) = 0$. The data are plotted in Fig.~\ref{h3u20}, We
observe a nice collapse of the data, again fully supporting the
multicritical $XY$ scenario.
\begin{figure}[t]
\includegraphics[width=0.95\columnwidth, clip]{u2zero_u1_H3A.eps}
\includegraphics[width=0.95\columnwidth, clip]{u2zero_u1_H3S.eps}
\caption{ Scaling plot of the third cumulants $A_{3}$ (top) and
$S_{3}$ (bottom) of the operators $A$ and $S$ along the $u_2=0$
line, using the $XY$ exponents $y_1=1.7639$ and $y_2=1.4888$, and
$\kappa^{\star}=0.7525$. }
\label{AS3u20}
\end{figure}
Finally, we also check the scaling behavior of the third cumulants of
$A$ and $S$ along the $u_2=0$ line, in Fig.~\ref{AS3u20}. The scaling
behavior of the cumulants of $A$ is given in Eq.~(\ref{HnAsca}). It
depends on $f_{0n}(x_1,0)$ which is always an even function of
$x_1$. The data shown in the top Fig.~\ref{AS3u20} are definitely
consistent within the errors. As for the cumulants of $S$, they scale
as reported in Eq.~(\ref{HnSsca}). Relation (\ref{SF-funf}) implies
that the odd (resp. even) cumulants are odd (resp. even) functions of
$x_1$. Again, this is confirmed by the data shown in the bottom
Fig.~\ref{AS3u20}. In particular, the ratio $S_3/L^{3 y_1}$ is
consistent with zero at the critical point $x_1=0$.
Note that statistical errors of the MC simulations along $u_2=0$ line
increase significantly in the region $x_1 \gtrsim 1$. The link update
algorithm for the model (\ref{HiggsHug}) becomes indeed less efficient
as $\kappa$ and $J$ are increased. Autocorrelation times increase by
more than one order of magnitude, likely due to a different dynamic
regime related to the presence of relevant nonlocal configurations,
which are hardly modified by local moves.
The results we have presented here complement those reported in
Ref.~\onlinecite{SSN-21}, which were already providing a remarkable evidence
of the multicritical $XY$ behavior (although the authors were quite
skeptical on its interpretation in terms of a multicritical $XY$
behavior). In particular, their estimates of the multicritical
exponents $y_1= 1.778(6)$ and $y_2=1.495(9)$ (other compatible, but
less precise, results were also reported in
Refs.~\onlinecite{VDS-09,DKOSV-11}) are in substantial agreement with the
$XY$ predictions (\ref{y1XY}) and (\ref{y2XY}). The small difference
in the estimate of $y_1$ can be easily explained by the very slowly
decaying scaling corrections predicted by the multicritical $XY$
scenario, that make a precise determination of the universal
asymptotic quantities very hard. The leading one vanishes as
$L^{-0.108}$, so that, to reduce it by a factor of two, the lattice
size must be increased by a factor of 600, which is unattainable in
practice.
Overall, we believe that the numerical results presented in this
paper, and those already reported in Ref.~\onlinecite{SSN-21}, provide
strong evidence of the multicritical $XY$ scenario put forward in the
previous sections.
\section{Conclusions}
\label{conclu}
We study the multicritical behavior of 3D ${\mathbb Z}_2$ gauge Higgs
model at the MCP, where one first-order transition line and two
continuous Ising transition lines meet, as sketched in
Fig.~\ref{phadia}. The duality properties of the model play a key role
in the phase diagram, and in determining the main features of the
multicritical behavior at the MCP located on the self-dual line.
We exploit duality to identify the scaling fields associated with the
relevant RG perturbations at the MCP, and outline the corresponding
multicritical scaling behaviors. Moreover, we present arguments
supporting the identification of the multicritical universality class
with the one controlled by the stable $XY$ fixed point of the 3D
multicritical LGW field theory (\ref{bicrHH}), with two competing
scalar fields associated with the continuous ${\mathbb Z}_2$
transition lines meeting at the MCP. The $XY$ nature of the MCP
implies an effective enlargement of the symmetry of the multicritical
modes, to the continuous group O(2).
We have also reported numerical FSS analyses of several energy
cumulants. The results are in good agreement with the theoretical
predictions based on the multicritical $XY$ scenario. We believe that
our numerical results, together with those already reported in the
literature, see, in particular, Ref.~\onlinecite{SSN-21}, provide a strong
evidence in favor of the multicritical $XY$ scenario at the MCP. Of
course, this picture calls for a deeper understanding of the
mechanisms that combine the local and nonlocal critical modes of the
${\mathbb Z}_2$ gauge Higgs model to give rise to the multicritical
$XY$ behavior, entailing an effective enlargement of the symmetry at
the MCP, to the continuous group O(2).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,287 |
About CBD Oil
Spreads Straight From The About Cbd Oil.
Health Regulations
Hotels Accommodations
City Guides and Information
Great Experiences and Great Holiday Specials in Perth, Western Australia
Perth, Western Australia, is a vibrant and attractive city with a unique selection of budget and exclusive accommodation with many things to do for the visitor such as attractions and tours.
Perth has always been a tourist destination in Australia due to its unique characteristics and gateway to outstanding natural scenery.
With many great accommodation specials and interesting tours, visitors get to enjoy the eclectic mix of modern and historical sites around the CBD whilst sampling the great selection of dining and bars.
Exploring Perth's CBD is an easy task. The city is compact and well planned, even though its location on a broad stretch of the Swan River gives it a spacious feeling.
There are tree-shaded walks to be found and picnics in the manicured gardens are a popular choice on hot afternoons.
With a lot to see and do, and an atmosphere of lazy sunny days, a good way to tour Perth is by public transport.
Activities close to accommodation in Perth include learning about the history of the area in the local museum or enjoying some of the local wine and cuisine in one of the many bistros or cafes.
Perhaps the best place to start is the western edge of the CBD, in Kings Park and the Botanic Gardens.
Other interesting attractions of the CBD area include the Fire Safety Education Centre and Museum in the original Perth City Fire Station; Francis Burt Law Education Centre and Museum near the Supreme Court Gardens.
On the banks of the Swan River, Barrack Square was originally built as a military parade facility. Today it is an attractively manicured garden square surrounding the unique Bell Tower , with a surrounding jetty of cafes, shops and a busy ferry terminal. The Swan Bells in the Tower include 12 original bells from St Martin-in-the-Fields Church in London, celebrated in the old nursery rhyme Oranges and Lemons.
Jump on a ferry across the Swan River to the South Perth Esplanade where the renowned Perth Zoo is located a few minutes away. It has one of Australia's best collections of native and exotic animals, which are in enclosures resembling their natural habitats. Wandering back along the Swan the length of the Sir James Mitchell Park , take a look at Heirisson Island, access is via the bicycle path from the City of Perth Causeway Carpark. The island contains a memorial to Yagan, an Aboriginal leader killed in 1833.
Returning to East Perth after visiting Heirisson Island, wander by Trinity College and the WACA Oval.
Due to its proximity to the airport and good public transport, visitors to Perth can enjoy both long stays and weekends in this cultural and enjoyable city.
More on Perth CBD, Maps, Accommodation and Tours: http://www.planbooktravel.com.au/regions/australia/wa/perth-and-fremantle
(Image Courtesy of Tourism Western Australia)
By giving anyone the opportunity to write about their town or experiences, the planbooktravel website is quickly becoming recognised as the trusted 'home' of user-generated content on Australian destination information.
Everything You Need To Know About Melbourne Nightlife
Great Experiences and Great Holiday Specials in Brisbane, Queensland
Comparison between CO2 extraction Methods and Ethanol extraction
How to Start a Home-Based CBD Business
Full Spectrum CBD A Complete Guide
How To Make A Great CBD Oil Bulk Wholesale Purchase?
Subscribe our email list | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,384 |
Produced by Al Haines
[Frontispiece: "'I HAVE A PRESENT FOR YOU--A SISTER'" See p. 45]
DR. LAVENDAR'S PEOPLE
BY
MARGARET DELAND
AUTHOR OF "OLD CHESTER TALES"
ILLUSTRATED BY
LUCIUS HITCHCOCK
NEW YORK AND LONDON
HARPER & BROTHERS PUBLISHERS
1903
Copyright, 1903, by HARPER & BROTHERS.
_All rights reserved._
Published October, 1903.
TO
DR. FRANCIS B. HARRINGTON
These Stories are
Dedicated
CONTENTS
The Apotheosis of the Reverend Mr. Spangler
The Note
The Grasshopper and the Ant
Amelia
"An Exceeding High Mountain"
At the Stuffed-Animal House
ILLUSTRATIONS
"I HAVE A PRESENT FOR YOU--A SISTER'" . . . . . . _Frontispiece_
"DAVID'S HEAD SWAM"
"SHE ALWAYS CAME INTO THE LIBRARY TO SAY GOOD-NIGHT TO HIM"
"LURCHED FORWARD INTO A CHAIR, BREATHING LOUDLY"
"MRS. BARKLEY ROSE, TAPPING THE TABLE WITH ALARMING LOUDNESS"
"MISS LYDIA, WATCHING HIM, GREW PALER AND PALER"
"THERE SHE TURNED AND LOOKED BACK"
"THOMAS DILWORTH GOT ON HIS FEET AND SWORE"
"'WHAT IS THE NAME OF THE KIND PERSON?'"
"SHE KNELT DOWN, AS USUAL, AT THE BIG CHINTZ-COVERED WINGED CHAIR"
MISS HARRIET WAS LEANING FORWARD
"'A HAPPY SLEEP,' MISS ANNIE REPEATED"
THE APOTHEOSIS
OF THE
REVEREND MR. SPANGLER
I
Miss Ellen Baily kept school in the brick basement of her old frame
house on Main Street.
The children used to come up a flagstone path to the side door, and
then step down two steps into an entry. Two rooms opened on this
entry; in one the children sat at small, battered desks and studied; in
the other Miss Baily heard their lessons, sitting at a table covered
with a red cloth, which had a white Grecian fret for a border and
smelled of crumbs. On the wall behind her was a faded print of
"Belshazzar's Feast"; in those days this was probably the only feasting
the room ever saw--although on a thin-legged sideboard there were two
decanters (empty) and a silver-wire cake-basket which held always three
apples. Both rooms looked out on the garden--the garden and, in fine
weather, _Mr. David Baily!_ ... Ah, me--what it was, in the dreary
stretches of mental arithmetic, to look across the flower-beds and see
Mr. David--tall and dark and melancholy--pacing up and down, sometimes
with a rake, oftener with empty hands; always with vague, beautiful
eyes fixed on some inner vision of heart-broken memory. Miss Ellen's
pupils were confident of this vision because of a tombstone in the
burial-ground which recorded the death of Maria Hastings, at the
romantic age of seventeen; and, as everybody in Old Chester knew, Mr.
Baily had been in love with this same seventeen-year-old Maria. To be
sure, it was thirty years ago; but that does not make any difference,
"_in real love_," as any school-girl can tell you. So, when David
Baily paced up and down the garden paths or sat in the sunshine under
the big larch we all knew that he was thinking of his bereavement.
In the opinion of the older girls, grief had wrecked Mr. David's life;
he had intended to be a clergyman, but had left the theological school
because his eyes gave out. "He cried himself nearly blind," the girls
told each other with great satisfaction. After that he tried one
occupation after another, but somehow failed in each; which was proof
of a delicacy of constitution induced by sorrow. Furthermore, he
seemed pursued by a cruel fortune--"Fate," the girls called it.
Elderly, unromantic Old Chester did not use this fine word, but it
admitted pursuing disaster.
For instance: there was the time that David undertook the charge of a
private library in Upper Chester, and three months afterwards the owner
sold it! Then Mr. Hays found a job for him, and just as he was going
to work he was laid up with rheumatism. And again Tom Dilworth got him
a place as assistant book-keeper; and David, after innumerable tangles
on his balance-sheet, was obliged to say, frankly, that he had no head
for figures. But he was willing to do anything else--"_any_ honest
work that is not menial," he said, earnestly. And Tom said, why, yes,
of course, only he'd be darned if he knew what to suggest. But he
added, in conjugal privacy, that David ought to be hided for not
turning his hand to something. "Why doesn't he try boot-blacking?
Only, I suppose, he'd say he couldn't make the change correctly. He
doesn't know whether two and two make five or three--like our Ned."
"Why, they make four, Tom," said Mrs. Dilworth. And Thomas stared at
her, and said, "You don't say so!"
There had been no end of such happenings; "and none of them my
brother's fault," Miss Ellen told the sympathetic older girls, who
glanced sideways at Mr. David and wished that they might die and be
mourned as Mr. David mourned Maria.
The fact was, the habit of failure had fastened upon poor David; and in
the days when Miss Ellen's school was in its prime (before the new
people told our parents that her teaching was absurdly inadequate), he
was depending on his sister for his bread-and-butter. That Miss Ellen
supported him never troubled the romantic souls of Miss Ellen's pupils
any more than it troubled Miss Ellen--or Mr. David. "Why shouldn't
she?" the girls would have demanded if any such rudely practical
question had been asked; "he is so delicate, _and he has a broken
heart!_" So that was how it happened that the pupils were able to have
palpitating glimpses of him, walking listlessly about the garden, or
dozing in a sunny window over an old magazine, or doing some pottering
bit of carpentering for Miss Ellen, but never losing his good looks or
the grieved melancholy of his expression.
Miss Ellen had been teaching for twenty years.
It is useless to deny that, unless one has a genius for imparting
knowledge, teaching is a drudgery. It was drudgery to Ellen Baily, but
she never slighted it on that account. She was conscientious about the
number of feet in the highest mountain in the world; she saw to it that
her pupils could repeat the sovereigns of England backward. Besides
these fundamentals, the older girls had Natural Philosophy every
Friday; it was not, perhaps, necessary that young ladies should know
that the air was composed of two gases (the girls who had travelled and
seen the lighted streets of towns knew what gas was), nor that rubbing
a cat's fur the wrong way in the dark would produce electric
sparks--such things were not necessary. But they were interesting,
and, as Mrs. Barkley said, if they did not go too far and lead to
scepticism, they would do no harm. However, Miss Ellen counteracted
any sceptical tendencies by reading aloud, every Saturday morning,
Bishop Cummings on the Revelation, so that even Dr. Lavendar was not
wiser than Miss Ellen's girls as to what St. John meant by "a time, and
a time, and a half of a time," or who the four beasts full of eyes
before and behind stood for. For accomplishments, there was fine
sewing every Wednesday afternoon; and on Mondays, with sharply pointed
pencils, we copied trees and houses from neat little prints; also, we
had lessons upon the piano-forte, so there was not one of us who, when
she left Miss Ellen's, could not play at least three pieces, viz., "The
Starlight Valse," "The Maiden's Prayer," and "The Last Rose of Summer."
Ah, well, one may smile. Compared to what girls know nowadays, it is,
of course, very absurd. But, all the same, Miss Ellen's girls knew
some things of which our girls are ignorant: reverence was one;
humility was another; obedience was a third. And poor, uneducated folk
(compared with our daughters) that we of Old Chester may be, we are, if
I mistake not, glad that we were taught a certain respect for our own
language, which, though it makes the tongue of youth to-day almost
unintelligible, does give us a joy in the wells of English undefiled
which our children do not seem to know; and for this, in our dull Old
Chester way, we are not ungrateful. However, this may all be sour
grapes....
At any rate, for twenty painstaking years Miss Ellen's methods fed and
clothed Mr. David. Then came the winter of Dr. Lavendar's illness, and
the temporary instalment of the Reverend Mr. Spangler, and Ellen Baily
realized that there were other things in the world than David's food
and clothes.
Dr. Lavendar, cross, unbelieving, protesting, was to be hustled down
South by Sam Wright; and the day before he started Mr. Spangler
appeared. That was early in February, and Dr. Lavendar was to come
back the first of May.
"Not a day sooner," said Sam Wright.
"I'll come when I see fit," said Dr. Lavendar. He didn't believe in
this going away, he said. "Home is the best place to be sick in. The
truth is, Willy King doesn't want me to die on his hands--it would hurt
his business," said Dr. Lavendar, wickedly; "I know him!"
But to Mr. Spangler Dr. Lavendar said other things about Willy, and Sam
Wright, too; in fact, about all of them. And he pulled out his big,
red silk pocket-handkerchief with a trembling flourish and wiped his
eyes. "I don't deserve it," he said. "I'm a dogmatic old fogy, and I
won't let the new people have their jimcrackery; and I preach old
sermons, and I've had a cold in my head for three months. And yet,
look at 'em: A purse, if you please! And Sam Wright is going down with
me. Sam ought to be ashamed of himself to waste his time; he's a busy
man. No, sir; I don't deserve it. And, if you take my advice, you'll
pray the Lord that your people will treat you as you don't deserve."
Mr. Spangler, a tall, lean man, very correctly dressed, who was
depended upon in the diocese as a supply, made notes solemnly while Dr.
Lavendar talked; but he sighed once or twice, patiently, for the old
man was not very helpful. Mr. Spangler wanted to know what
Sunday-school teachers could be relied upon, and whether the choir was
very thin-skinned, and which of the vestry had chips on their shoulders.
"None of 'em. I knocked 'em all off, long ago," said Dr. Lavendar.
"Don't you worry about that. Speak your mind."
"I have," said Mr. Spangler, coughing delicately, "an iron hand when I
once make up my mind in regard to methods; firmness is, I think, a
clergyman's duty, and duty, I hope, is my watchword; but I think it
best to canvass a matter thoroughly before making up my mind."
"It is generally wise to do so," said Dr. Lavendar, very meekly.
"Of course," Mr. Spangler said, kindly, "you belong to a somewhat older
period, and do not, perhaps, realize the value of our modern ways of
dealing with a parish--I mean in regard to firmly carrying out one's
own ideas. I suppose these good people do pretty much as they please,
so far as you are concerned?"
"Perhaps they do," said Dr. Lavendar, very, very meekly.
"So, not wishing to offend, I will ask a few questions: I have heard
that the parish is perhaps a little old-fashioned in regard to matters
of ritual? I have wondered whether my cassock would be misunderstood?"
"Cassock?" said Dr. Lavendar. "Bless your heart, wear a pea-jacket if
it helps you to preach the Word. It will only be for ten Sundays," he
added, hopefully.
The Reverend Mr. Spangler smiled at that; and when he smiled one saw
that his face, though timid, was kind.
So Dr. Lavendar, growling and scolding, fussing about Danny and his
little blind horse Goliath, and Mr. Spangler's comfort, was bundled
off; and Mr. Spangler settled down in the shabby rectory. His iron
will led him to preach in his surplice, and it was observed that a
silver cross dangled from his black silk fob. "But it's only for ten
weeks," said Old Chester, and asked him to tea, and bore with him, and
did nothing more severe than smile when he bowed in the creed--smile,
and perhaps stand up a little straighter itself.
This, of the real Old Chester. Of course the new people were pleased;
and one or two of the younger folk liked it. Miss Ellen Baily was not
young, but she liked the surplice better than Dr. Lavendar's black gown
and bands, and the sudden sparkle of the cross when Mr. Spangler knelt
gave her a pang of pleasure. David, too, was not displeased. To be
sure, David was rarely stirred to anything so positive as pleasure.
But at least he made no objections to the cross; and he certainly
brightened up when, on Saturday afternoon, Mr. Spangler called. He
even talked of Gambier, to which he had gone for a year, and of which,
it appeared, the clergyman was an alumnus. Miss Ellen had a pile of
compositions on the table beside her, and she glanced at one
occasionally so that she might not seem to expect any share in the
conversation. But, all the same, Mr. Spangler noticed her. He was not
drawn to the brother; still, he talked to him about their college, for
Mr. Spangler believed that being agreeable was just as much a
clergyman's duty as was changing the bookmarks for Advent or Lent; and
duty, as Mr. Spangler often said, was his watchword. Furthermore, he
was aware that his kindness pleased the silent, smiling woman seated
behind the pile of compositions.
It pleased her so much that that night, after David had gone to bed,
she went over to Mrs. Barkley's to talk about her caller.
"Well, Ellen Baily," Mrs. Barkley said, briskly, as Miss Baily came
into the circle of lamplight by the parlor-table, "so you had a visitor
to-day? I saw him, cross and all."
"It was a very small one," Miss Baily protested, "and only silver."
"Would you have had it diamonds?" demanded Mrs. Barkley, in a deep
bass. "Oh, well; it doesn't really matter; there are only nine more
Sundays. But Sam Wright says he shall mention it when he writes to Dr.
Lavendar."
"I suppose Dr. Lavendar saw it before he went away," Ellen said, with
some spirit.
"Well, if he doesn't take his religion out in crosses, I suppose it's
all right. But he's not a very active laborer in the vineyard. I
suppose you know about him?"
"Why, no," Ellen said; "nothing except that he supplies a good deal."
"Supplies? Yes, because his mother left him a house in Mercer, and
enough to live on in a small way; so he likes supplying better than
taking a charge where he'd have to work hard and couldn't have his
comforts."
"Why doesn't he take a charge where he could have his comforts?"
"Can't get the chance," Mrs. Barkley explained, briefly. "Not enough
of a preacher. And, besides, he likes his ease in Zion. Rachel
Spangler's old house, and her Mary Ann, and his father's library,
and--well, the flesh-pots of Mercer!--and supplying, just enough to buy
him his ridiculous buttoned-up coats. That's what he likes. I suppose
he uses the same old sermons over and over. Doesn't ever have to write
a new one. However, he's here, and maybe Old Chester will do him good.
Ellen Baily, did you know that we have a new-comer in Old Chester? A
widow. I don't like widows. Her name's Smily. Foolish name! She's
staying at the Stuffed Animal House. She's Harriet Hutchinson's
cousin, and she's come down on her for a visit."
"Maybe she'll make her a present when she goes away," said Ellen,
hopefully.
"Present! She needs to have presents made to her. She hasn't a cent
but what her husband's brother gives her. He's a school-teacher, I
understand; and you know yourself, Ellen Baily, how much a
school-teacher can do in that way?"
Miss Ellen sighed.
"Well," proceeded Mrs. Barkley, "I just thought I'd tell you about her,
because if we all invite her to tea, turn about, it will be a relief to
Harriet--(she isn't well, that girl; I'm really uneasy about her). And
I guess the Smily woman won't object to Old Chester food, either," said
Mrs. Barkley, complacently. "I've asked her for Tuesday evening, and I
thought I'd throw in Mr. Spangler and get him off my mind."
"David likes him so much," Miss Ellen began.
"Does he?" said Mrs. Barkley. "Well, tell him to come; he can talk to
Mr. Spangler. I'm afraid I might hurt the man's feelings if I had to
do all the talking. I seem to do that sometimes. Did you ever notice,
Ellen, that the truth always hurts people's feelings? But I knew his
mother, so I don't want to do anything to wound him. I won't ask you,
Ellen; I don't like five at table. But just tell David to come, will
you?"
And Miss Baily promised, gratefully. David was not often asked out in
Old Chester.
II
The supper at Mrs. Barkley's was a great occasion to David Baily.
Right after dinner he went up to the garret, and Ellen heard him
shuffling about overhead, moving trunks. After a while he came down,
holding something out to his sister.
"Guess I'll wear this," he said, briefly. It was an old black velvet
waistcoat worked with small silk flowers, pink and blue and yellow.
"I haven't seen gentlemen wear those waistcoats lately," Miss Ellen
said, doubtfully.
Mr. David spread the strange old garment across his narrow breast, and
regarded himself in the mirror above the mantel. "Father wore it," he
said.
Then he retired to his own room. When he reappeared he wore the
waistcoat. His old black frock-coat, shiny on the shoulders and with
very full skirts, hung so loose in front that the flowered velvet
beneath was not conspicuous; but Mr. David felt its moral support when,
at least ten minutes before the proper time, he started for Mrs.
Barkley's.
His hostess, putting on her best cap before her mirror, glanced down
from her window as he came up the path. "Ellen ought not to have sent
him so early," she said, with some irritation. "Emily!" she called, in
her deep voice, "just go to the front door and tell Mr. Baily to go
home. I'm not ready for him. Or he can sit in the parlor and wait if
he wants to. But he can't talk to me."
Emily, a mournful, elderly person, sought, out of regard for her own
feelings, to soften her mistress's message; but David instantly
retreated to walk up and down the street, keeping his eye on Mrs.
Barkley's house, so that he could time his return by the arrival of Mr.
Spangler.
"He'll come at the right hour, I presume," he said to himself. Just
then he saw Mrs. Smily stepping delicately down the street, her head on
one side, and a soft, unchanging smile on her lips. As they met she
minced a little in her step, and said:
"Dear me! I'm afraid I've made a mistake. I'm looking for Mrs.
Barkley's residence."
"Mrs. Barkley resides here," said Mr. David, elegantly.
She looked up into his sad, dark eyes with a flurried air. "Dear me,"
she said, "I fear I am late."
"Oh, not _late_," said poor David. "Perhaps we might walk up and down
for a minute longer?"
Mrs. Smily, astonished but flattered, tossed her head, and said, Well,
she didn't know about _that_! But, all the same, she turned, and they
walked as far as the post-office.
"I'm afraid you are very attentive to the ladies," Mrs. Smily said,
coquettishly, when David had introduced himself; and David, who had
never heard a flirtatious word (unless from Maria), felt a sudden
thrill and a desire to reply in kind. But from lack of experience he
could think of nothing but the truth. He had been too early, he said,
and had come out to wait for Mr. Spangler--"and you, ma'am," he added,
in a polite after-thought. But his hurried emphasis made Mrs. Smily
simper more than ever. She shook her finger at him and said:
"Come, come, sir!" And David's head swam.
[Illustration: "DAVID'S HEAD SWAM"]
At that moment Mr. Spangler, buttoned to his chin in a black waistcoat,
came solemnly along, and, with his protection, David felt he could face
Mrs. Barkley.
But, indeed, she met her three guests with condescension and kindness.
"They are all fools in their different ways," she said to herself, "but
one must be kind to them." So she made Mrs. Smily sit down in the most
comfortable chair, and pushed a footstool at her. Then she told Mr.
Spangler, good-naturedly, that she supposed he found Old Chester very
old-fashioned. "Don't you be trying any candles on us," she threatened
him, in a jocular bass. As for David, she paid no attention to him
except to remark that she supposed time didn't count with him. But her
bushy eyebrows twitched in a kindly smile when she said it. Then she
began to talk about Dr. Lavendar's health. "It is a great trial to
have him away," she said. "Dear me! I don't know what we will do when
the Lord takes him. I wish he might live forever. Clergymen are a
poor lot nowadays."
"Why, I heard," said Mrs. Smily, "that he didn't give entire
satisfaction."
"What!" cried Mrs. Barkley. "Who has been talking nonsense to you?
Some of the new people, I'll be bound."
Mrs. Smily, very much frightened, murmured that no doubt she was
mistaken. Wild horses would not have drawn from her that she had heard
Annie Shields that was, say that Dr. Lavendar had deliberately advised
some one she knew to be bad; and that he had refused to help a very
worthy man to study for the ministry; and that the Ferrises said he
ought to be tried for heresy (or something) because he married Oscar
King to their runaway niece; and that he would not give a child back to
its repentant (and perfectly respectable) mother--"And a mother's claim
is the holiest thing on earth," Mrs. Smily said--and that he had
encouraged Miss Lydia Sampson in positively _wicked_ extravagance.
After hearing these things, Mrs. Smily had her opinion of Dr. Lavendar;
but that was no reason why she should let Mrs. Barkley snap her head
off. So she only murmured that no doubt she had made a mistake.
"I think you have," said Mrs. Barkley, dryly; and rose and marshalled
her company in to supper. "She's a perfect fool," she told herself,
"but I hope the Lord will give me grace to hold my tongue." Perhaps
the Lord gave her too much grace, for, for the rest of the evening, she
hardly spoke to Mrs. Smily; she even conversed with David rather than
look in her direction.
For the most part the conversation was a polite exchange of views upon
harmless topics between Mrs. Barkley and Mr. Spangler, during which
Mrs. Smily cheered up and murmured small ejaculations to David Baily.
She told him that she was scared nearly to death of the stuffed animals
at Miss Harriet's house.
"They make me just scream!" she said.
David protectingly assured her that they were harmless.
"But they are so dreadful!" Mrs. Smily said. "Isn't it strange that my
cousin likes to--to do that to animals? It isn't quite ladylike, to my
mind."
Mr. Baily thought to himself how ladylike it was in Mrs. Smily to
object to taxidermy. He noticed, too, that she ate almost nothing,
which also seemed very refined. It occurred to him that such a
delicate creature ought not to go home alone; the lane up to Miss
Harriet's house was dark with overhanging trees, and, furthermore,
half-way up the hill it passed the burial-ground. In a burst of fancy
David saw himself near the low wall of the cemetery, protecting Mrs.
Smily, who was shivering in her ladylike way at the old head-stones
over in the grass. He began (in his own mind) a reassuring
conversation: "There are no such things as spectres, ma'am. I assure
you there is no occasion for fear." And at these manly words she would
press closer to his side. (And this outside the burial-ground--oh,
Maria, Maria!)
But this flight of imagination was not realized, for later Emily
announced that Miss Harriet's Augustine had come for Mrs. Smily.
"Did she bring a lantern?" demanded Mrs. Barkley. "That lane is too
dark except for young folks."
Augustine had a lantern, and was waiting with it at the front door for
her charge; so there was no reason for Mr. David to offer his
protection. He and Mr. Spangler went away together, and David twisted
his head around several times to watch the spark of light jolting up
the hill towards the burial-ground and the Stuffed-Animal House. When
the two men said good-night, Mr. Spangler had a glimpse of a quickly
opened door and heard an eager voice--"Come in, dear brother. Did you
have a delightful evening?"
"How pleasing to be welcomed so affectionately!" said the Reverend Mr.
Spangler to himself.
III
The gentle warmth of that welcome lingered persistently in Mr.
Spangler's mind.
"I suspect that she _kissed_ him," he said to himself; and a little
dull red crept into his cheeks.
Miss Ellen, dark-eyed, gentle, with soft lips, made Mr. Spangler
suddenly think of a spray of heliotrope warm in the sunshine. "That is
a very poetical thought," he said, with a sense of regret that it
probably could not be utilized in a sermon. But when he entered the
study he banished poetry, because he had a letter to write. It was in
answer to an offer of the secretaryship of a church publishing-house in
a Western city.
Dr. Lavendar, it appeared, had mentioned Mr. Spangler's name to one Mr.
Horatius Brown, stating that in his opinion Mr. Spangler was just the
man for the place--"exact, painstaking, conscientious," Mr. Brown
quoted in his letter; but forbore to add Dr. Lavendar's further remark
that Mr. Spangler would never embarrass the management by an original
idea. "He'll pick up pins as faithfully as any man I know," said Dr.
Lavendar, "and that's what you religious newspapers want, I believe?"
Mr. Spangler was not without a solemn pride in being thus sought out by
the ecclesiastical business world, especially when he reflected upon
the salary which Mr. Brown was prepared to offer; but acceptance was
another matter. To leave his high calling for mere business! A
business, too, which would involve exact hours and steady
application;--Compared with that, and with the crude, smart bustle of
the Western city, the frugal leisure of his placid days in Mercer
assumed in his mind the sanctity of withdrawal from the world, and his
occasional preaching took on the glow of missionary zeal. "No," said
Mr. Spangler, "mercenary considerations do not move me a
hair's-breadth." Mr. Spangler did not call his tranquil life in
Mercer, his comfortable old house, his good cook, his old friends, his
freedom from sermon-writing, mercenary considerations. On the
contrary, he assured himself that his "circumstances were far from
affluent; but I must endure hardness!" he used to add cheerfully. And
very honestly his declination seemed to him something that Heaven would
place to his credit. So he wrote to the publishing-house that he had
given the proposition his most prayerful consideration, but that he
believed that it was his duty to still labor at the sacred desk--and
duty was, he hoped, the watchword of his life. And he was Mr. Brown's
"obedient servant and brother in Christ--Augustus Spangler."
Then he settled down in Dr. Lavendar's armchair by the fire in the
study; but he did not read the ecclesiastical paper which every week
fed his narrow and sincere mind. Instead he wondered how often Dr.
Lavendar called upon his female parishioners. Would twice in a
fortnight be liable to be misunderstood? Mr. Spangler was terribly
afraid of being misunderstood. Then he had a flash of inspiration: he
ought, as rector, to visit the schools. That was only proper and could
not possibly be misunderstood. "For an interest in educational affairs
is part of a priest's duty," Mr. Spangler reflected.
If he was right, it must be admitted that Dr. Lavendar was very remiss.
So far as we children could remember, he had never visited Miss Ellen's
school and listened to recitations and heard us speak our pieces.
Whether that was because he did not care enough about us to come, or
because he saw us at Collect class and Sunday-school and church, and in
the street and at the post-office and at home, until he knew us all by
heart, so to speak, may be decided one way or the other; but certainly
when Mr. Spangler came, and sat through one morning, and told us
stories, and said we made him think of a garden of rosebuds, and took
up so much of Miss Ellen's time that she could not hear the mental
arithmetic, it was impossible not to institute comparisons. Indeed,
some hearts were (for the moment) untrue to Mr. David. When Miss Ellen
called on us to speak our pieces, we were so excited and breathless
that, for my part, I could not remember the first line of "Bingen on
the Rhine," and had to look quickly into the Fourth Reader; but before
I could begin, Lydia Wright started in with "Excelsior," and she got
all the praise; though I'm sure I--well, never mind! But Dr. Lavendar
wouldn't have praised one girl so that all the others wanted to scratch
her! All that first half, the pupils, bending over their copy-books,
writing, "_Courtesy to inferiors is true gentility_," glanced at the
visitor sideways, and if they caught his eye, looked down, blushing to
the roots of their hair--which was not frizzled, if you please, or
hanging over their eyes like the locks of Skye-terriers, but parted and
tied with a neat ribbon bow on the tops of all the small heads. But
Mr. Spangler did not look often at the pupils; instead he conversed in
a low voice with Miss Ellen. Nobody could hear what he said, but it
must have been very interesting, for when Miss Ellen suddenly looked at
the clock she blushed, and brought her hand hurriedly down on the bell
on her desk. It was ten minutes after the hour for recess!
For the rest of that day Miss Ellen Baily moved and looked as one in a
dream. Her brother, however, did not seem to notice her
absent-mindedness. Indeed, he was as talkative as she was silent.
"Sister," he said, as they sat at tea, "I need a new hat. One with a
blue band about it might be--ah--becoming."
"Blue is a sweet color," said Miss Ellen, vaguely.
"Mrs. Smily remarked to me that before her affliction made it improper,
she was addicted to the color of blue."
"Was she?" Ellen said, absently.
"Don't you think," David said, after a pause, "that my coat is somewhat
shabby? You bought it, you may remember, the winter of the long frost."
"Is it?" Miss Ellen said.
"Yes; and the style is obsolete, I think. Not that I am a creature of
fashion, but I do not like to be conspicuous in dress."
"You are not that, dear David," Miss Ellen protested. "On Sunday I
often think nobody looks as handsome as you."
David blushed. "You are partial, Ellen."
"No, I'm not," cried Miss Ellen, coming out of her reveries. "Only
yesterday I heard some one say that you were very fine-looking."
"Who said it?"
"Never mind," Ellen said, gayly.
"Do tell me, sister," he entreated; "that's a good girl."
"It was somebody whose opinion you care a great deal about."
"I think you might tell me," said Mr. David, aggrieved. "Not that I
care, because it isn't true, and was only said to please you. People
know how to get round you, Ellen. But I'd just like to know."
"Guess," said Miss Ellen.
"Well, was it--Mrs. Smily?"
"Oh, dear, no! It was somebody very important in Old Chester. It was
Mrs. Barkley."
"Oh," said Mr. David.
"A compliment from her means so much, you know," Miss Ellen reminded
him.
David was silent.
"But all the same," Ellen said, "you do need a coat, dear brother. I'm
afraid I've been selfish not to notice it."
Mr. David made no reply.
Miss Ellen beamed at him. "You always look well, in my eyes: but it
pleases me to have you well dressed, too."
"Well, then, to please you, I'll dress up," said Mr. David, earnestly.
IV
"Does not Mr. Baily take any part whatever in his sister's work?" Mr.
Spangler said. He was calling upon Mrs. Barkley, and the conversation
turned upon the guests whom he had met at the tea-party.
"That is a very foolish question," said Mrs. Barkley; "but of course
you don't know poor David, or you wouldn't have asked it. David means
well, but he has no mind. Still, he has tried, poor fellow." Then she
recited the story of David's failures. "There is really nothing that
he is capable of doing," she ended, thoughtfully; "though I think, if
his eyes hadn't given out, he might have made a good minister. For
David is a pious man, and he likes to visit."
A faint red came into Mr. Spangler's cheeks; although he had been in
Old Chester nearly a month, he had not yet become acclimated to Mrs.
Barkley. The watchword of duty made him call, but he closed her front
door behind him with an emphasis which was not dutiful.
"That's done!" he said; and thought to himself how much pleasanter than
parochial visits were educational matters.
Mr. Spangler felt their importance so deeply that he spent two more
mornings watching Miss Ellen's pupils work out examples on the
blackboard and hearing them read, turn about, in the Fourth Reader. In
fact, the next month was a pretty happy time for Miss Ellen's girls.
"I skipped to the bottom of the page in 'Catiline's Reply,'" Lydia
Wright said, giggling, "and she never knew it!"
The girls were tremendously interested but not very sympathetic, for
"she's so dreadfully old!" they told each other. Had Miss Ellen been
Maria's age and had a beau (by this time they called Mr. Spangler Miss
Ellen's beau, the impudent little creatures!), how different it would
have been! But Miss Ellen was forty. "Did you ever know anything so
perfectly absurd?" said the older girls. And the second-class girls
said they certainly never did. So when Mr. Spangler came and listened
to recitations we poked one another, and put out our tongues behind our
Readers, and made ourselves extremely obnoxious--if dear Miss Ellen had
had the eyes to see it, which, indeed, she had not. She was very
absent in those days; but she did her work faithfully, and saw to
David's new coat, and asked Mrs. Smily to tea, not only to help out
Miss Harriet at the Stuffed-Animal House, but because David told her a
piteous tale of Mrs. Smily's loneliness and general forlornness. David
had had it directly from Mrs. Smily herself, and had been greatly moved
by it; she had told him that this was a sad and unfriendly world.
"But I am sure your brother-in-law's family is much attached to you?"
David said, comfortingly.
Then poor Mrs. Smily suddenly began to cry. "Yes; but I am afraid I
can't live at my brother-in-law's any longer. His wife is--is tired of
me," said the poor little creature.
David was thunderstruck. "Tired? Of you! Oh, impossible!"
Then she opened her poor foolish heart to him. And David was so
touched and interested that he could hardly wait to get home to pour it
all into Ellen's ears. Ellen was very sympathetic, and made haste to
ask Mrs. Smily to tea; and when she came was as kind and pitiful as
only dear, kind Ellen could be. But perhaps she took Mrs. Smily's
griefs a little less to heart than she might have done had she heard
the tale a month before. Just then she was in the whirl of Old Chester
hospitality; she was asked out three times in one week to meet the
Supply!--and by that time the Supply had reached the point of hoping
that he was going to meet Miss Ellen.
Yet, as Mr. Spangler reflected, this was hardly prudent on his part.
"For I might become interested," he said to himself, and frowned and
sighed. Now, as everybody knows, the outcome of "interest" is only
justified by a reasonable affluence. "And," Augustus Spangler reminded
himself, "my circumstances are not affluent." Indeed, that warm,
pleasant old house in Mercer, and Mary Ann, and his books, and those
buttoned-up coats needed every penny of his tiny income. "Therefore,"
said Mr. Spangler, "it is my duty to put this out of my head with an
iron hand." But, all the same, Ellen Baily was like a spray of
heliotrope.
For a week, the second week in April, while Old Chester softened into a
mist of green, and the crown-imperials shook their clean, bitter
fragrance over the bare beds in the gardens--for that week Mr. Spangler
thought often of his income, but oftener of Miss Ellen. Reason and
sentiment wrestled together in his lazy but affectionate heart; and
then, with a mighty effort, sentiment conquered....
"It seems," said Mr. Spangler, nervously, "a little premature, but my
sojourn in Old Chester is drawing to a close; I shall not tarry more
than another fortnight; so I felt, my dear friend, that I must, before
seeking other fields of usefulness, tell you what was in my mind--or
may I say heart?"
"You are very kind," Ellen Baily said, breathlessly.
.... Mr. Spangler had invited Miss Ellen to walk with him on Saturday
afternoon at four. Now, as everybody knows in Old Chester, when a
gentleman invites you to walk out with him, you had better make up your
mind whether it is to be "yes" or "no" before you start. As for poor
Ellen, she did not have to make up her mind; it was made up for her by
unconquerable circumstances. If she should "seek other fields of
usefulness," she could not take David with her. It was equally clear
that she could not leave him behind her. Where would he find his
occasional new coat, or even the hat with the blue band, if there were
no school in the basement? Compared to love-making and romance, how
sordid are questions about coats! Yet, before starting on that
Saturday-afternoon walk, poor, pretty Miss Ellen, tying the strings of
her many-times retrimmed bonnet under her quivering chin, asked them,
and could find no answer except that if he should "say anything," why,
then, she must say "no"; but, of course, he wasn't going to say
anything. So she tied her washed and ironed brown ribbons into a neat
bow, and started down the street with the Reverend Mr. Spangler.
David Baily, watching them from the gate, ruminated over obvious
possibilities. Mrs. Barkley had opened his eyes to the fact that Mr.
Spangler "was taking notice," and David was not without a certain
family pride in a ministerial proposal. "He'll do it this afternoon,"
said David; and went pottering back into the empty school-room to mend
a bench that Ellen told him needed a nail or two. But the room was
still and sunny, and Ellen's chair was comfortable; and sitting there
to think about the bench, he nodded once or twice, and then dozed for
an hour. When he awoke it seemed best to mend the bench the next day;
then, yawning, and staring vacantly out of the window, he saw Mrs.
Smily, and it seemed only friendly to go out and tell her
(confidentially) what was going to happen.
"It will make quite a difference to you, won't it?" Mrs. Smily said.
"Oh," David said, blankly, "that hadn't occurred to me. However," he
added, with a little sigh, "my sister's happiness is my first thought."
Mrs. Smily clasped her hands. "Mr. Baily, I do think you are real
noble!" she said.
Mr. David stood very erect. "Oh, you mustn't flatter me, ma'am."
"Mr. Baily, I never flatter," Mrs. Smily said, gravely. "I don't think
it's right."
And David thought to himself how noble Mrs. Smily was. Indeed, her
nobility was so much in his mind that, strangely enough, he quite
forgot Ellen's exciting afternoon. He remembered it the next morning,
but when he essayed a little joke and a delicate question, the asperity
with which the mild Ellen answered him left him gaping with
astonishment. Evidently Mr. Spangler had not spoken. David would have
been less (or more) than a human brother if he had not smiled a very
little at that. "Ellen expected it," he said to himself. "Well, I did
myself, and so did Mrs. Barkley." It never occurred to him that the
Reverend Mr. Spangler might also have had expectations which left him
disappointed and mortified. Yet when a gentleman of Mr. Spangler's
age--one, too, whose income barely suffices for his own comfort, and
who, added to this, has had his doubts whether the celibacy of the
clergy may not be a sacrament of grace--when such a gentleman does make
up his mind to offer himself--to offer himself, moreover, to a lady no
longer in her first youth, who is pleasing perhaps to the eye, but not,
certainly, excessively beautiful, and whose fortune is merely (and most
meritoriously, of course) in her character and understanding--it is a
blow to pride to be refused. Mr. Spangler found it hard to labor at
the sacred desk that morning; yet no one would have thought it, to see
the fervor with which, as Old Chester said, he "went through his
performances."
But he read the service, hot at heart and hoping that Miss Baily
observed how intensely his attention was fixed on things above. When
he stood in the chancel waiting for the collection-plates, and saying,
in a curious sing-song, absolutely new to Old Chester, "_Zaccheus stood
forth, and said, Behold, Lord--_" his glance, roving over the
congregation, rested once on Ellen Baily, and was as carefully
impersonal as though she were only a part of the pew in which she sat.
Miss Ellen thrilled at that high indifference; it occurred to her that
even had David's circumstances been different, she could scarcely have
dared to accept the hand of this high creature.
"_--the half of all my goods--_" said Mr. Spangler. Yes, it was
inconceivable, considering what he was offering her, that Ellen Baily
could let her brother stand in the way!
All that long, pleasant spring Sunday, Augustus Spangler was very
bitter. All that week he was distinctly angry. He said to himself
that he was glad that Dr. Lavendar was soon to return; he would, after
making his report of the parish, shake the dust of Old Chester from off
his feet as witness against Miss Baily, and depart. By the next Sunday
he had ceased to be angry, but his pride was still deeply wounded. By
Wednesday he had softened to melancholy; he was able to say that it all
came from her sense of duty. Unreasonable, of course, but still duty.
Then, on Thursday, suddenly, he was startled by a question in his own
mind: Was it unreasonable? If she gave up her teaching--"what would
that fellow live on?"
That was a very bad moment to the Reverend Mr. Spangler. Pride
vanished in honest unhappiness. He began to think again about his
income; he had known that to marry a wife meant greater economy; but
sacrifices had not seemed too difficult considering that that wife was
to be Miss Ellen Baily. But if the wife must be Miss Baily
_plus_--"that fellow"!
"It is out of the question," said poor Mr. Spangler, and arose and
paced up and down the study. He was very miserable; and the more
miserable he became, the more in love he knew himself to be. "But it
is madness to think of the matter further," he told himself,
sternly--"madness!"
Yet he kept on thinking of it--or of Miss Ellen's dark eyes, and her
smile, and the way her hair curled in little rings about her temples.
"But it's impossible--impossible!" he said. Then, absently, he made
some calculations: To meet the support of David Baily he would have to
have an increase of so much in his income or a decrease of so much in
his expenses. "Madness!" said Augustus Spangler, firmly. "But how her
eyes crinkle up when she smiles!"
Yet it took another day before the real man conquered. His expenses
should be decreased, _and David should live with them_.
Yes, it would mean undeniable pinching; he must give up this small
luxury and that; his Mary Ann could not broil his occasional
sweetbread; and the occasional new book must be borrowed from the
library, not purchased for his own shelves. He must push about to get
more supplying. He had meant to come down one step when he got
married; well, he would have to come down two--yes, or three. But he
would have Miss Baily. And warmed with this tender thought, he sat
down, then and there, at nearly midnight, and wrote Miss Ellen a
letter. It was a beautiful letter, full of most beautiful sentiments
expressed with great elegance and gentility. It appreciated Miss
Ellen's devotion to her family, and acknowledged that a sense of duty
was a part of the character of a Christian female. It protested that
it was far from the Reverend Mr. Spangler to interfere with that sense
of duty; on the contrary, he would share it; nay more, he would assist
it, for duty was, he hoped, the watchword of his life. If Miss Baily
would consent to become his wife, Mr. Baily, he trusted, would make his
home with his sister.
Mr. Spangler may have been addicted to petticoats (in his own toilet)
and given to candles and other emblems of the Scarlet Woman, but his
letter, beneath its stilted phrase, was an honest, manly utterance, and
Ellen Baily read it, thrilling with happiness and love.
That was Friday, and she had only time to read those thin, blue pages
and thrust them into the bosom of her dress, when it was time to go to
school and hear her girls declare that the Amazon was the largest river
in South America; but we might have said it was the largest river in
Pennsylvania, and Miss Ellen would have gone on smiling at us. At
recess we poured out into the garden, eager to say, "Goodness! do you
suppose he's popped?" The older girls were especially excited, but
they took their usual furtive look about the garden before sitting down
on the steps to eat their luncheons. Alas, He was not there!
"Perhaps," said Lydia Wright, "he has gone to the tomb."
This, for the moment, was deliciously saddening; but, after all, real
live love-making, even of very old people, is more fascinating than
dead romance. Through the open window we could see Miss Ellen sitting
at her desk, writing. There were some sheets of blue paper spread out
in front of her, and she would glance at them, and then write a little,
and then glance back again, and smile, and write. But she did not look
troubled, or "cross," as the girls called it; so we knew it could not
be an exercise that she was correcting. But when she came out to us,
and said, in a sweet, fluttered voice, "Children, will one of you take
this letter to the post-office?" we knew what it meant--for it was
addressed to the Reverend Mr. Spangler. How we all ran with it to the
post-office!--giggling and palpitating and sighing as our individual
temperaments might suggest. In fact, I know one girl who squeezed a
tear out of each eye, she was so moved. When we came back, there was
Miss Baily still sitting at her desk, her cheek on one hand, her
smiling eyes fastened on those sheets of blue paper. "Gracious," said
the girls, "what a long recess!" and told each other to be quiet and
not remind her to ring the bell.
Then suddenly something happened....
An old carry-all came shambling along the road; there were two people
in it, and one of them leaned over from the back seat and said to the
driver: "This is my house. Stop here, please." The girls, clustering
like pigeons on the sunny doorstep, began to fold up their
luncheon-boxes, and look sidewise, with beating hearts, towards the
gate--for it was _He_! How graceful he was--how elegant in his
manners! Ah, if our mothers had bidden us have manners like Mr.
David!--but they never did. They used to say, "Try and behave as
politely as Miss Maria Welwood," or, "I hope you will be as modest in
your deportment as Miss Sally Smith." And there was this model before
our eyes. It makes my heart beat now to remember how He got out of
that rattling old carriage and turned and lifted his hat to a lady
inside, and gave her his hand (ah, me!) and held back her skirts as she
got out, and bowed again when she reached the ground. She was not much
to look at; she was only the lady who was visiting at the
Stuffed-Animal House, and she was dressed in black, and her bonnet was
on one side. They stood there together in the sunshine, and Mr. David
felt slowly in all his pockets; then he turned to us, sitting watching
him with beating hearts.
"Little girls," he said--he was near-sighted, and, absorbed as he
always was with sorrow, we never expected him to know our
names--"little girls, one of you, go in and ask my sister for two coach
fares, if you please."
We rose in a body and swarmed back into the school-room--just as Miss
Ellen with a start looked at the clock and put out her hand to ring the
bell. "Mr. David says, please, ma'am, will you give him money for two
coach fares?"
Miss Ellen, rummaging in her pocket for her purse, said: "Yes, my love.
Will you take this to my brother?" Just why she followed us as we ran
out into the garden with her purse perhaps she hardly knew herself.
But as she stood in the doorway, a little uncertain and wondering, Mr.
David led the shabby, shrinking lady up to her.
"My dear Ellen," he said, "I have a present for you--a _sister_."
Then the little, shabby lady stepped forward and threw herself on Miss
Ellen's shoulder.
"A sister?" Ellen Baily said, bewildered.
"We were married this morning in Upper Chester," said Mr. David, "and I
have brought her home. Now we shall all be so happy!"
V
That evening Dr. Lavendar came home. Of course all the real Old
Chester was on hand to welcome him.
When the stage came creaking up to the tavern steps, the old white head
was bare, and the broad-brimmed shabby felt hat was waving tremulously
in the air.
"Here I am," said Dr. Lavendar, clambering down stiffly from the
box-seat. "What mischief have you all been up to?"
There was much laughing and hand-shaking, and Dr. Lavendar, blinking
very hard, and flourishing his red silk pocket-handkerchief, clapped
Mr. Spangler on the shoulder.
"Didn't I tell you about 'em? Didn't I tell you they were the best
people going? But we mustn't let 'em know it; makes 'em vain," said
Dr. Lavendar, with great show of secrecy. "And look here, Sam Wright!
You fellows may congratulate yourselves. Spangler here has had a fine
business offer made him, haven't you, Mr. Spangler? and it's just your
luck that you got him to supply for you before he left this part of the
country. A little later he wouldn't have looked at Old Chester. Hey,
Spangler?"
"Oh, that's settled," Mr. Spangler said. "I declined--"
"Oh," said Dr. Lavendar, "have you? Well, I'm sorry for 'em."
And Augustus Spangler smiled as heartily as anybody. He had a letter
crushed up in his hand; he had read it walking down from the
post-office to the tavern, and now he was ready to say that Old Chester
was the finest place in the world. He could hardly wait to get Dr.
Lavendar to himself in the rectory before telling him his great news
and giving him a little three-cornered note from Ellen Baily which had
been enclosed in his own letter.
"Well, well, _well_," said Dr. Lavendar.
He had put on a strange dressing-gown of flowered cashmere and his
worsted-work slippers, and made room for his shaggy old Danny in his
leather chair, and lighted his pipe. "Now tell us the news!" he said.
And was all ready to hear about the Sunday-school teachers, and the
choir, and Sam Wright's Protestantism, and many other important things.
But not at all:--
"_I'm engaged to be married._"
"Well, well, well," said Dr. Lavendar, blinking and chuckling with
pleasure; then he read Ellen's little note. "I had to tell you
myself," Ellen wrote him, "because I am so happy." And then there were
a dozen lines in which her heart overflowed to this old friend. "Dear
child, dear child," he murmured to himself. To no one but Dr.
Lavendar--queer, grizzled, wrinkled old Dr. Lavendar, with never a
romance or a love-affair that anybody had ever heard of--could Miss
Ellen have showed her heart. Even Mr. Spangler did not know that heart
as Dr. Lavendar did when he finished Ellen's little letter.--And Dr.
Lavendar didn't tell. "I am so happy," said Miss Ellen. Dr. Lavendar
may have looked at Mr. Spangler and wondered at the happiness. But,
after all, wonder, on somebody's part, is a feature of every
engagement. And if the wonder is caused only by the man's coat, and
not by his character, why be distressed about it? Mr. Spangler was an
honest man; if his mind was narrow, it was at least sincere; if his
heart was timid, it was very kind; if his nature was lazy, it was clean
and harmless. So why shouldn't Ellen Baily love him? And why
shouldn't Dr. Lavendar bubble over with happiness in Ellen's happiness?
"She's the best girl in the world," he told Mr. Spangler. "I
congratulate you. She's a good child--a good child."
Mr. Spangler agreed, in a somewhat solemn manner.
"But David--how about David?"
"My house shall always be open to Mrs. Spangler's relatives," said Mr.
Spangler, with Christian pride.
"You are a good fellow, Spangler," Dr. Lavendar said; and listened,
chuckling, to Mr. Spangler's awkward and correct expressions of bliss.
For indeed he was very happy, and talked about Miss Ellen's virtues
(which so eminently qualified her to become his wife), as fatuously as
any lover could.
"Hi, you, Danny," said Dr. Lavendar, after half an hour of it, "stop
growling."
"There's somebody at the door," said Augustus Spangler, and went into
the entry to see who it was. He came back with a letter, which he
read, standing by the table; then he sat down and looked white. Dr.
Lavendar, joyously, was singing to himself:
"'Ten-cent Jimmy and his minions
Cannot down the Woolly Horse.'
"Spangler, we must drink to your very good health and prospects. Let's
have Mary bring the glasses."
"I fear," said Mr. Spangler--he stopped, his voice unsteady. "I
regret--"
"Hullo!" said Dr. Lavendar, looking at him over his spectacles; "what's
wrong?"
"I'm extremely sorry to say," said poor Mr. Spangler, "that--it can't
be."
"A good glass of wine," said Dr. Lavendar, "never hurt--"
"I refer," said Mr. Spangler, sighing, "to my relations with Miss Ellen
Baily."
Dr. Lavendar looked at him blankly.
"I have just received a letter," the poor man went on, "in which she
informs me that it can never be." His lip trembled, but he held
himself very straight and placed the letter in his breast-pocket with
dignity.
"Spangler, what are you talking about?"
"It appears," said Mr. Spangler, "that her brother--"
"Fiddlesticks!" said Dr. Lavendar. "Has Ellen started up some
fantastic conscientiousness? Spangler, women's consciences are
responsible for much unhappiness in this world. But I won't have it in
my parish! I'll manage Ellen; trust me." He pulled at his pipe, which
had gone out in these moments of agitation. "I tell you, sir," he
said, striking a match on the bottom of his chair, "these saintly,
self-sacrificing women do a fine work for the devil, if they only knew
it, bless their hearts."
"You misapprehend," said Mr. Spangler, wretchedly; and then told Miss
Ellen's news. It was brief enough, this last letter; there was no
blame of David; indeed, he had displayed, Miss Baily said, "a true
chivalry; but of course--" "Of course," said Mr. Spangler.
But Dr. Lavendar broke out so fiercely that Danny squeaked and jumped
down out of the chair. "Upon my word; upon my word, Spangler, what
were you thinking of to let it go on? If I had been at home, it would
never--upon my _word_!" This was one of the times that Dr. Lavendar
felt the limitations of his office in regard to language. Mr.
Spangler, his elbows on his knees, his chin on hands, was staring
miserably at the floor.
"I shall, I trust, meet it in the proper spirit," he said.
Dr. Lavendar nodded. "Of course," he said. "Fortunately, she is
dealing with a man who has backbone--perhaps."
Mr. Spangler sighed. "I regret to say that her presence in her school
under the circumstances does seem imperative."
Dr. Lavendar lighted his pipe. "Do you mean on account of money,
Spangler?"
"The support of Mr. David Baily and this--this _female_, must be met, I
suppose, by Miss Baily's school."
"You are not so situated that you--" began Dr. Lavendar, delicately.
"My circumstances," said Augustus Spangler, "are not affluent. I have
my residence in Mercer; and I supply, as you know. But my income
barely suffices for one. Four--would be out of the question."
Dr. Lavendar looked at Ellen's little, happy note, lying half open on
the table. "Poor old jack-donkey of a David!" he groaned.
"His selfishness," said Augustus Spangler, between his teeth, his voice
suddenly trembling with anger, "is perfectly incomprehensible to
me--perfectly incomprehensible! I endeavor always to exercise charity
in judging any human creature; but--really, _really_!"
"It isn't selfishness as much as silliness. David hasn't mind enough
to be deliberately selfish. The poor fellow never thought. He never
has thought. Ellen has always done the thinking for the family. Well,
the harm's done. But, Spangler--" the old man stopped and glanced
sharply at the forlorn and angry man opposite him. Yes, he certainly
seemed very unhappy;--and as for Ellen! Dr. Lavendar could not bear
that thought. "Spangler, I'll stand by you. I won't let her offer you
up as well as herself. There must be some way out."
Mr. Spangler shook his head hopelessly. "The support of four persons
on my small stipend is impossible."
"Spangler, my boy!" said Dr. Lavendar, suddenly, "there is a way out.
What an old fool I am not to have thought of it! My dear fellow"--Dr.
Lavendar leaned over and tapped Mr. Spangler's knee, chuckling
aloud--"_that secretaryship_!"
"Secretaryship?" Mr. Spangler repeated, vaguely.
"You declined it? I know. But I don't believe Brown's got a man yet.
I heard from him on another matter, yesterday, and he didn't say he
had. Anyway, it's worth trying for. We can telegraph him to-morrow,"
said Dr. Lavendar, excitedly.
Mr. Spangler stared at him in bewilderment. "But," he said,
breathlessly, "I--I don't think--I fear I am not fit." He felt as if
caught in a sudden wind; his face grew red with agitation. "I declined
it!" he ended, gasping.
"Fit?" said Dr. Lavendar. "My dear man, what fitness is needed?
There's nothing to it, Spangler, I assure you." Dr. Lavendar was very
much in earnest; he sat forward on the edge of his chair and
gesticulated with his pipe. "Don't be too modest, my boy."
"Business entails such responsibilities," Mr. Spangler began, in a
frightened voice.
"Oh, but this is mere routine," Dr. Lavendar interrupted; "they want a
clergyman--somebody with tact. There's a good deal of church politics
in it, I suppose, and they've got to have somebody who would never step
on anybody's toes."
"I would never do that," said Mr. Spangler, earnestly, "but--"
"No," said Dr. Lavendar, abruptly, his voice changing--"no, Spangler,
you never would." Then he was silent for a moment, pulling on his
pipe, wondering perhaps, in spite of himself, at Ellen. "No, you never
would. You see, you are just the man for the place. Brown said they
wanted somebody who was presentable; he said they didn't need any
particular abil--I mean any particular business ability."
"But," said Mr. Spangler, "to give up my sacred calling--"
"Spangler, come now! you don't 'call' very loudly, do you? There, my
dear boy, let an old fellow have his joke. I merely mean you don't
preach as often as if you had a regular parish. And you can supply,
you know, there just as well as here."
"The Master's service is my first consideration," said Augustus
Spangler.
Dr. Lavendar looked at him over his spectacles. "Mr. Spangler, the
Christian business-man serves the Master just as well as we do."
"I should wish to reflect," said Mr. Spangler.
"Of course."
"Miss Baily would, I fear, object to going so far away."
"If the place is still open, I'll manage Ellen," said Dr. Lavendar; but
he looked at Mr. Spangler narrowly. "And your own entreaties will, of
course, weigh with her if you show determination. I think you told me
you were pretty determined?"
"I have," said Mr. Spangler, "an iron will; but that would not justify
me in insisting if Miss Baily--" His voice trailed off; it rose before
him--the far-off, bustling city, the office, the regular hours, the
people whose toes must not be stepped upon, the letters to write and
read, the papers to file, all the exact minutiae the position involved.
And his comfortable old house? his leisure? his ease? And Mary Ann?
Mary Ann would never consent to go so far! "I--I really--" he began.
Dr. Lavendar frowned. "Mr. Spangler, I would not seem to urge you.
Ellen is too dear to us for that. But if you appreciate her as I
suppose you do--"
"I do indeed!" broke in poor Augustus Spangler, fervently.
"The way is probably open to you."
"But--" said Mr. Spangler, and then broke out, with marked agitation;
"I--I really don't see how I could possibly--" Yet even as he spoke he
thought of Ellen's sweet eyes. "Good Heavens!" said Mr. Spangler,
passionately; "what shall I do?"
But Dr. Lavendar was silent. Mr. Spangler got up and began to walk
about.
"My affection and esteem," he said, almost weeping, "are unquestioned.
But there are other considerations."
Dr. Lavendar said nothing.
"It is a cruel situation," said Mr. Spangler.
Dr. Lavendar looked down at his pipe.
There was a long silence. Augustus Spangler walked back and forth.
Dr. Lavendar said never a word.
"A man must consider his own fitness for such a position," Mr. Spangler
said, pleadingly.
"Perhaps," Dr. Lavendar observed, mildly, "Ellen's affections are not
very deeply engaged? It will be better so."
"But they are!" cried Mr. Spangler. "I assure you that they are! And
I--I was so happy," said the poor man; and sniffed suddenly, and tried
to find the pocket in his coat-tails.
Dr. Lavendar looked at him out of the corner of his eye.
Mr. Spangler stood stock-still; he opened and shut his hands, his lips
were pressed hard together. He seemed almost in bodily pain, for a
slight moisture stood out on his forehead. He was certainly in
spiritual pain. The Ideal of Sacrifice was being born in Mr.
Spangler's soul. His mild, kind, empty face grew almost noble;
certainly it grew very solemn.
"Dr. Lavendar," he said, in a low voice, "_I will do it._"
Dr. Lavendar was instantly on his feet; there was a grip of the hand,
and, for a moment, no words.
"I'll telegraph Mr. Brown," said Mr. Spangler, breathlessly.
"So will I!" said Dr. Lavendar.
Mr. Spangler was scarlet with heroism. "It means giving up my house
and my very congenial surroundings, and I fear Mary Ann will feel too
old to accompany me; but with--with Ellen!"
"She's worth six Mary Anns, whoever Mary Ann may be," said Dr. Lavendar.
"You may have thought me hesitant," said Mr. Spangler, "but I felt that
I must weigh the matter thoroughly."
"Why, certainly, man. It was your duty to think what was best for
Ellen."
"Exactly," Mr. Spangler said, getting his breath again, and beginning
to feel very happy. "And duty is, I hope, my watchword; but I had to
reflect," he ended, a little uncomfortably.
But Dr. Lavendar would not let him be uncomfortable. They sat down
again, and Dr. Lavendar filled another pipe, and until long after
midnight they talked things over--the allowance to be made to David and
his bride, the leasing of the house in Mercer, the possible obduracy of
Mary Ann, and, most of all, the fine conduct of the Reverend Mr.
Spangler.
But when they had said good-night, Dr. Lavendar sat awhile longer by
his fireside, his pipe out, his old white head on his breast.
"The minute I get back," he said to himself after a while,
sheepishly--"the minute I get back I poke my finger into somebody
else's pie. But I think 'twas right: Ellen loves him; and he's not a
bad man.--And Brown don't want brains."
Then he chuckled and got up, and blew out the lamp.
THE NOTE
I
Of course everybody in Old Chester knew that there was something queer
about Mary Gordon's marriage--not the mere fact of the man, queer as he
was; for, to Old Chester's ideas, he was very queer.... A
"travelling-man," to begin with--and the Gordons had a line of scholars
and professional men behind them--a drummer, if you please. In theory,
Old Chester was religiously democratic; it plumed itself upon its
Christian humility, and every Sunday it publicly acknowledged that Old
Chesterians were like the rest of humanity to the extent of being
miserable sinners. But, all the same, that Mary Gordon should marry a
"person" of that sort--
"Dear me!" said Old Chester.
However, travelling-men may be worthy; they need not necessarily use
perfumery or put pomade upon their shiny, curly, black hair. But Mr.
Algernon Keen was obviously not worthy, and he was saturated with
perfumery, and his black, curly hair was sleek with oil. Furthermore,
he was very handsome: his lips were weak and pouting and red; his eyes
liquid and beautiful; his plump cheeks slightly pink. One may believe
that such physical characteristics do not imply moral qualities; but
only youth has such a belief. When one has lived a little while in the
world, one comes to know that a human soul prisoned in such pretty
flesh is piteously hampered. Yet Mary Gordon, meeting this poor
creature by chance, fell deeply in love with him. Of course such
falling in love was queer--it was inexplainable; for Mary was a nice
girl--not, of course, of the caliber of some Old Chester girls; she had
not the mind of Alice Gray nor the conscience of Sally Smith; but she
was a quiet, biddable, good child--at least so far as anybody knew.
But nobody knew much about her. In the first place, the Gordons lived
just far enough out of Old Chester to miss its neighborliness. Mary
was not often seen in town, and in her own home her brother Alex's loud
personality crushed her into a colorless silence. Her father did not
crush her--he merely did not notice her; but he was fond of her--at
least he had the habit of indifferent affection. She always came into
the library to say good-night to him; and he, sitting by the fire in a
big, winged chair, a purple silk handkerchief spread over his white
locks, to keep off possible draughts, would turn his cheek up to her
mechanically; but the soft touch of her lips never made him lift his
eyes from his book. She never kissed Alex good-night; she was openly
afraid of him. Alex was rude to her and made her wait on him, throwing
her a curt "thank you" once in a while, generally coupled with some
sarcastic reference to her slowness or stupidity--for, indeed, the
child was both slow and stupid. Perhaps, had she been loved-- But no
one can tell now how that would have been. At any rate, there was a
pathetic explanation of loneliness to account for the fact that she was
drawn to this Algernon Keen, who had nothing to recommend him except a
cheap and easy kindliness that cost him no effort and was bestowed on
everybody.
[Illustration: "SHE ALWAYS CAME INTO THE LIBRARY TO SAY GOOD-NIGHT TO
HIM"]
Of course the two men, her father and brother, refused to consider Keen
as Mary's suitor at all. Alex nearly had a fit over it; in his rage
and mortification he took all Old Chester into his confidence. He went
to the Tavern--this was the day after Mary had, trembling and crying,
told her little love affair to her father and begged his consent--Alex
went to the Tavern and ordered the snickering, perfumed youth out of
town.
"Well, I guess not," said Algy. "This town doesn't belong to you, does
it?"
Alex stammered with passion: "If--if you dare to address Miss Gordon
again, I'll--I'll--I'll horsewhip you," he said, his pale eyes bulging
from his crimsoning face.
"I guess Mary has a right to let me talk to her if she wants to; this
is a free country," the other blustered. And Alex, loudly, on the
Tavern steps, cursed him for a skunk, a-- Well, Old Chester was never
able to quote Alex. He came to his senses after this dreadful
exhibition of himself, and was horribly mortified. But
post-mortification cannot undo the deed, and before night everybody in
Old Chester knew that Mary Gordon had fallen in love with--"the person
who brings samples to Tommy Dove's apothecary shop."
Old Chester was truly sorry for Mary; "for," as Mrs. Barkley said,
"love's love, whether it's suitable or not; and Mary has such a lonely
life, poor child! Well, it will take time for her to get over it."
It seemed to take a good deal of time. That winter she grew pale and
was often ill. The poor little thing seemed to creep into her shell to
brood over her blighted hopes. Once she was downright sick for a week,
and Mr. Gordon sent for William King. Willy said at first that Mary
had something on her mind (which certainly Mary's family did not need
to be told).
"I believe she's thinking about that scoundrel yet," said Alex. "But
she has just got to understand that we'll never allow it, Willy. You
may as well make that clear to her, and let her get over her moping."
William King looked thoughtful and said he would call again.
However, any of us Old Chester girls could have enlightened the doctor.
"Mary was pining away for her lover;" that was all there was to it.
But the lover never appeared, being engaged in offering samples of
pomade and perfumery to apothecary stores in other regions. And then,
suddenly, the queer thing happened....
The _Globe_ announced: "Married--by Dr. Lavendar, Mary Gordon to
Algernon Keen"--and the date, which was the night before.
"_What!_" said Old Chester at the breakfast-table, and gaped out of its
windows to see Mary, crying very much, get into the stage, not at her
father's house, but at the Tavern door, if you please, and drive away
with the Person. What did it mean? "Was Alex at home? Did he
consent?" demanded Old Chester; for Alex had been away from home for a
week. By noon it was decided that Alex had consented; for it came out
that he had returned to Old Chester the previous afternoon, and with
him, shrinking into the corner of the stage, was Mr. Algy Keen.
"Get out," Alex said to him when the stage drew up at the Gordon house.
The man got out, shambling and stumbling, with a furtive look over his
shoulder, for Alex Gordon walked behind him to the front door, his
right hand gripped upon his walking-stick, his left clinched at his
side.
"He kep' just behind the feller," the stage-driver told Van Horn at the
Tavern afterwards--"just behind him, like as if he was afraid the
feller'd run away from him. But the feller, he stopped right at the
steps, and he turned around, and he says, 'Mind you,' he says (mad as a
hatter)--'mind you,' he says, 'I'm not _brought_, I've
_come_';--whatever that means," the stage-driver ruminated.
So much Old Chester knew the day after Mary Gordon's wedding. And it
naturally sought to know a little more.
"I suppose her father feels it very much?" ventured Mrs. Barkley to Dr.
Lavendar.
"Any man feels the marriage of his only girl," said Dr. Lavendar,
briefly. And Mrs. Barkley held her tongue. But Mrs. Drayton, who was
just then anxious about her soul and found it necessary to consult Dr.
Lavendar as to the unpardonable sin--Mrs. Drayton was not so easily
squelched. "My Jean says that the Gordon's Rachel told her that Alex
brought the man into the house by the ear, and then sent her for you,
running, and--"
"She didn't bring me into the house by the ear," said Dr. Lavendar.
"But why, do you suppose, was it all so sudden?" said Mrs. Drayton; "it
almost looks--"
"How do you know it was sudden?" said Dr. Lavendar.
"Well, my Jean said--"
"It may have been sudden to Jean," said the old man; "possibly Mary had
not taken Jean into her confidence. Some folks don't confide in
servants, you know."
But Mrs. Drayton was proof against so delicate a thrust. "Well, I only
hope she won't repent at her leisure;--if there's nothing but haste to
repent of. If there's anything else--"
"I'll say good-day, Mrs. Drayton," interrupted Dr. Lavendar; "and as
for your question about the unpardonable sin, ma'am, why, just be ready
to forgive other folks and you needn't be afraid of the unpardonable
sin for yourself."
He took his hat and stick and went thumping down-stairs. In the hall
he met William King going up to see the invalid, and said, with a gasp:
"Willy, my boy, a good, honest murderer is easier to deal with than
some milder kinds of wrong-doing."
"Dr. Lavendar," said William, "I'd rather have a patient with small-pox
than treat some lighter ills that I could name."
As for Mrs. Drayton, she told her daughter that Dr. Lavendar was very
unspiritual, and did not understand the distress of a sensitive
temperament. "Even the slightest error fills me with remorse," said
Mrs. Drayton. "Dear me! I should think Mary Gordon would know what
remorse is--for, of course, there is only one thing to think."
II
Old Chester thought the one thing. No evasions of Dr. Lavendar's, no
miserable silence on the part of the disgraced father and the
infuriated brother, could banish that one thought. But nothing
definite was known. "Although," as everybody said to everybody else,
"of course, Dr. Lavendar knows the whole thing, and probably Willy King
does, too." If they did, they kept their knowledge to themselves. But
Dr. Lavendar went often to the Gordon house that winter. "They're
pretty lonely, those two men," he told Willy once--perhaps six months
afterwards.
"Would either of them have softened if the baby had lived, do you
think, sir?" William said. And Dr. Lavendar shook his head.
"Perhaps her father might. But Alex will never forgive her, I'm
afraid."
And Alex never did forgive her--not even when she died, as, happily,
she did six or seven years later. She died; and life closed over the
miserable little tragedy as water closes, rippling, over some poor,
broken thing flung into its depths.
"_Thank God!_" Alex said, when he heard she was gone.
"You may thank God for her," Dr. Lavendar said, turning upon him
sternly, "but ask mercy for yourself, because this door of opportunity
is shut upon you forever."
Dr. Lavendar had brought them the news. They did not ask how it had
come to him; it was enough to hear it. The two men, Mary's father and
brother, listened while he told them, briefly: "She died yesterday.
The funeral will be to-morrow, at twelve."
"Thank God!" Alex said, hoarsely, and lifted his hand and cursed the
man who had dishonored them.
And Dr. Lavendar turned upon him in solemn anger. "Your opportunity is
gone--so far as she is concerned. There yet remains, however, the
poor, foolish sinner whom she loved--"
"Damn him!" said Alex.
"_--and who loved her._"
Old Mr. Gordon dropped his face in his hands and groaned.
"Who loved her," Dr. Lavendar repeated.
"For that, at least, he cannot be indifferent to us, whatever he has
made us suffer."
Neither of his listeners spoke. It was growing dark in the long room,
walled to the ceiling with books and lighted only by a fire sputtering
in the grate. Mr. Gordon, sitting in his big, winged chair close to
the hearth, said, after a long pause: "You said--to-morrow, Edward?
Where?"
"In Mercer. I shall go up on the morning stage."
Again the silence fell. Alex got up and walked to the window and
looked out. "Why didn't you bring Danny in, Dr. Lavendar?" he said,
carelessly; "the little brute will freeze out there in your buggy.
I'll call him in." He turned to leave the room, and then stopped.
"Alexander, _sit down_," said Dr. Lavendar.
Alex sat down with involuntary quickness; then he threw his legs out in
front of him and thrust his hands down into his pockets. "Dr.
Lavendar, this is our affair. I'm obliged to you for your kind
intentions; but this is our affair. You've told your news, and we have
listened respectfully--if I should say gladly you might be shocked. So
I only say respectfully. But you have spoken; we have listened. That
is all there is to it. The thing is finished. The book is closed. I
say thank God! I don't know what my father says. If he takes my
advice, for I've been a good son to him; I never gave him any cause to
be ashamed;--if he takes my advice, he'll forget the whole affair.
That's what I mean to do. The book is closed. I shall never think of
it again." He got up and walked about with affectation of vast
indifference.
"Alex, you will probably never think of anything else," Dr. Lavendar
said, half pitifully; and then, sternly, again: "I can't make you
accept the opportunity that still is open to you; but I will point it
out to you: Come up to Mercer to-morrow with your father and me."
"Mercer!" the younger man cried out, furiously; "you mean to see her
buried? To dance on her grave and pull the man out and spit in his
face and--" He stopped, his face suddenly purpling, his light eyes
staring and rolling; then he stumbled and jerked himself together, and
lurched forward into a chair, breathing loudly. The two old men,
trembling with horror, ran to him. "Oh, Edward," John Gordon
said--"oh, Edward, why did you rouse him? He can't speak of it, he
can't think of it. Alex--there!--we'll say no more about it."
[Illustration: "LURCHED FORWARD INTO A CHAIR, BREATHING LOUDLY"]
Alex stared at them with glassy eyes, in silence; his father kept
bemoaning himself and imploring his old friend to say no more. "You
won't speak of it again, Edward? He goes out of his head with rage.
Promise me not to speak of it any more."
"No, John; no," Dr. Lavendar said, sadly; and as Alex's eyes cleared
into bewildered consciousness, the old minister stood a little aside
while the father helped the son to his feet and led him away. When he
came back, shuffling feebly down the long, darkening room, Dr. Lavendar
was still sitting by the fire. "He's quiet now; I--I think he's
ashamed. I hope so. But he won't come out of his room."
Dr. Lavendar nodded.
John Gordon spread his purple handkerchief over his white locks, with
shaking hands, and then sat down, tumbling back in his chair in a
forlorn heap. "Edward," he said, feebly, "tell me about it. It was on
Thursday? Had she been sick long?" Then, in a low voice, "She--didn't
lack for comforts?"
"No; I think not. The man was as tender with her as--as you might have
been. She was sick--I mean in bed--two weeks. She had been ailing for
a long time; you remember I spoke to you about it about a month ago.
And again last week."
"You--saw her?"
"Yes."
"More than once?"
"Oh, many times," Dr. Lavendar said, simply; "many times, of course."
John Gordon put out his hand; Dr. Lavendar shook it silently. Then
suddenly the old man broke out, in weak, complaining anger: "He
wouldn't let me write to her. I would have sent her some money. He
wouldn't hear of it. He was awful, Edward. I--I didn't dare."
Dr. Lavendar was silent. It had grown so dark that he could not see
the father's face. Suddenly, from behind the leafless trees at the
foot of the garden, a smouldering yellow glow of sunset broke across
the gloom of the room, and touched the purple cowl and the veined hands
covering the aged face. Dr. Lavendar sighed.
"What can I do, Edward? I can't go to-morrow. You see I can't."
"Yes, you can, John."
"He would die; he'd have another attack. His heart is bad, Edward."
"Oh, I'm afraid it is, I'm afraid it is. But John, you do your duty.
Never mind Alex's heart. That isn't your affair."
"Oh, I couldn't possibly go--not possibly," the father protested,
nervously.
The glow died out. The room grew dusk and then dark. Mr. Gordon got
up and reached to the mantel-shelf for a spill. "Mary used to make the
spills for me," he said, vaguely. "Now our Rachel does it, and she
doesn't half bend the end over." He lighted the spill, the little
flame flickering upon his poor old face peering out from under his
purple handkerchief. "Oh, Alex ought not to be so hard. I would go
with you to-morrow, Edward, but I can't, you know. I can't." Then,
with a shaking hand, he took off the ground-glass globe and lighted the
tall lamp that stood among a litter of papers on the library-table.
"You see how it is, Edward, don't you? I can't possibly go."
"You will be sorry if you don't, John."
"I'll be sorry anyhow," he burst out. "I'm always sorry. I've been
sorry all my life. My children are my sorrow."
III
Algy Keen, his face swollen with crying, his black hair limp and
uncurled, sat on the edge of the bed in the back room of a dingy Mercer
lodging-house. The windows had been left open after Mary had been
taken away, so that the room was cold; and there were still two chairs
facing each other,--a certain distance apart. The room was in dreary
order, and there was the scent of flowers in the chill air. The bed
was tumbled, for the forlorn man had dropped down upon it to rest. But
he was too tired to rest, and was sitting up again, dangling his
stockinged feet on the shabby carpet and talking to Dr. Lavendar. He
snuffled, and his poor, weak lips shook, and he rubbed the back of his
trembling hand across his nose. Algy had had broken nights for a
fortnight, and the last three days and nights of Mary's life he had
almost no sleep at all; these two days when she lay dead in their bare
room he had slept and wept and slept again; and now, when he and Dr.
Lavendar had come back from the funeral, he sat on the edge of the bed
and whimpered with weakness and grief.
"Well, sir, she was a good girl," he said. "I don't care what anybody
says, she was a good girl. I ain't saying that things was just right,
to begin with. But that wasn't Mary's fault. No; she was a good girl.
And her folks treated her bad. They'd always treated her mean bad. My
goodness! if they'd 'a' let me come to see her respectable, as you
would any of your lady friends, 'stead of skulkin' 'round--... _I can't
stand the smell of those flowers_," he broke out, in a high, crying
voice; "I left them all out there at the cemetery, and I smell them
here--I smell them here," he moaned, trembling.
"I like to smell them," Dr. Lavendar said. "They mean the old
friendship for Mary. Mrs. King sent them. She's our doctor's wife in
Old Chester. She always liked Mary."
"I don't see how she could help it," Algy said, his face crumpling with
tears. "Well, she was a good girl. And she was a good wife, sir, too.
I tell you, you never saw a better wife. I used to come home tired,
and there'd be my slippers out for me. Yes, sir; she never missed it.
And she was always pleasant, too; you mayn't call just being pleasant,
religion, but I--"
"I do," Dr. Lavendar interposed.
"Well, so do I," Algy said, his face lightening a little. "I call it a
better religion than her folks showed. Well, now, sir, I loved
Mary"--he stopped and cried, openly--"I loved her (I didn't need that
hell-hound of a brother to come after me)--yes, I was just as fond of
her; and yet there was times when I come home at night--not--not
quite--well, maybe a little--you know?"
"Yes," said Dr. Lavendar.
"But, my God, sir, Mary was pleasant. It isn't every woman that would
be pleasant then, is it?"
"No, it isn't, Algy."
"Course, next day she'd tell me I done wrong. (She never told me so at
the time--Mary had sense.) And I always said: 'Well, yes, Mary, that's
so. And I'll never do it again.' But she was pleasant. Course I
don't mean she was lively. She used to remember--well, that we'd made
a mistake. _You_ know? And she used to kind a brood on it. She
talked to you considerably about it, I guess. She said you comforted
her. She said you said that maybe her--her mistake had brought her to
be kind o' more religious--saved her, as you might say."
"I said that she had come to know her Saviour through His forgiveness."
"I don't think Mary needed any forgiveness," the poor husband said,
with tearful resentment; "I think her folks needed it."
"I'm sorry for them," Dr. Lavendar said. "They have got to remember
that they might have been kinder. That's a hard thing to have to
remember."
The young man nodded. "I hope they'll remember it, hard!"
"They will," said Dr. Lavendar, sighing.
"I spent my last cent on Mary," Algernon rambled on. "I got her a good
coffin--a stylish coffin. The plate was solid silver. The man wanted
me to take a plated one. I says 'no,' I says; 'I don't get plated
things for my wife if it takes my last cent.' Well, it just about took
it. But I don't care. Her people threw her off, and I did for her. I
spent my last cent."
"You took her from them in the first place, Algernon," the old minister
said. "Don't forget that you sinned."
"Well, you said she was forgiven," the other broke out, angrily. "I
guess God's more easy than some people."
"He is."
"Well, then," Algy said, resentfully; "what's the use of talking?"
Dr. Lavendar was silent.
"I don't begrudge a cent I spent on her," Algy went on. "I had laid by
$1140 to set up a place of my own here in Mercer. At least, it wasn't
me; I'm not one to save much; it was Mary did it. But these last eight
months have taken it all, 'cause I 'ain't done hardly any work;
couldn't be away from her on the road, you know; so we had to live on
that money. I could 'a' got a cheaper coffin; but I wouldn't. As for
the doctor, I got the best in town. I don't believe in economizing on
your wife. And I paid him. I paid him $204 yesterday morning, though
it seems high, considering he didn't cure her. But I wasn't going to
let Mary get buried owing the doctor. And I paid for the coffin.
'Spot cash,' I says to the man, 'make it spot cash, and name your
figure.' He took off $17. Well, how much do you suppose I've got left
now, Dr. Lavendar, out of $1140? Just $23, sir. I don't care; I don't
begrudge Mary a cent. I thought the coffin looked handsome, didn't
you?--_Oh, I wish somebody had 'a' moved those chairs when we were
gone!_" he cried, his voice shrill and breaking.
Dr. Lavendar got up and pushed one of the chairs back against the wall
and brought the other to Algy's side. The young man laid his hand on
it and began to cry.
IV
"No, I suppose you don't care to hear about it, John. But I want to
tell you; so I guess you'll listen to please me?"
John Gordon said nothing.
"It isn't a long story," Dr. Lavendar said, and told him briefly of the
funeral. When he ended there was silence. Then, "John," Dr. Lavendar
said.
"Yes, Edward."
"The man is in need."
"What's that to me?" the other burst out.
"Much," said Dr. Lavendar; "it gives you a chance."
"You mean a chance to give him some money?" said the other. "Good God!
To pay the scoundrel for what he did to us? Edward, you don't
understand human nature."
"He spent his last cent making Mary comfortable, John. She told me so
herself."
"I will never give that--creature one penny of my clean money."
Dr. Lavendar said nothing.
The older man bent forward, shivering, and stirred the fire. The coal
broke into sputtering fragments and the flames roared up into the soot.
"Alex would never listen to giving him any money."
"Don't ask him to listen to it. Haven't you got your own check-book?"
"Let him rot. That's what Alex says."
"I don't believe it's what you say, John, because he was good to
Mary;--and you were not."
Mr. Gordon groaned.
"Well, I won't give him anything; I'll lend it, possibly."
Dr. Lavendar frowned and got up.
Mr. Gordon put out a trembling, detaining hand.
"Edward, you don't understand.... How much do you want for him?"
"He had saved about $1200 to go into some business. It's all gone."
"Well, I won't give it to him," the other repeated, with feeble
sharpness; "I'll lend it--to please you."
"I'm sorry you haven't a better motive."
John Gordon got up and went over to his library-table and fumbled about
in one of the drawers for his check-book. "I'm a fool," he said,
fretfully; "I don't know but what I'm worse. Lending money to-- But
you say he was good to her? Poor Mary! Oh!" he ended, half to
himself, "I don't know why Alex is so hard." Then he took his quill
and began to scrawl his check. "I'd rather see him starve," he said.
"No, you wouldn't," Dr. Lavendar said, calmly.
"Well, there! Take it! Get a receipt."
"Johnny, think better of it."
"You needn't take it if you don't want to," the other said, sullenly.
Dr. Lavendar took it, and John Gordon called after him,
"You won't tell Alex?"
Dr. Lavendar shook his head and sighed. As he drove home he said to
himself that a loan was better than nothing. "But, Danny, my boy," he
added, "what a chance he had! Well, he'll take it yet--he'll take it
yet. The trouble with me, Daniel, is, I'm in too much of a hurry to
make folks good. I must reform."
Danny blinked a grave agreement, and Dr. Lavendar, dropping his
shortcomings joyfully from his mind, began to sing to himself:
"Oh! what has caused this great commotion--motion--motion
Our country through?"
When, however, a day or two later, Dr. Lavendar went up to Mercer to
take the check to Algernon Keen, he found to his astonishment that it
was not so easy to secure to his old friend even the smaller and meaner
opportunity of lending, much less giving.
At first, Algernon looked at him open-mouthed. "_Him_--offering to
lend money to--?" His astonishment robbed him of words. Then into his
poor, shallow face came the first keen touch of shame. But instantly
he was ashamed of his shame,--ashamed, like so many of us strange human
creatures, of the stirring of God within him. He didn't want their
dirty money, he said. They thought themselves so good, they couldn't
stomach Mary. Well, then, they were too good for him to touch their
money. His voice shook with angry grief. His bitterness was genuine,
even though he used it to hide that first regenerative pang of shame.
No; Dr. Lavendar could take their money back to them. "I spent my last
cent, just about, on Mary," he said; "and I didn't begrudge it, either."
"I'm sure you didn't begrudge it."
Algy's weak mouth shook and his eyes filled; he turned away and stared
out of the window. "He better have offered to lend her some money than
me," he said. "I bet he's glad she's dead."
(Dr. Lavendar thought of Alex.) "He wants to help you now for her
sake," he said.
"I don't want his money," the younger man insisted, brokenly; "he let
her die."
"I think that it would please her to have you take it."
"I don't want to be under obligations to those people," Algernon said,
doggedly.
"If Mr. Gordon has your note, it's business."
Algy hesitated. "I suppose he thinks I'd never pay it back?"
"If he takes your note, it looks as if he expected to be repaid."
"It's treating me white, I'll say that," Algernon said. And again his
face reddened slowly to his forehead and he would not meet Dr.
Lavendar's eye. "But I don't want their favors," he cried,
threateningly.
"It's business, if you give your note," Dr. Lavendar repeated. "Come,
Algernon, let her father do something for her sake. And as for
you--it's a chance to play the man; don't you see that?"
Algy caught his breath. "Damn!--if I borrowed his money I'd pay
it--I'd pay it, if it took the blood out of me."
"I will make your feeling clear to him," Dr. Lavendar said. "Let's
make out the note now, Algy."
The old man got up and hunted about for pen and paper. "Here's a
prescription blank," he said; "that will do." An ink-bottle stood on
the narrow mantel-shelf, a rusty pen corroding in its thickening
depths; but Dr. Lavendar, in a very small, shaky old hand, managed to
scrawl that "Algernon Keen, for value received, promised to pay to John
Gordon--"
--"in a year," Algy broke in; "I ain't going to have it run but a
year--and put in the interest, sir. I'll have no favors from 'em.
I'll pay interest; I'll pay six per cent.--like anybody else would."
--"and interest on same," Dr. Lavendar added. "Now, you sign here,
Algy. There! that will please Mary."
"Oh, my!" said Algernon, his poor, red-rimmed eyes filling--"oh, my!
my! what will I do without her?"
V
The next day Dr. Lavendar carried the note back to old John Gordon, who
took it, his mouth tightening, and glanced at it in silence. Then he
shuffled over to a safe in the corner of his library and pulled out a
japanned tin box. Dr. Lavendar watched him fumble with the combination
lock, holding the box up to catch the light, and shaking it a little
until the lid clicked open. "He'll never pay it," John Gordon said.
"He'll try to," Dr. Lavendar said; "but it's doubtful, of course. He's
a sickly fellow, and he hasn't much gumption. But if there's any good
in him, your trusting him will bring it out."
"There isn't any good in him," the other said, violently.
And that was the last they said about it; for the time Algernon Keen
dropped out of their lives.
He set up his little store in Mercer, and struggled along, advertising
his samples of perfumery and pomade upon his own person; trying to
drink a little less, for Mary's sake; whimpering with loneliness and
sick-headache in his grimy room in the hotel where Mary had died; and
never forgetting for a day that promise to pay on the back of the
prescription paper in John Gordon's possession. But when the year came
round, on the 2d of December, he had not a cent in hand to meet his
obligation. And that was why Dr. Lavendar heard of him again. Would
the doctor--this on perfumed paper, ruled, and with gilt edges--would
the doctor "ask him if he would extend?" Algernon could pay the
interest now; but that was all he could do. He wasn't in very good
shape, he said. He'd been in the hospital for a month, and had had to
hire a salesman. "I guess he cheated me; he was a kind of fancy
talker, and got me to let him buy some stock; he got off his slice, I
bet." That was the reason, Algy said, that he could not make any
payment on the principal. But he was going to introduce a new article
for the lips (no harmful drugs in it), called Rosebloom--first-class
thing; and he expected he'd do first rate with it. And in another year
he'd surely pay that note. It hung over him, he said, like a ton. "I
guess he don't want it paid any more than I want to pay it," Algy
ended, simply.
Of course Dr. Lavendar asked for an extension, and got it, though John
Gordon's lip curled. "I never expected to hear from him or his note
again," he said. "Probably his honesty won't last over another year."
Dr. Lavendar went up to Mercer to see Algy, and they talked things over
in the store between the calls of two customers. Algy's hair was sleek
and curly as before, for business is business; but he looked draggled
and forlorn; his color had gone and he was thinner, and there were
lines on his forehead, and his bright, hazel eyes, kind and shallow as
those of some friendly animal, had come into their human birthright of
worry. "It's this note that takes the spunk out of me," he said. "If
I could only get it paid! Then I'd hire a house and have the shop in
front. I've thought some I'd get married, too. It's hard on your
digestion living in one of these here cheap hotels. But I can't get
over thinking of Mary. I don't seem to relish other ladies. I suppose
they're all right; but Mary was so pleasant." And his eyes reddened.
"And, anyway, it would cost more to keep a wife, and I don't propose to
spend money that way. _He's_ treated me white, I'll say that for him;
and I propose to show him--Dr. Lavendar, I haven't drunk too much only
three times in the last year--honest, I haven't. I thought you'd think
that would please Mary?"
"I'm sure it does," said Dr. Lavendar.
"I suppose you think," the drummer said, sheepishly, "that it was
pretty darned foolish to drop three times?"
"I think pretty soon it won't be even three times," Dr. Lavendar
declared; "but it's hard work; I know it is."
Algernon looked at him eagerly. "You know how it is yourself, maybe?"
"Well, I never happened to want to take too much," Dr. Lavendar said,
gently; "if I had, it would have been hard, I'm sure."
"Well, you bet," Algy told him, knowingly. Then they talked the
business over, and Dr. Lavendar clapped Algy on the shoulder and said
he believed he'd have that house and shop yet. "Rosebloom may be a
gold-mine," said Dr. Lavendar. Then he gave Algy some advice about the
window display, and suggested a little gas-jet on the counter where
gentlemen might light their cigars; and he told Algy what brand he
smoked himself, and recommended it, in spite of its price. Algy
smacked his thigh at that, and said Dr. Lavendar had the making of a
smart business man in him. Indeed, Algy felt so cheered that he opened
his show-case and displayed a box of his new cosmetic.
"Look here, doctor," he said, earnestly; "I'll give you a box.
Yes--yes! I will. I'd just as lief as not. You maybe wouldn't want
to use it yourself; gentlemen don't, often. But give it to one of your
lady friends. Do, now, doctor. It don't cost me much of anything--and
I'm sure you've been kind to me."
And Dr. Lavendar accepted the lip-salve, and thanked Algy warmly; then
he said that the picture on the lid of the tight-waisted lady was very
striking.
"That's so!" cried Algy. "She's a beauty. She makes me think of Mary."
Algernon had presented Dr. Lavendar with a cigar, and the old minister
was smoking it in great comfort, his feet on the base of a rusty,
melon-shaped iron stove; Algy was leaning back against the counter, his
elbows on the show-case behind him. "Dr. Lavendar," he said, looking
at the toe of his boot, "I--got something on my mind."
"Well, off with it, quick as you can."
"I've been thinking about the Day of Judgment."
"Ho!" said Dr. Lavendar.
"Well, sir, I get to thinking: if everybody's sins are to be read out
loud before all the world--standing up, rows and rows and rows of 'em.
Can't see the end of 'em--so many. I kind a' hate to think that Mary
might hear--things about me."
"Well, Keen," said Dr. Lavendar, slowly, "I don't believe it will be
that way." He hesitated a little. After all, it is a risk to take
away even a false belief, unless you can put a true one in its place.
Algy stopped looking at the toe of his boot. "_What!_" said he.
"Now just look at it," said Dr. Lavendar. "Who would be the better for
that kind of publicity? Good people wouldn't like it; it would pain
them. You say yourself that Mary wouldn't like to hear that you did
wrong three times."
"No; she wouldn't," Algernon said.
"Wicked people might enjoy it," Dr. Lavendar ruminated, "but--"
--"but God don't cater to the wicked?" Algy finished, quickly.
"That's just it," said Dr. Lavendar. "He doesn't. But I tell you what
it is, Algy, it is painful enough to just have your Saviour tell you
your sins when you're sitting all alone--or, maybe, lying awake in the
dark; that's a dreadful time to hear them. It's worse than having rows
of people listening."
Algernon nodded. "Maybe you're right," he said, sighing.
The birth of a soul is a painful process. But when he went away Dr.
Lavendar's eyes were full of hope.
And he grew more hopeful when, as the next year came round and Algernon
again asked for extension, he was able to carry back, not only the note
and the interest to John Gordon, but a payment of $24. What that $24
meant of self-denial and perseverance Dr. Lavendar knew almost as well
as Algy himself.
"I don't know whether you meant it, John," he said, as the old man took
the note and locked it up in the japanned box--"I don't know that it
was your intention, but I believe the responsibility of debt is going
to make a man of Mary's husband."
"Debt doesn't generally work that way," Mr. Gordon said.
"No; it doesn't. But He maketh the wrath of man to praise Him, once in
a while, Johnny."
"It's nothing to me. I'm done with him."
"'If the court knows itself, which it think it do,'" said Dr. Lavendar,
chuckling, "you're just beginning with him."
"I'd rather have him decent, if that's what you mean. But I despise
him."
"I don't," said Dr. Lavendar. "I tell you, John, we're poor, limited
critters, you and I. We felt that no good could possibly come out of
Nazareth. I must confess that when I got you to send him that money I
was thinking more of the benefit to you than any effect it might have
on him. I thought he didn't amount to two cents. To my shame I say
it. But I was blind as a bat; the Lord had sent him a great
experience--_Mary's death_. Well, it was like a clap of thunder on a
dark night; the lightning showed up a whole landscape I didn't know.
There was honesty; and there was perseverance; and there was love, mind
you, most of all. Love! I tell you, Johnny, only the Lord knows what
is lying in the darkness of human nature. In fact," said Dr. Lavendar,
reflectively, "as I get older there is nothing more constantly
astonishing to me than the goodness of the Bad;--unless it is the
badness of the Good. But that's not so pleasant. No, sir; I don't
despise Mr. Keen."
Nor did he despise Algy when the note had to be extended still again,
although again Algy was ready not only with the interest, but with
$37.50 of the principal.
VI
As Algernon struggled along with Rosebloom and cheap cigars and bright
red and green perfumed soaps, the debt was lessened and lessened; and
the back of the note was almost covered with extensions, yet only $317
had been paid off. In spite of himself John Gordon grew interested; he
would not have admitted it for the world, but he wanted to hear about
Dr. Lavendar's annual visits to Mercer; and Dr. Lavendar used to drive
out to smoke a pipe with him and tell him what Algy had said and done.
One day--it was seven years after the note had been drawn--a clear,
heartless winter day, with a cold, high wind that made the old minister
look so blue that John Gordon mixed a glass of whiskey-and-water and
made him drink it before they began to talk--that day Mr. Gordon went
so far as to ask a question about Algy. "Has he given you anything
more for your complexion, Edward?" he said, with a faint grin.
"He gave me a smelling-bottle this time. I handed it over to Mary, and
told her not to let me get a sniff of it; and she said, 'Sakes! it's
beautiful!' But I'll tell you something he said, Johnny: he said that
his debt to you was a millstone round his neck. And yet the truth is,
it's a life-buoy!"
John Gordon looked at the soiled, crumpled paper, with its dates of
extensions, and smiled grimly. "Well, I won't deprive him of his
life-buoy."
"The store is doing pretty well," Dr. Lavendar went on--and stopped,
because Alex entered.
"Whose store is doing pretty well," he asked, civilly enough--for Alex.
"Algernon Keen's," said Dr. Lavendar.
Alex's face changed; he looked from one to the other of the old men by
the fire, and he saw his father's hand open and close nervously. But
he restrained himself until their visitor had gone. He even went out
into the sharp, bright wind and unhitched Dr. Lavendar's little blind
horse Goliath, backing the buggy close to the steps and helping the old
man in with what politeness he could muster. Then he hurried back into
the library to his father.
"I should like to know, sir," he said, standing up with his back to the
fire, his legs, in their big, mud-stained top-boots, wide apart, his
hands under his coat-tails--"I should like to know, sir, why Dr.
Lavendar sees fit to refer to a subject which is most offensive to us?"
He fixed his motionless, pale eyes on his father, shrinking back in the
winged chair.
"I don't know--I don't know," said John Gordon. Then, suddenly, he put
out his hand and caught at the crumpled note on the table beside him
and put it in his pocket. Instantly suspicion flamed into Alex's eyes.
His face turned dully red, almost purple. He made a step forward as
though to interpose and grasp at the paper, restrained himself, and
said, with laborious politeness:
"If that is a note, sir--I thought I saw indorsements of
interest--sha'n't I put it into the safe for you?"
"I won't trouble you, Alex."
Alex stood silent; then suddenly he struck the table with his fist: "My
God! I believe you've been lending money to that--to that--"
Mr. Gordon began to shake very much.
"Did Dr. Lavendar presume to ask you to lend money to--to--"
Mr. Gordon passed his hand over his lips; then he said, faintly, "No;
he didn't."
Alex, like a boat brought suddenly up into the wind, stammered
uncertainly. "Oh; I--I--thought--" And then suspicion broke out
again. "Has the creature asked you for a loan?"
"No," Mr. Gordon said.
And Alex gaped at him, silenced. Yet he was certain that that strip of
paper had some connection with Algernon Keen. "I beg your pardon," he
said; "I thought for an instant that you were dickering with the man
who seduced your daughter. I am sure I beg your pardon for the
thought," he ended, with elaborate and ironical courtesy, for his
father's obvious agitation assured him that he was right. "I only felt
that if it was his note, it must be kept carefully--carefully." He
smiled in a deadly way he had, and opened and shut his hand as though
he would close it on the hilt of a knife. "But, of course, I was
mistaken. You would press it if you had his note--although 'sue a
beggar.' And, besides, if we had got as far as lending him money, we
would be asking him to dinner next."
Mr. Gordon cringed.
"So I beg your pardon," Alex ended, sardonically.
"Very well--very well," his father said; and got up and began to potter
about among his books, as much as to say that the subject was ended.
"It _is_ a note," Alex said to himself, and smiled.... So far the
creature had gone scot-free. In these days of lawfully accepted
dishonor revenge is not talked about. But perhaps it would come to his
hand. Not the revenge of the instincts--not the shedding of blood, man
fashion; but the revenge of inflicting misery. Not much of a revenge,
of course, but the best that he could get. And so he smiled to
himself....
He said no more at the time; but months later his father realized that
the incident was not forgotten when Alex said, suddenly, sneering: "So
your son-in-law is prospering in his business? I saw his establishment
to-day in Mercer. If he owes you any money he will be able to pay
cash. I congratulate you, sir."
Old Mr. Gordon made no reply. He was very feeble that autumn. Willy
King told Alex that another attack of bronchitis would be the end. "He
can't stand it," said Dr. King. "I'd take him South, Alex, if I were
you."
Alex did not like to leave his mill in Upper Chester, but, as he told
Willy, he was a good son, and always did his duty to his father. "I
play dominoes with him every night," he said;--so he would take the old
man South, though to go and come would keep him from business almost a
week.
It was then that John Gordon told Dr. Lavendar that Alex suspected him
of lending money to Mr. Keen. "And if I die," he said, "Alex will
squeeze the poor devil--he'll squeeze him till he ruins him. I--I
suppose I'm a great fool, but I almost thought maybe, sometime, I'd
destroy that note, Edward?"
Dr. Lavendar chuckled: "I knew you'd come to it, Johnny; but--" he
stopped and ruminated. "You've come to it; so that's all right. But
do you know--I don't believe he can do without it quite yet awhile."
"Poor devil!" John Gordon said again, kindly. "Well, I'll let him gnaw
on it awhile longer. I suppose he'll want another extension?"
"Probably," said Dr. Lavendar. "He is just holding his own this year;
he will be able to pay the interest, he told me, but not very much
more."
Extension was necessary, as Dr. Lavendar had foreseen; and when he
wrote to Mr. Gordon about it the old man replied in obvious fear of his
son. The note was in his safe, he said; Edward knew where it was; it
was in the japanned box. "But I don't care to ask Alex to get it," he
explained. "He doesn't know of its existence; so I'll give you power
of attorney to see to it. You'd better just have Ezra Barkley put it
in shape for you, because it will be necessary to go up to the house
and open the safe to get it and put it back again. Alex is never at
home until late in the afternoon, but Rachel is there and will let you
in. You'll find some very good Monongahela in the chimney closet."
Then he added the combinations of the locks on the safe and the
japanned box.
"Stick that in, Ezra, will you, about going up to the house?" Dr.
Lavendar said.
And Ezra stuck it in solemnly, and then held his pen between his teeth
and blotted his paper. "It is estimated," he observed, through his
shut teeth, "that the amount of ink used in the United States of
America, in signatures to wills, since the year when the independence
of the colonies was declared, would be sufficient in bulk to float a--"
"Well, Ezra," said Dr. Lavendar, chuckling, "this paper seems rather
liberal. Suppose I take some cash out of the safe to repair the roof
of the vestry? It leaks like a sieve."
"Your construction of liberality is at fault, sir," Mr. Ezra corrected
him, gently; "this paper defines just exactly what you may do, up to
the moment when the principal reclaims the paper--or dies."
"Well, I hope he won't reclaim it, or die, either, till he gets an
affair we are both interested in patched up," Dr. Lavendar said; then
he listened politely while Mr. Ezra told him how many times the word
"ink" occurred in Holy Writ.
Dr. Lavendar went away with his power of attorney in his pocket. And
when he sent it to John Gordon to sign, he seemed to take it for
granted that he and Mr. Gordon were equally interested in the
development and well-being of Mary's husband. He said in his letter
such things as, "You'll make a man of him yet;" and, "Your patience has
given the best elements in him time to come out." Dr. Lavendar had a
perfectly unreasonable way of imputing good motives to people; the
consequence was he was not very much astonished when they displayed
goodness. He was not astonished when, some two months later, another
letter came from old Mr. Gordon, saying that on the whole he thought
the note had better not run any longer. "I am going to forgive him his
debt," Mary's father wrote, in a feeble scrawl; "and I'll be obliged to
you if you will go up to my house and get that note and send it to me.
I'm pretty shaky on my pins, and I don't want to run risks, so I wish
you'd tear the signature out and burn it before you mail the note.
I'll send it along to Mr. Keen. I mean to write to him and tell him I
think he is honest, anyway. The fact is, I half respect the poor
fellow. It's been a long winter, and I can't say I'm much better.
Willy King doesn't know everything. These doctors are too confoundedly
ready to send a man away from home. I should have been just as well
off in Old Chester. _Be sure and destroy that signature_."
Dr. Lavendar read this letter joyfully, but without surprise. "I'm
glad he didn't take my advice and let it go on any longer," he said to
himself; "I guess I'll risk the effect on Algy now."
Then he wondered if there would be any danger of meeting Alex if he
went up to the house right after dinner. "I can't manage it this
morning," he said to himself. "I've got to go and see Mrs. Drayton.
Well, I wish the Lord would see fit to cure her--or something."
So he went plodding out into a still, gray February day, and called on
Mrs. Drayton, and stopped at the post-office to hear the news, and then
went home to his dinner. "Ye're not going out _again_?" his Mary
cried, in shrill remonstrance, when in the afternoon she saw him muffle
himself up for the drive out into the country; "it's beginning to snow!"
"I am," said Dr. Lavendar; "and see you have a good supper for me when
I get back." He got into his buggy, buttoning the apron up in front of
him, for it was a wet snow. He had on a shabby old fur cap, which he
pulled well down over his forehead, furrowed by other people's sins and
troubles; but his eyes peered from under it as bright and happy as a
squirrel's.
His little blind horse pulled slowly and comfortably up the hill,
stopping to get his breath on a shaky bridge over a run. In the
silence of the snow Dr. Lavendar did not hear the stage coming down the
hill until it was almost on the bridge; then he had to pull over to let
it pass. As he did so the single passenger inside rapped on the
window, and then opened it and thrust his head out, calling to the
driver to stop.
"Dr. Lavendar! you have heard, I suppose? Very sad. A great shock.
Of course I'm going on at once to bring the body back. It is difficult
to get off at this season, but a son has a sacred duty." Alex's pale
eyes were bulging from his red, excited face.
"What news?" Dr. Lavendar said. "You don't mean--Alex! John
isn't--your father isn't--"
"My father is dead," Alex said, with ponderous solemnity. "It is a
great grief, of course; but I trust I shall be properly resigned. His
age rendered such an event not altogether unexpected."
Dr. Lavendar could not speak; but as the stage-driver began to gather
up his reins from the steaming backs of his horses, he said, brokenly:
"Wait--wait. Tell me about it, Alex; your father and I have been
friends all our lives." Alex told him briefly: He had just had a
despatch; his father had died that morning; he had been less well for a
fortnight. "I had a letter from him this morning," Alex said, "in
which he referred to his health--"
"So had I--so had I."
"I cannot get back with the body for six days--three to go, three to
come," Alex said, "but I will be obliged if you will arrange for the
obsequies next Thursday."
"Yes, yes. I will make any arrangements for you," Dr. Lavendar said.
He took out his big red silk pocket-handkerchief and blew his nose with
a trembling flourish. "We were boys together; your father was the big
boy, you know; I was the youngster. But we were great friends. Alex,
I am afraid my own grief has made me forgetful of yours; but you have
had a loss, my boy--a great loss."
"Very much so--very much so," Alex agreed, with a proper sigh, and
pulled up the window of the stage, then lowered it abruptly: "Oh, Dr.
Lavendar, are you going on as far up as--as _my_ house?"
"As _your_ house?" Dr. Lavendar repeated. "Oh--oh yes; I didn't
understand. Yes, I am."
"Would it inconvenience you," Alex said, "to stop there? I am going to
ask Mr. Ezra Barkley to come up at once and put seals on various
things. I am the sole executor, as well as the heir, of course; but I
sha'n't be able to attend to things for a week; and the forms of law
must be observed. If you could be on hand when Barkley is there--not
that I do not trust him."
Dr. Lavendar stared at him blankly; for an intelligent man, Alex was
sometimes a great fool. But he only nodded gravely, and said he would
stop at the house and wait for Mr. Ezra; Alex signed to the driver, and
the stage went rolling noiselessly on into the storm. When, at the
foot of the hill, Alex glanced back through the little oblong of bubbly
glass in the leather curtain of the coach, he saw Dr. Lavendar's buggy
standing motionless where he had passed it on the bridge; then the snow
hid it.
Under the bridge the creek ran swiftly between edges of ice that here
and there had caught a dipping branch and held it prisoner, or had
spread in agate curves--snow white, clear black, faint white
again--around a stone in mid-stream. On the black current, silent
except for a murmurous rush of bubbles under the ice, the snowflakes
melted instantly, myriads of them--hurrying, hurrying, hurrying; then,
as they touched the water, gone. Dr. Lavendar, in the buggy, sat
looking down at them:
"_In an instant--in the twinkling of an eye, we shall be changed._" ...
"He was my oldest friend." ("Was": with what an awful promptitude the
mind adjusts itself to "he _was_"!) Yet as he sat there, peering out
over the top of the apron and making, heavily, those plans familiar to
every clergyman, Dr. Lavendar did not really believe that the plans
were for Johnny. The snow fell with noiseless steadiness; the top of
the buggy was white; thimbles of down heaped themselves on the hubs,
tumbling off when the horse moved restlessly a step forward or backed a
little and stamped. Suddenly Goliath shook himself, for the snow was
cold upon his shaggy back, and the harness clattered and the shafts
rattled. Dr. Lavendar drew a long breath. "G'on!" he said. And
Goliath went on with evident relief. He knew the road well, and turned
in at the Gordon gateway, as a matter of course. When he stopped at
the front steps, the door opened and Rachel stood there, her eyes red.
"Sam will take him round to the stable, sir," she said, as Sam shambled
out from the back of the house to stand at Goliath's head. "Oh, my!
sir; I suppose you've heard?"
"Yes, Rachel; I've heard," the old man said, unbuttoning the apron and
climbing out. Rachel took his hand and wept audibly. "I knew he'd
never come back; he was marked for death. I've lived here eighteen
years, and I always said it was a privilege to work for a gentleman
like him."
"Yes--yes," he said, kindly. He was plainly agitated, and Rachel saw
that he was trembling.
"Course you feel it, sir, being about of an age," she said,
sympathetically. "Dr. Lavendar, sir, won't you have a glass of
something?" With the hospitality of an old servant, she would have
opened the little closet in the chimney-breast, but he checked her.
"Not yet; not now, Rachel. Leave me here awhile by myself, my girl.
I'll come out to the kitchen and see you before I go. When Mr. Barkley
comes, ask him to step into the library."
"Yes, sir," Rachel said, obediently; and went away sniffling and
sighing.
Dr. Lavendar stood looking about him at the emptiness of the room: the
winged chair, with the purple silk handkerchief hanging over the back;
the table heaped with books; the fire drowsing in the grate; the old
safe in the corner by the window. Outside, the snow drove past,
blotting the landscape. Ezra would probably arrive within a half-hour;
he had better get the note before he came. Then there need be no
explanations.
When Mr. Ezra came in he found the old minister sitting by the fire,
quite calm again, and even cheerful. "Yes," he said, in answer to the
lawyer's very genteel expressions of sympathy--"yes, I'll miss him. We
were boys together. He used to call me Bantam. I hadn't thought of it
for years."
"Nicknames," said Mr. Ezra, "were used by the ancients as long ago as
300 B.C."
"Well, I'm not as ancient as 300 B.C.," said Dr. Lavendar, "but I
called him Storkey; I can't imagine why, for he was only an inch and a
half taller; he always said it was two inches, but it wasn't. It was
an inch and a half."
"We are here," said Mr. Ezra, pulling off his gloves and coughing
politely, "for indeed a solemn and an affecting task. It is my duty,
sir, to seal the effects of the deceased, so that they may be
delivered, intact, to the executor."
Dr. Lavendar nodded.
"In all my professional career I have never happened to be called upon
for this especial duty. It is quite unusual. But Alex seemed to think
it necessary. Alex is a good son."
"So he says," said Dr. Lavendar.
"Are you aware, sir," proceeded Mr. Ezra, producing from his bag the
paraphernalia of his office, "that such is the incredible celerity of
bees (belonging to the _Hymenoptera_) that they can within twenty-four
hours manufacture four thousand cells in the comb? This interesting
fact is suggested by the use of wax for sealing."
Dr. Lavendar watched him in a silence so deep that he hardly heard the
harmless stream of statistics; but at last he was moved to say, with
his kind, old smile, "How _can_ you know so many things, Ezra?"
"In my profession," Mr. Ezra explained, "it is necessary to keep the
mind up to the greatest agility; I, therefore, exercise it frequently
in matters of memory." He lit a candle and held his wax sputtering in
the flame. "I recall," he said, "with painful interest, that at one of
our recent meetings I had the honor of drawing the power of attorney
for you, from the deceased."
"So you did," said Dr. Lavendar.
"Did you ever reflect," said Mr. Barkley, "that should that power be
used after the death of the donor, to carry out a wish of said donor,
expressed an hour, nay, a moment, before the instant of
dissolution--such act would be an offence in the eye of the law?"
"I've always thought the law ought to put on spectacles, Ezra," said
Dr. Lavendar; "it has mighty poor eyesight once in a while."
Mr. Barkley was shocked. "The law, Lavendar, is the deepest expression
of the human sense of justice!"
"But, Ezra," Dr. Lavendar said, suddenly attentive, "that is very
interesting. I remember you referred to the lapsing of the power of
attorney when you made out that paper for me; but I didn't quite
understand. Do you mean that carrying out, now, directions given
before the death of my old friend would be against the law? Suppose he
had asked me--last week, perhaps, to destroy--well, say that old
account-book there on the table, couldn't I do it to-day?"
"Dr. Lavendar, you do not, I fear, apprehend the majesty of the law!
Why," said Mr. Ezra, standing up, very straight and solemn, "such a
deed--"
"But suppose I didn't want--suppose Johnny didn't want, for reasons of
his own, to have anybody--say, even his executor--see that
account-book; suppose it might be put to some bad purpose--used to
injure some third person (of course that is an absurd supposition, but
it will do for an illustration); if he had asked me last week to
destroy it, do you mean to say, Ezra, I couldn't destroy it
to-day?--just because he happened to die this morning!"
"My dear sir," said Mr. Ezra, "such conduct on your part would be
perilously near a criminal offence."
Dr. Lavendar whistled. "Well, Ezra, I won't destroy it."
"I hope not, sir--I hope not, indeed," cried Mr. Ezra.
Dr. Lavendar laughed; he had the impulse to turn round and wink at
Johnny, to take him into the joke. But it was only for an instant, and
his face fell quickly into puzzled lines.
"A moment's reflection," Mr. Ezra continued, "will convince you, Dr.
Lavendar, that the aforesaid account-book is now the property, not of
the deceased, but of the estate. Its destruction would be the
destruction of property belonging to the heirs. Furthermore, your
belief that the herein before mentioned account-book might be put to an
improper use, for the injury of a third person--such belief would no
more justify you in destroying it than would your belief in its
unfairness towards said third person justify you in destroying a will."
Dr. Lavendar thrust out his lower lip and stared at him, frowning.
"Yes," he said, slowly--"yes; I see. I did not quite understand. But
I see."
Mr. Ezra solemnly began to pour forth a stream of statistics; he
referred to the case of Buckley vs. Grant, and even mentioned chapter
and page of _Purdon's Digest_ where Dr. Lavendar could find further
enlightenment. Dr. Lavendar may have listened, but he made no comment;
he sat staring silently at the old purple handkerchief on the top of
John's chair.
When Mr. Ezra had finished his work and his statistics, the two men
shook hands; then Dr. Lavendar said good-bye to Rachel and climbed into
his buggy, buttoning the apron high up in front of him; the lawyer
mounted his horse, and they plodded off into the snow, single file.
But Dr. Lavendar's eyes, under his old fur cap, had lost their
squirrel-like brightness....
So Algy's note belonged to the estate; and the estate belonged to Alex;
and Alex was the executor. And upon Alex Gordon his father's
intentions in regard to Algy's note would make no more impression than
the flakes of snow on running water. A vision of Alex's mean and cruel
mouth, his hard, light eyes, motionless as a snake's in his purpling
face, made Dr. Lavendar wince. The note--the poor, shabby, worn
note,--that stood for the best there was in Algy, that stood for
perseverance and honesty and courage; the note, which had weighed so
heavily that he had had to stand up in his pitiful best manhood to bear
it: the note that John had meant to "forgive"--Alex would use to
humiliate and torture and destroy. Under the pressure which he would
bring to bear that note would be poor Algy's financial, and perhaps his
moral, ruin. "And if I had not objected, John would have cancelled
it," Dr. Lavendar thought, frowning and blinking under his fur cap. He
saw the smoking flax quenched, the bruised reed broken; he saw Algy
turning venomously upon his enemy--for he knew him well enough to know
that his code of defence would not include any conventional delicacy;
he saw the new and hardly won integrity crumbling under the assault of
Alex's legal wickedness. Dr. Lavendar groaned to himself. Alex could,
lawfully, murder Algernon Keen's soul.
When Mary saw the old minister come into the house she was much
displeased. "There, now, look at him," she scolded; "white as a sheet.
What did I tell you? I'll bet ye he won't eat them corn dodgers, and I
never made 'em finer."
It must be admitted that Mary was right. Dr. Lavendar did not eat much
supper. He went shuffling back to his study, Danny slinking at his
heels; but for once he did not notice his little, grizzled friend.
When he got into his flowered cashmere dressing-gown and put on his
slippers and stirred his fire, he sat a long time with his pipe in his
hand, forgetting to light it. When he did light it, it went out,
unnoticed. Once Danny tried to scramble into his chair, but, receiving
no encouragement, curled up on the rug. The fire burned low and
smouldered into ashes; just one sullen, red coal blinked in a corner of
the grate; Dr. Lavendar watched this red spot fixedly for a long time.
Indeed, it was well on towards twelve before he suddenly reached over
for the bellows and a couple of sticks, and, bending down, stirred and
blew until the sticks caught and the cinders began to sparkle under the
ashes. This disturbed Danny, who sat up, displeased and yawning. But
when at last the flames broke out, sputtering and snapping, and caught
a piece of paper--a shabby, creased piece of paper covered with
dates--caught it, ran over it, curling it into brittle blackness, and
then whirled it, a flimsy, crumbling ghost, up the chimney, Dr.
Lavendar's face shone with a light that was not only from the fire.
"Ha, Danny, you scoundrel," he said, cheerfully, "I guess you are
_particeps criminis_!"
Then he went over to his study-table and rooted about for a thin,
shabby, blue book, over which he pored for some time, stopping once or
twice to make some calculations on the back of an envelope, then
turning to the book again. He covered the envelope with his small,
neat figuring, and turned it over to begin on the other side--and
started: "Johnny's letter!" he said. But when the calculations were
made, the rest was easy enough: first, his check-book and his pen. (At
the check he looked with some pride. "Daniel," he said, "look at that,
sir. You never saw so much money in your life; and neither did I--over
my own signature.") Next, a letter to Alex Gordon:
"MY DEAR ALEXANDER,--I owe your father's estate to the amount of the
enclosed check. No papers exist in regard to it, as the matter was
between ourselves. I will ask you for a receipt. Yours truly,
"EDWARD LAVENDAR."
THE GRASSHOPPER AND THE ANT
I
When William Rives and Lydia Sampson quarrelled and broke their
engagement, Old Chester said that they were lucky to fall out two weeks
before their wedding-day instead of two weeks after it. Of course, Old
Chester said many other things: it said it had always known they could
never get along. William, who had very little money, was careful and
thrifty, as every young man ought to be; Lydia, who was fairly well
off, was lavish and no housekeeper. "What could you expect?" demanded
Old Chester. Old Chester never knew exactly what the trouble between
them had been, for they kept their own counsel; but it had its
suspicions: it had something to do with William's father's will. By
some legal quibble the Orphan's Court awarded to William a piece of
property which everybody knew old Mr. Rives supposed he had left to his
daughter Amanda. Lydia thought (at least Old Chester thought she
thought) that William would, as a matter of course, at once turn the
field over to his sister. But William did no such thing. And, after
all, why should he? The field was his; the law allowed it, the Court
awarded it. Why should he present a field to Amanda? Old Chester said
this thoughtfully, looking at William with a sort of respectful regret.
Very likely Lydia's regret was not respectful. Lydia was always so
outspoken. However, it was all surmise. About the time that Amanda
did not get the field the engagement was broken--and you can put two
and two together if you like. As for Old Chester, it said that it
pitied poor, dear Lydia; and it was no wonder William left town after
the rupture, because, naturally, he would be ashamed to show his face.
But then it also said it pitied poor, dear William, and it should think
Lydia would be ashamed to show her face; for, of course, her obstinacy
made the trouble--and a young female ought not to be obstinate, ought
not, in fact, to have opinions on such matters. Legal affairs, said
Old Chester, should be left to the gentlemen. In fact, Old Chester
said every possible thing for and against them both; but gradually, as
years passed, conflicting opinions settled down to the "poor Lydia"
belief.
This was, probably, for two reasons: first, because William had never
seen fit to come back to Old Chester, and that, quite apart from his
conduct to his lady-love, was a reason for distrust; and, secondly,
Lydia had, somehow, become Old Chester's one really poor person--that
is, in a genteel walk of life. After the crumbling of the Sampson
fortune, Old Chester had to plan for Lydia, and take care of her, and
give her its "plain sewing"; so, naturally, William was reprobated.
Besides, she may have quarrelled and broken her engagement two weeks
before her wedding, but all these years afterwards she had been
faithful to the memory of Love! Old Chester knew this, for the simple
reason that Miss Lydia, during all these years, had kept in her
sitting-room a picture of William Rives, adorned with a sprig of box;
furthermore, it knew (Heaven knows how!) that she kissed this slender,
tight-waisted picture every night before she went to bed. Of course,
Old Chester softened! Lydia may have broken her engagement and all
that, but she kept his picture, and she kissed it every night. "But he
ought to be ashamed of himself," said Old Chester--"that is, if he is
alive." Then it added, reflectively, that he must be dead, for he had
never returned to Old Chester. Yet as time went on people forgot even
to disapprove of William; they had enough to do to take care of poor
Lydia, "for she is certainly very poor--and very peculiar," said Old
Chester, sighing.
"Peculiar!" said Martha King; "I call it something worse than peculiar
to spend money that ought to go towards rent on a present for Rachel
King's Anna. She gave that child a picture-book. I'm sure _I_ can't
afford to go round giving children picture-books. I told her so flatly
and frankly. And then it was so trying, because, right on top of my
scolding, she gave me a present--a cup all painted with roses, and
marked 'Friendship's Gift,' in gilt. I didn't want it; I could have
shaken her," Mrs. King ended, helplessly.
It was not only Martha whose patience was tried by Miss Lydia; the
experience was common to all Old Chester. Even Dr. Lavendar had felt
the human impulse to shake her. When he had, very delicately, asked
"as an old friend, the privilege of assisting her," it was exasperating
to have a lamp-shade made of six porcelain intaglios set in a tin frame
come to him the next day, with the "respectful compliments of L.S."
But somehow, when, beaming at him from under her shabby bonnet, Miss
Lydia had asked him if he liked that preposterous shade, he could not
speak his mind,--at least to her. He spoke it mildly to Mrs. Barkley.
"We must restrain her; she brought me $2 for Zenanna Missions
yesterday."
"What did you do?" Mrs. Barkley said, sympathetically.
"I made her take it back. I pointed out that her first duty was to her
landlord."
"Her landlord has some duties to her," Mrs. Barkley said, angrily.
"The stairs are just crumbling to pieces, and that chimney is dreadful.
She says that Davis said the flue would have to be rebuilt, and maybe
the whole chimney. He couldn't be sure about that, but he thought it
probable. He said it would cost $100 to put all the things in
repair--floor and roof and everything. But he would do it for $85,
considering. He thinks the flue has broken down inside somehow. She
might burn up some night; and then," said Mrs. Barkley, in a deep bass,
"how would that Smith person feel?"
"He says," Dr. Lavendar explained, "that by the terms of the lease the
tenant is to make repairs."
Mrs. Barkley snorted. "And how is poor Lydia to make repairs? She
hasn't two cents to bless herself with. I told him so."
Mrs. Barkley's face grew very red at the recollection of her interview
with Mr. Smith (he was one of the new Smiths, of course). "I don't mix
philanthropy and business," he had said; "the lease says the tenant
shall make repairs. And, besides, I do not wish to be more attractive
than I am. With that chimney, some other landlord may win her
affections. Without it, she will never desert Mr. Micawber."
"I am not acquainted with your friend Mr. Micawber," said Mrs. Barkley,
"neither, I am sure, is Miss Sampson; and if you will allow me to say
so, sir, we do not in Old Chester consider it delicate to refer to the
affections of an unmarried female."
Upon which Mr. Smith laughed immoderately. (None of the new people had
any manners.)
"So there is no use asking him to do anything," Mrs. Barkley told Dr.
Lavendar.
"The only thing I can think of," the old minister said, "is that we all
join together and give her the price Davis named, as a present."
"Eighty-five dollars!" Mrs. Barkley exclaimed, startled; "that's a good
deal of money--"
"Well, yes; it is. But something has got to be done."
"And to take up a collection for Lydia! It's--charity."
"It isn't taking up a collection," Dr. Lavendar protested, stoutly.
"And it isn't charity. Miss Lydia's friends have a right to make her a
present if they feel like it."
Mrs. Barkley agreed, doubtfully.
"Mrs. Dale would contribute, I'm sure," said Dr. Lavendar. "And
perhaps the Miss Ferises."
"I wouldn't like to ask them."
"Don't ask 'em. Offer them the chance."
"No," Mrs. Barkley insisted; "they've no right. They are not really
her friends. Lydia doesn't call them by their first names." But she
went away very much encouraged and full of this project of a present
for poor Lydia, who, happily, had no idea that she was "poor" Lydia.
She was not poor to herself (except, of course, in purse, which is a
small matter). She lived in a shabby and dilapidated cottage at the
Smith gates, and every month squeezed out a few dollars rent to Mr.
Smith; she was sorry for the Smiths, for they were new people; but she
always spoke kindly to them, for she never looked down on anybody. So,
as far as position went, she was not "poor." She had no relations
living, but she called all Old Chester of her generation by its first
name; so, as to friendship, there was nothing "poor" about her. And,
most of all, she was not "poor," but very rich, in her capacity for
interest.
Now, no one who has an interest is poor; and Miss Lydia had a hundred
interests. A hundred? She had as many interests as there were people
in the world or joys or sorrows in Old Chester; so she was really very
rich.... Of course, there are different degrees of this sort of
wealth: there are folk who have to manufacture their interests; with
deliberation they are philanthropic or artistic or intellectual, or
even, if hard put to it, they are amused. Such persons may be said to
be in fairly comfortable circumstances, although they live anxiously
and rather meagrely, because they know well that when interest gives
out they are practically without the means to support life. Below this
manufacturing class come the really destitute--the poor creatures who
do not care vitally for anything and who are without the spiritual
muscle to manufacture an interest. These pathetic folk are
occasionally made self-supporting by a catastrophe--grief or even
merely some uncomfortable surgery in regard to their bank account may
give them a poor kind of interest; but too often they exist
miserably--sometimes, with every wish gratified, helplessly poor.
Above the manufacturing class comes the aristocracy, to which Miss
Lydia Sampson belonged, the class which is positively rolling in
wealth. Every morning these favored creatures arise with a zest for
living. You hear them singing before breakfast; at the table they are
full of eager questions: Is it going to rain? No; it is a fair day;
delightful!--for it might have rained. And the sun will bring up the
crocuses. And this was the day a neighbor was to go to town. Will she
go? When will she come back? How pleasant that the day is pleasant!
And it will be good for the sick people, too. And the moment the
eager, simple mind turns to its fellows, sick or well, the field of
interest widens to the sky-line of souls. To sorrow in the sorrows of
Tom and Dick and Harry and their wives, to rejoice in their joys--what
is better than that? And then, all one's own affairs are so vital: the
record of the range of the thermometer, the question of turning or not
turning an alpaca skirt, the working out of a game of solitaire--these
things are absorbing experiences.
No wonder we who are poor, or even we who work hard at philanthropy or
art or responsibility to manufacture our little interests--no wonder we
envy such sky-blue natures. Certainly there were persons in Old
Chester who envied Miss Lydia; at least, they envied her her unfailing
joyousness--but they never envied her her empty purse. Which was like
envying a rose its color, but despising the earth from which by some
divine chemistry the color came.
Miss Lydia's eyes might smart from the smoke puffing out into her room,
but she was able to laugh at the sight of her bleared visage in the
narrow mirror over the mantel. Nor did the fact that the mirror was
mottled and misty with age, the frame tarnished almost to blackness,
cause her the slightest pang. What difference does it make in this
world of life and death and joy and sorrow, if things are shabby? The
fact is, the secret of happiness is the _sense of proportion_;
eliminate, by means of that sense, trouble about the unimportant, and
we would all be considerably happier than kings. Miss Lydia possessed
this heaven-born sense, as well as the boundless wealth of interest
(for to him that hath shall be given). "I don't want to brag," she
used to say, "but I've got my health and my friends; so what on earth
more do I want?" And one hesitated to point out a little thing like a
shabby mirror, or even a smoky chimney. When the chimney smoked, Miss
Lydia merely took her rocking-chair and her sewing out into a small
room that served as a kitchen--and then what difference did the smoking
make?
And as it turned out, one shadowy April day, it was the best thing she
could have done, because, when Dr. Lavendar dropped in to see her, she
could make him a cup of tea at once, without having to leave him alone.
She was a little, bustling figure, rather dusty and moth-eaten, with a
black frizette, always a little to one side, and eager, gentle, blue
eyes.
"What's the news?" she said. She had given Dr. Lavendar an apple, and
put on the kettle, and taken up her hemming.
"I never saw anybody so fond of sewing," the old man ruminated, eating
his apple. "I believe you'd sew in your grave."
"I believe I would. Dear me! I am so sorry for the poor women who
don't like to sew. Amelia Dilworth told me that Mrs. Neddy can't bear
to take a needle in her hand. So Milly does Ned's mending just as she
did before he was married."
"Aren't you sorry for the poor men that don't like to sew?" Dr.
Lavendar said, looking about for a place to deposit his core--("Oh,
drop it on the floor; I'll sweep it up sometime," Miss Lydia told him;
but he disposed of it by eating it).
"Well, as for sewing," said Miss Lydia, "it's my greatest pleasure.
Why, when I get settled down to sew, my mind roves over the whole
earth. I don't want to brag, but I don't believe anybody enjoys
herself more than I do when I'm sewing. If you won't tell, I'll tell
you something, Dr. Lavendar."
"I won't tell."
"Well, then: Sunday used to be an awful day to me. I couldn't sew, and
so I couldn't think. And I really couldn't go to church all day. So I
just bought some beautiful, fine nainsook and cut out my shroud. And I
work on that Sundays, because a shroud induces serious thoughts."
"I should think it might," said Dr. Lavendar.
"You don't think it's wrong, do you?" she asked, anxiously; and added,
joyously, "I'm embroidering the whole front. I declare I don't know
what I'll do when I get it done."
"Embroider the whole back."
"Well, yes. I can do that," Miss Lydia assented. "There! there's your
tea."
Dr. Lavendar took his tea and stirred it thoughtfully. "Miss Lydia,"
he said, and looked hard at the tea, "what do you suppose? Mr. William
Rives--" Dr. Lavendar stopped and drank some tea. "How many years ago
was it that he went away from Old Chester? I don't exactly remember."
"It was thirty-one years ago," she said; she put down her own cup of
tea and stared at him. "What were you going to say about him, sir?"
"Well, only," said Dr. Lavendar, scraping the sugar from the bottom of
his cup, "only that--"
"There! my goodness! I'll give you another lump," cried Miss Lydia;
"don't wear my spoon out. What about him, sir?"
Dr. Lavendar explained that he had come back on the stage from Mercer
the night before with a strange gentleman--"stout man," Dr. Lavendar
said, "with a black wig. I was rooting about in my pocket-book for a
stamp--I wanted to post a letter just as we were leaving Mercer; and
this gentleman very politely offered me one. I took it. Then I looked
at him, and there was something familiar about him. I asked him if we
had not met before, and he told me who he was. He has changed a good
deal."
Miss Lydia drank her tea excitedly. "Where is he going to stay? Is he
well? Has he come back rich?" She hoped so. William was so
industrious, he deserved to be rich. She ran into the smoky front room
and brought out his picture, regarding it with affectionate interest.
"Did you know I was engaged to him, years ago, Dr. Lavendar? We
thought it best to part. But--" She stopped and looked at the
picture, and a little color came into her face. But in another moment
she was chattering her birdlike questions.
"I declare," Dr. Lavendar said, at last, "you are the youngest person
of my acquaintance."
Miss Lydia laughed. "I hope you don't think it's wrong to be young?"
she said.
"Wrong?" said Dr. Lavendar; "it's wrong not to be young. I'd be
ashamed not to be young. My body's old, but that's not my fault. I'm
not to blame for an old body, but I would be to blame for an old soul.
An old soul is a shameful thing. Mind, now, don't let me catch you
getting old!"
And then he said good-bye, and left her sitting by the stove. She
turned her skirt back over her knees to keep it from scorching and held
the picture in her left hand and warmed the palm of the right; then in
her right hand and warmed the left. Then she put it down on her knees
and warmed both hands and smiled.
II
When Mrs. Barkley heard the news of the wanderer's return, she hurried
to Dr. Lavendar's study. "Do you suppose we need go on with the
present?" she demanded, excitedly.
"Why not?" said Dr. Lavendar.
Mrs. Barkley looked conscious. "I only thought, perhaps--maybe--Mr.
Rives--"
"William Rives's presence in Old Chester won't improve draughts, will
it?" Dr. Lavendar said, crossly. And that was all she could get out of
him.
Meantime, Old Chester began to kill the fatted calf. Mr. Rives liked
fatted calves; and, furthermore, he had prudently arranged with Van
Horne at the Tavern for a cash credit for each meal at which he was not
present. "For why," he had said, reasonably enough, "should I pay for
what I don't get?" So he went cheerfully wherever he was bidden. Old
Chester approved of him as a guest, for, though talkative, he was
respectful in his demeanor, and he did not, so Old Chester said, "put
on airs." He was very stout, and he wore a black wig that curled all
around the back of his neck; his eyes were somewhat dull, but
occasionally they glanced out keenly over his fat cheeks. He had a
very small mouth and a slight, perpetual smile that gave his face a
rather kindly look, and his voice was mild and soft.
He had come back rich (his shabby clothes to the contrary); "and poor
Lydia is so poor," said Old Chester; "perhaps--" and then it paused and
smiled, and added that "it would be strange, after all these years,
_if_--"
When somebody said something like this to Dr. Lavendar he grew very
cross. "Preposterous!" he said. "I should feel it my duty to prevent
anything so dreadful."
And there were romantic hearts in Old Chester who were displeased with
him for this remark. Mrs. Drayton said it showed that he could not
understand love; "though he can't be blamed for that, as he never
married. Still," said Mrs. Drayton, "he ought to have married. I
don't want to make any accusations, _but I always look with suspicion
on an unmarried gentleman_." Mrs. Barkley did not go as far as that,
but she did say to herself that Dr. Lavendar was unromantic. "Dear
me!" she confided to Jane Jay--"if anything _should_ happen! Well, I'd
be glad to do anything I could to bring it about."
And Mrs. Barkley, who had not only the courage but the audacity of her
convictions, invited the parted lovers to tea, so they met for the
first time at her house. Mrs. Barkley was the last person one would
accuse of being romantic, and yet Dr. Lavendar saw fit to stop at her
door that morning and say, "Matches are dangerous playthings, ma'am!"
and Mrs. Barkley grew very red, and said that she couldn't imagine what
he meant.
However, the party went off well enough. Miss Jane Jay, who made a
conscious fourth, expected some quiverings and blushings; but that was
because she was young--comparatively. If she had been older she would
have known better. Age, with shamefaced relief, has learned the
solvent quality of Time. It is this quality which makes possible the
contemplation of certain embarrassing heavenly reunions--where
explanations of consolation must be made.... Thirty-one years of days,
days full of personal concerns and interests, had blurred and softened
and finally almost blotted out that one fierce day of angry parting;
those thirty-one years of days had made this man and woman able to meet
with a sort of calm, good-natured interest in each other. Miss
Lydia--her black frizette over one smiling eye, her hands encased in
white cotton gloves, a new ribbon at the throat of her very old
alpaca--called him "William," with the most commonplace friendliness.
He began with "Miss Sampson," but ended before supper was over with her
first name, and even, once, just as they were going home, with "Lydy,"
at which she did start and blink for an instant, and Jane Jay thought a
faint color came into her cheek. However, he did not offer to walk
home with her, but bowed politely at Mrs. Barkley's gate, and would
have betaken himself to the Tavern had not Mrs. Barkley, when he was
half-way across the street, called after him. There was a flutter of
uncertainty in her voice, for those words of Dr. Lavendar's (which she
did not understand) "stuck," she said to herself, "in her crop." Mr.
Rives came back and paused in the moonlight, looking up at Mrs. Barkley
standing in the doorway. "I should be pleased, sir," she said, "to
have a few words with you."
"Certainly, ma'am," said Mr. Rives, in his soft voice, and followed her
into the parlor.
"Sit down," said Mrs. Barkley.
William Rives sat down thoughtfully. A tall lamp on the heavy,
claw-footed table emitted a feeble light through its ground-glass
globe, and Mrs. Barkley stared at it a moment, as though for
inspiration; then she said, in a deep bass: "Mr. Rives, I thought you
might be interested in a certain little project. Some of us have
thought that we would collect--a--a small sum--"
Mr. Rives bowed; his smiling lips suddenly shut tight.
"Perhaps you have not heard that our old friend Lydia Sampson is in
reduced circumstances; and some of us thought that a small present of
money--
"Ah--" said Mr. Rives.
Mrs. Barkley felt the color come up into her face at that small, cold
sound. "Lydia is very poor," she blurted out.
"Really?" murmured Mr. Rives, with embarrassment; and fell to stroking
his beaver hat carefully. Then he added that he deeply regretted Mrs.
Barkley's information.
"I knew you would," she said, in a relieved voice. "Lydia is a dear
girl. So kind and so uncomplaining! And--and faithful in her
affections, William."
"Ah!" said Mr. Rives again; his smile never changed, but his eyes were
keen.
"Yes," Mrs. Barkley said, boldly. "Why, William--I don't know that I
ought to tell you, but do you remember a sketch of yourself that you
gave her in--in other days? William, she has kept it ever since. It
hangs in her parlor, (horrid, smoky room!) And she keeps a sprig of
fresh box stuck in the frame."
"Really?" said Mr. Rives; and his face grew a little redder.
"That's all," Mrs. Barkley said, abruptly. "Now go. I just thought
I'd mention it."
"Yes," said Mr. Rives; then added that it was a beautiful night, and
politely bowed himself out.
"But he didn't say anything about giving anything," Mrs. Barkley told
Dr. Lavendar the next day. And whatever romantic hopes she may have
had withered under the blighting touch of such indifference.
III
Mrs. Barkley's hopes withered and then revived; for as she climbed the
hill to the Stuffed-Animal House a day or two later whom should she see
wandering through the graveyard (of all places!) but Lydia and William.
"Of course, I pretended not to see them," she told Harriet Hutchinson,
"but I believe they've begun to take notice."
They had not seen her; the graveyard was on the crest of the hill, and
the road lay below the bank and the stone wall, wherein were set two or
three iron doors streaked and eaten with rust, each with its name and
its big ring-bolt. There was a bleached fringe of dead grass along the
top of the wall, but the bank above was growing green in the April
sunshine. There were many trees in this older part of the cemetery,
and even now, when the foliage was hardly more than a mist, the tombs
and low mounds and old headstones were dappled with light shadows.
Miss Lydia and William had met here, by some chance; and Mrs. Barkley,
climbing the road before it dipped below the bank, had caught sight of
them just where the <DW72> broke into sunshine beyond the trees. Behind
them, leaning sidewise over a sunken grave, was a slate headstone, its
base deep in a thatch of last year's grass; there were carved cherubs
on the corners, and the inscription was blurred with lichen. A still
older tomb, a slab of granite on four pedestals, made a seat for Miss
Lydia. She had been deciphering its crumbling inscription:
"Mr. Amos Sm ... Sr.
Born ...... 1734
Die ... May 7th, 1802
Aged 68
"Base body, thou art faint and weak--
(How the sweet moments roll!)
A mortal paleness on thy cheek,
But glory in thy soul!"
William, reading it, had remarked that he thought people lived longer
nowadays. "Don't you?" he added, anxiously.
"We live long enough," Miss Lydia said. "I don't want to live too
long."
"You can't live too long," he told her, with his sharp smile.
Miss Lydia laughed and looked down at the crumbling stone. "I think
sixty-eight was just about long enough. I'm like Dr. Lavendar; he says
he 'wants to get up from the banquet of life _still hungry_.' That's
the way I feel. I don't want to lose my appetite for life by getting
too much of it."
"I couldn't get too much," Mr. Rives said, nervously. "Let us proceed.
This place is--is not cheerful. I like cheerfulness. You always seem
cheerful, Lydy?"
"Course I am," she said, getting up. "Why shouldn't I be? I haven't a
care in the world."
"You don't say so!" said William Rives. "I was under the impression
that your circumstances--"
"My circumstances?" said Miss Lydia. "Bless you! I haven't any.
Father didn't leave much of anything. I had $2000, but Cousin Robinson
invested it and lost it. He felt so badly, I was just distressed about
him."
"He should have been prosecuted!" Mr. Rives said, angrily.
Miss Lydia shook her head in horrified protest, but she beamed at him
from under her black frizette, grateful for his sympathy.
"I remember," he said, thoughtfully, "that you were always
light-hearted. I recall your once telling me that you began to sing as
soon as you got up in the morning."
"Oh yes," Miss Lydia said, simply. "I always sing the morning hymn.
You know the morning hymn, William?
"'Awake, my soul, and with the sun
Thy daily course of duty run--'"
William nodded. "Vocal exercises (if in tune and not too loud) are
always cheerful," he said.
Gossiping thus of simple things, they walked back to Lydia's house and
sat down in her parlor. There William told her, with a sort of
whimper, that his health was bad. "I sent for Willy King--he is so
young, he ought not to charge the full fee. I remember him as a very
impudent boy," Mr. Rives said, growing red at some memory of William's
youth; "however, he seems a respectable young man."
"Oh, indeed he is," said Miss Lydia; "he is a dear, good boy. I hope
he is doing you good?" she ended, with eager kindness.
"Yes, I think so," he said, anxiously. And then he gave his symptoms
with a detail that made poor Miss Lydia get very red. "And I don't
sleep very well," he ended, sighing. "Willy told me to try repeating
the kings of England backward, but I couldn't remember them; so it
didn't do any good."
"When I don't sleep," said Miss Lydia, "I just count my blessings.
That's a splendid thing to do, because you fall asleep before you get
to the end of 'em."
William sighed. "The kings of England was a foolish prescription; yet
I paid Willy $1.50 for that call. Still, I must say I think he is
doing me good; but he recommends many expensive things--perhaps because
he is young. He wished me to hire a vehicle and drive every day. Now
just think of the expense of such a thing! I suggested to him that
instead of hiring a conveyance, I would go out with him in his buggy
whenever he calls. He is a very young man to treat an important case,"
William ended, sighing. Then he asked Lydia about her health, with an
exactness which she thought very kind.
"Yes, I'm always well; and _so_ sorry for the poor people who are
sick," she said.
"You are a good nurse, aren't you, Lydy?" he asked.
"I'm always glad when I can do anything for a sick person. I'm so
sorry for 'em," Miss Lydia said, kindly.
"And you are economical, aren't you, Lydy?" Mr. Rives inquired, in his
mild voice, "and not fond of dress?"
"Bless you!" said Lydia, "how can I be anything but economical? And as
for being fond of dress--I'm fond of my old dresses, William."
"That is an excellent trait," said William Rives, solemnly. Then,
catching sight of his own portrait--the slim, anaemic young person in a
stock and tight-waisted coat, with very small feet and very large hat,
he got up to look at it. "I--have changed a little," he said,
doubtfully.
"It's more becoming to be heavier," Miss Lydia said. And this remark
gave him such obvious satisfaction that when he went away his perpetual
smile had deepened into positive heartiness.
It was after this talk that he finally added his offering to the
"Present" which just then was occupying Old Chester's attention. "And
how much do you suppose I got out of him?" Mrs. Barkley asked Dr.
Lavendar. "_$1.50!_"
However, other friends were more liberal, and by the end of May the $85
(grown now into the round sum of $100) was ready for Miss Lydia. A
little silk bag, with a scrap of paper twisted about its ribbon
drawing-string, was thrust one evening by an unknown hand into Miss
Lydia's door. In it were twenty five-dollar gold pieces. "From old
friends," Dr. Lavendar had written on the scrap of paper.
"Sha'n't we say--'for repairs'?" Mrs. Barkley asked, doubtfully.
"No," Dr. Lavendar declared; "I'd rather say 'to buy curl-papers.' Of
course she'll use it for repairs; but we mustn't dictate."
Nobody saw Miss Lydia gasp when she opened the bag, and sit down, and
then cry and laugh, but probably every friendly heart in Old Chester
was busy imagining the scene, for every friend had contributed. They
had all done it in their different ways--and how character confesses
itself in this matter of giving! ... Mrs. Dale, who gave the largest
sum, did it with calm, impersonal kindness. Martha King said that she
had so many calls upon her charity that she couldn't give much, but was
glad to do what she could. Miss Harriet Hutchinson said it was a
first-rate idea, and she was obliged to Mrs. Barkley for letting her
have a hand in it; as for Mrs. Drayton, she said it was a great trial
not to contribute, but she could not do so conscientiously. "_I_ make
such things a matter of prayer," she said; "some do not. I do not
judge them. I never judge any one. But I take all such matters to the
Throne of Grace, and as a result I feel that such things are injurious
to a poor person, and so I must deny myself the pleasure of charity."
William Rives said that he would be pleased to contribute, and Mrs.
Barkley had a moment of intense excitement when she read his
check--$150. But her emotion only lasted until she put on her
spectacles.
And yet, when Lydia, sitting at the kitchen table, wiped her eyes and
counted her gold by the light of a candle in a hooded candlestick, she
felt, somehow, William's hand in it. For, by this time, William's
friendliness was beyond any question. He came to see her every other
day, and he told her all his symptoms and talked of his loneliness and
forlornness until they were both moved to tears.
"Poor William!" she said, her eyes overflowing with sympathy. "Well,
I'm glad you have plenty of money, anyhow. It would be hard to be poor
and have bad health, too."
"But I haven't plenty of money," William said, with agitation. "How
did you get such an idea? I haven't!"
And then Miss Lydia was sorrier for him than ever. "Although," she
said, cheerfully, "poverty is the last thing to worry about. Look at
me. I don't want to brag, but I'm always contented, and I'll tell you
why: _I don't want things_. Don't want things, and then you're not
unhappy without 'em."
"Oh, Lydy, that's so true," Mr. Rives said, earnestly. "I'm so glad
you feel that way." And he began to call every day.
"It's plain to be seen what's going to happen," said Mrs. Barkley,
excitedly, and whispered her hopes (in secret) to almost everybody in
Old Chester--except Dr. Lavendar. He became very ill-tempered the
moment she approached the subject. But she was jocose, in a deep bass,
to Miss Lydia herself; and Miss Lydia did not pretend to misunderstand.
She reddened and laughed; but her eyes were not clear; there was a
puzzled look at the back of them. Still, when she sat and looked at
her gold the puzzle lightened, and her face, under her black
frizette--in her excitement fallen sidewise over one ear--softened
almost to tears. "William is kind," she said to herself.
And, indeed, at that very moment William was referring to her in most
kindly terms. He was sitting in Mrs. Barkley's gloomy parlor, on the
edge of the horse-hair sofa, and Mrs. Barkley was regarding him with
romantic interest. "I have been much saddened, ma'am," he was saying,
"to observe the destitution of Miss Lydia Sampson."
Mrs. Barkley beamed. Was he going to do something, after all? She
spoke in an amiable bass, twitching her heavy eyebrows. "Our little
gift, which has gone to her to-night, will make her more comfortable.
I could wish it had been larger," she ended, and looked sidewise at Mr.
Rives, who bowed and regretted that it was not larger. He then coughed
behind his hand.
"Mrs. Barkley, I wish to approach a subject of some delicacy."
("He _is_ going to do something," she thought, excitedly; "or perhaps
he means marriage!")
"Mrs. Barkley, in past years there were passages of affection between
Miss Sampson and myself" (Mrs. Barkley bowed; her heart began to glow
with that warmth which stirs the oldest of us at the sight of a lover).
"We were younger in those days, ma'am," William said, in his soft voice.
"Oh no!" she protested, politely. "Why, you are very well preserved,
I'm sure."
"Yes," said William, "I am. Yet I am not as young as I once was."
This drifting away from Miss Lydia disturbed Mrs. Barkley. She lowered
her chin and glared at him over her spectacles, saying, in a rumbling
bass: "Neither is Lydia; and it's hard for her to be destitute in her
old age."
"Just so," Mr. Rives said, eagerly--"exactly. She is not as young as
she once was, which, for many reasons, is desirable. But I think she
is healthy?"
"Why, yes," Mrs. Barkley admitted; "but I don't know that that makes it
easier to be poor."
"But I infer that poverty has taught her economy?" William Rives said.
"Yes; but poverty is a hard teacher."
"But thorough--thorough!" said Mr. Rives; "and some people will learn
of no other."
Mrs. Barkley was growing impatient; she gave up marriage and thought of
a pension.
"Yes," said William; "she is economical, and has good health, and is
fond of old clothes, and is kind-hearted, and doesn't have any wants.
Excellent traits--excellent. I have looked very carefully at the items
of expense in regard to a housekeeper or nurse."
Mrs. Barkley stared at him in bewilderment. Was he going to offer
Lydia a position as housekeeper? She was fairly dizzy with this seesaw
of possibilities; and she was perplexed, too, for, after all, badly as
Lydia needed assistance, propriety must be considered, and certainly
this housekeeping project was of doubtful propriety. "Because you know
you are neither of you very old," she explained.
Mr. Rives looked disturbed. "Yes, we are," he said, sharply. "Quite
old enough. I would not wish a youthful wife, for--many reasons.
There might be--results, which would interfere with my comfort. No,
Lydia is no longer young; yet she is sufficiently robust to make me
extremely comfortable." The light was breaking slowly on Mrs. Barkley.
Her face flushed; she sat up very straight and tapped the table with
her thimble. "The expense of an extra person is not very considerable,
is it?" Mr. Rives said, doubtfully. "It was in regard to this that I
wished to consult you."
"Not more than the wages of a housekeeper or a nurse," Mrs. Barkley
said, in a restrained voice.
"Exactly!" cried Mr. Rives--"granted that her health is good."
Mrs. Barkley opened and closed her lips. Her impulse to show him the
door battled with her common-sense. After all, it would mean a home
for Lydia; it would mean comfort and ease and absence from worry--plus,
of course, Mr. Rives. But if Lydia liked him, that wouldn't make any
difference. And she must like him--her faithfulness to the picture
proved it--and he was an agreeable person; amiable, too, Mrs. Barkley
thought, for he always smiled when he spoke.
"Would you live in Old Chester?" she managed to say, after a pause.
"Yes."
"You would build, I suppose?" Mrs. Barkley said, trying, in the
confusion of her thoughts, to make time.
"No," Mr. Rives said; "we would reside in Lydia's present abode."
"_In Lydia's house_? You couldn't!--why, it would be impossible!"
Mrs. Barkley, her mouth open with astonishment, saw, suddenly, that
this project was not comfort plus William, but William minus comfort.
"You couldn't! The chimney in the parlor is dreadful; it smokes
whenever the wind is from the west."
"But, as I understand, Lydia has been provided with the means of
mending the chimney?" William said, anxiously.
At this the rein broke. Mrs. Barkley rose, tapping the table with
alarming loudness and glaring down at her guest. "William Rives, I
have been a perfect fool. But you are worse--you are a mean person.
I'd rather live with a murderer than a mean man!"
[Illustration: "MRS. BARKLEY ROSE, TAPPING THE TABLE WITH ALARMING
LOUDNESS"]
Mr. Rives was unmoved. His little, steely smile never wavered; he rose
also, bowed, and said: "Possibly Miss Sampson does not agree with you.
I will bid you good-night, ma'am."
"I was a perfect fool," she said again, as the door closed softly
behind him.
But William Rives was no fool.... He said to himself that it behooved
him to see Miss Lydia before Mrs. Barkley had a chance to impart to her
those impolite views regarding himself. And that was why, as she was
still sitting at her kitchen table, twinkling with happiness over the
kindness of her world and piling her gold pieces in a little leaning
tower, William knocked at the door.
Miss Lydia threw an apron over the small, glittering heap and ran to
let her caller in. When she saw who it was she whipped off the apron
to display her wealth; the tears stood in her eyes, and her happy heart
burst into words: "How good people are! Just think--$100! Why, it
takes my breath away--"
"It is a large sum of money," William said, solemnly, touching the gold
with respectful fingers. "I would suggest a bank until you pay for the
mending of your chimney. And you will get some interest if you defer
payment for ninety days."
"Mending my chimney?" Miss Lydia said, thoughtfully. "Well--that
wouldn't take nearly all this."
William's face brightened. "You are right to be prudent, Lydia," he
said. "I admire prudence in a female; but still, masons and
carpenters--in fact, all persons of that sort,--are--thieves!" Then he
coughed delicately. "Lydia," he said, "I--I have been thinking--"
"Yes?" said Miss Lydia, calmly.
"We are so situated--each alone, that perhaps we might--we might,
ah--marry--to our mutual advantage?"
"_Marry?_"
"Yes," William said, earnestly; "I should be pleased to marry, Lydy. I
need a home. My health is not very good, and I need a home. You need
a home, also."
"Indeed I don't!" she said; "I've got a home, thank you."
"I haven't," William said; and Lydia's blue eyes softened. "I am not
very strong," he said ("though I see no reason why I should not live to
old age); but I want a home. Won't you take me, Lydy?"
Miss Lydia frowned and sighed. "I am very well satisfied as I am," she
said; "but perhaps that is a selfish way to look at it."
"Yes, it is," he told her, earnestly; "and you didn't use to be
selfish, Lydia."
Miss Lydia sighed again. "I suppose I could make you comfortable,
William."
"Do take me, Lydy," he entreated.
And somehow or other, before she quite knew it, she had consented.
As soon as the word was spoken, William arose with alacrity. "I don't
like to be out in the night air," he said, "so I'll say good-night,
Lydy. And, Lydy--shall we, for the moment, keep this to ourselves?"
"Oh yes," said Miss Lydia, getting very red, "I'd rather, for the
present." Then, smiling and friendly, she went out with him,
bare-headed, to the gate. There William hesitated, swallowed once,
rubbed his hands nervously, and then suddenly gave her a kiss.
Miss Lydia Sampson jumped. "Oh!" she said; and again, "_Oh!_"
And then she ran back into the house, her eyes wet and shining, her
face flushed to her forehead. She sat down by the table and put her
hands over her eyes; she laughed, in a sort of sob, and her breath came
quickly.
"I hadn't thought of it--that way," she whispered to herself. And
somehow, as she sat there by her kitchen table, she began to think of
it that way--Miss Lydia was very young! ... Oh, she would try and make
him happy; she would try and be more orderly; she would try to be good,
since her Heavenly Father had given back to her the old happiness.
And that night she did not bid the picture good-night.
Mr. Rives was himself not without emotion. It was many years, he
reflected, since his lips had touched those of a female, and the
experience was agreeable--so agreeable that he wished to repeat it as
soon as possible; and, furthermore, he felt anxious to know that Lydia
had put the gold in a safe place. But when he called the next day he
was a little late, because, as he explained to Miss Lydia, he had had
to wait for the mail. She met him with a new look in her innocent,
eager eyes, and her face was shy and red. As she sat sewing, listening
vaguely, she would glance at him now and then, as if, until now, she
had not seen him since that day of parting, thirty-one years ago--the
thirty-one years which had blotted Amanda's field from her memory. The
old happiness, like a tide long withdrawn, was creeping back, rising
and rising, until it was overflowing in her eyes. This puffy
gentleman, with his tight, smiling mouth, was the William of her
youth--and she had never known it until last night! She had thought of
him during the last month or two only as an old friend who needed the
care which her kind heart prompted her to give; and lo! suddenly he was
the lover who would care for her.
"I was sorry, my dear Lydia, to be late," said Mr. Rives, in his soft
voice; "I was detained by waiting for the mail."
Miss Lydia said, brightly, that it didn't matter.
"But it was worth waiting for," William assured her. "I have done a
good piece of business. (Not that it will make me richer; I have so
many obligations to meet!) But it was a fortunate stroke."
"That is good," said Miss Lydia.
"A female in a distant city, where I own a poor little bit of real
estate--nothing of any value, Lydia; I am a poor man--"
"That's no difference," she told him, softly.
"--this female, a widow, and foolish (as widows always are)," William
said, with a little giggle, "asked me to sell her a house I owned. She
wished, for some reason, to purchase in that locality. I named the
market price. I did so, by letter, a fortnight ago. I believe she
thought it high; but that was her affair. She would have to sell
certain securities to purchase it, she said. But as I wrote her--'my
dear madam, that's your business.'" Mr. Rives laughed a little. Miss
Lydia looked up, smiling and interested. "Yes," said Mr. Rives--"I
didn't urge it. I never urge, because then I can't be blamed if things
go wrong. But I held my price. That is always good policy--not to
drop a dollar on price. So she's bought it. She made a payment
yesterday to bind the sale. Not that I feel any richer, for I must
immediately apply the money to the purchase of other things."
"That's nice," Miss Lydia said.
"I guess it is," William agreed; "I happen to know that a boiler
factory is to be erected on the rear lot."
"But will she like that--the poor widow?" Miss Lydia said.
Mr. Rives laughed comfortably. "Ah, Lydy, my dear, in business we do
not ask such questions before making a sale. _I_ like it. In three
months that bit of property will have shrunk to an eighth of its
selling price to-day." Mr. Rives's eyes twinkled with satisfaction.
"But--_William!_" said Miss Lydia. Suddenly she grew pale. "William,"
she said, "it seems to me you ought to have told the poor widow."
"Lydia, a lady cannot understand business," William said, with kindly
condescension, but with a slight impatience. "Don't you see, if I had
told her, she would not have made the purchase?"
Miss Lydia was silent, stroking the gathers of her cambric with a
shaking needle. Then she said, in a low voice, "I suppose she
wouldn't."
William nodded encouragingly. "You'll learn, Lydia. A married lady
learns much of business methods through her husband. Though they don't
profit by it, I notice; widows are always foolish. Not that--that you
will be likely to be--to be foolish," he ended, hastily, frowning very
much.
Lydia went on sewing in silence. The color did not come back into her
face, which caused William to ask her anxiously how she was.
"You are sure you are healthy, Lydia, aren't you?" he said.
Miss Lydia, without looking at him, said she was. When he had gone,
she stopped sewing and glanced about her in a frightened way; then she
put her hands over her eyes and drew in her breath, and once she
shivered. She sat there for a long time. After a while she got up and
went over to the picture of Mr. William Rives and stood looking at it;
and as she looked her poor, terrified eyes quieted into tears and she
straightened the bit of box with a tender hand, and then she suddenly
bent down and kissed the slim gentleman behind the misty glass.
The next day when she met her lover she was cheerful enough. It was at
the front door of the Tavern; Dr. Lavendar was there, too, waiting for
the morning stage for Mercer.
"Well! well! So I am going to have company, am I?" he said, for Miss
Lydia was waiting for it, too. Her bonnet was on one side, her shabby
jacket, fading from black to green on the shoulders, was split at the
elbow seams, and the middle finger of each glove was worn through; but
her eyes were shining with pleasure.
"Yes," she said, nodding; "I'm going."
Her presence seemed to be a surprise to Mr. Rives, who had strayed
forth from the breakfast-room to see the stage start.
"You are going to Mercer?" he said, his small smile fading into an
astonished question.
"Yes," Miss Lydia said, laughing, and suddenly she gave a little jump
of happiness. "I haven't been to Mercer for nine years. Oh, dear!
isn't it just delightful!"
"But, why?" William persisted, in an amazed aside.
"Oh, that's the secret!" cried Miss Lydia, clambering into the stage;
"you'll know sometime."
"I suppose you wish to arrange for the alterations of your house?"
William said; "but considering the stage fares back and forth-- Oh,
there is Dr. Lavendar."
He came round to the other side of the stage, smiling very much.
"Well, sir, good-morning! good-morning, sir!"
"Hello," Dr. Lavendar said.
Mr. Rives rubbed his hands. "I--I was about to say, Dr. Lavendar--that
little matter between us--it's of no importance, of course; quite at
your convenience, sir; I don't mean to press you--but at your
convenience, sir."
"What are you talking about?" Dr. Lavendar said, with a puzzled blink.
"Well," William said, smiling, "there's no haste, only I thought I'd
just remind you. I'm always business-like myself; and that little
matter of accommodation--"
Dr. Lavendar stared at him. "I am afraid I'm a stupid old fellow; I
don't understand."
The stage-driver gathered up his reins; Miss Lydia nodded joyously on
the back seat, the two other passengers frowned at the delay; so
William Rives made haste to explain: "Merely, sir, the stamp I had the
pleasure of lending you. But pray don't incommode yourself; I merely
remind you; it's of no--"
Dr. Lavendar pulled out his shabby leather pocket-book, his hands
fairly trembling with haste, and produced the stamp; then he pulled the
door to, and as the stage sagged forward and went tugging up the hill,
he turned his astonished eyes on Miss Lydia. She had grown very pale,
but she said nothing, only looking out of the window and rubbing her
little cotton gloves hard together.
"Would you have asked him for a receipt?" Dr. Lavendar said, under his
breath, chuckling. But when she tried to answer him, there was
something in her face that turned Dr. Lavendar grave.
The stage jolted on; the two other passengers chatted, then one fell
asleep and the other read an almanac. Suddenly Miss Lydia turned
sharply round. "It just kills me!" she said.
"Nonsense!" Dr. Lavendar told her. "He is a man of business, and I'm a
forgetful old codger. I knew William, and I ought to have remembered."
But Miss Lydia's face had fallen into such drawn and anxious lines that
Dr. Lavendar had to do his best to cheer her. He began to ask
questions: How long was it since she had been in Mercer? Was she going
to call on friends? Was she going to shop? "I believe you ladies
always want to shop?" said Dr. Lavendar, kindly. And somehow Miss
Lydia brightened up. Yes; she was going to shop! It was a secret: she
couldn't tell Dr. Lavendar yet, but he should know about it first of
all. She was so happy, so important, so excited, that her pain at
William's business-like ways seemed forgotten; and when in Mercer they
separated at the Stage House, she went bustling off into the sunshine,
waving a shabby cotton glove at him, and crying, "I haven't a minute to
lose!"
Dr. Lavendar stood still and shook his head. "Pity," he said--"pity,
pity. But I suppose it can't be helped. There's no use telling
William about her; he must see it. And there's no use telling her
about William; she must see it. No--no use. But it's a pity--a pity."
Which shows that Dr. Lavendar had reached that degree of wisdom which
knows that successful interference in love affairs must come from the
inside, not from the outside.
He did not see Miss Lydia again until they met in the afternoon at the
Stage House, and for a minute he did not recognize her. She came
running and panting, laden with bundles, to the coach door. Indeed,
she was so hurried that one of her innumerable packages, a long, slim
bundle, slipped from her happy, weary arms, and, hitting the iron
drop-step, crashed into fragments and splashed her dress with its
contents. "Oh! that's one of my bottles of Catawba," said Miss Lydia.
"Dear, dear! Well, never mind; I'll order another."
The fragrance of the wine soaking her gloves and the front of her faded
dress, filled the stage (in which they were the only passengers), and
Miss Lydia joyously licked the two bare finger-tips. "Too bad!" she
said; "but accidents will happen."
Dr. Lavendar helped her pile her bundles on the front seat, and then he
unhooked the swinging strap so that certain parcels could be put on the
middle bench. Miss Lydia leaned back with a happy sigh. "The rest
will come down to-morrow," she said.
"The _rest_?" said Dr. Lavendar.
"Oh, I've just got to tell somebody!" she said. "Promise you won't
tell?"
"I won't tell," he assured her.
"Well," said Miss Lydia, "look here--do you see that?" She tore a
little hole in a long, flat package, and Dr. Lavendar saw a gleam of
blue. "That's a dress. Yes, a blue silk dress--for myself. I'm
afraid it was selfish to get a thing just for myself, but that and a
pair of white kid gloves and some lace are all I did get; and I've
wanted a silk dress, a blue silk dress, ever since I got poor."
Dr. Lavendar looked at her and at the hole in the package, and at her
again. "Lydia!" he said, "is it possible that you--? _Lydia!_" he
ended, speechless with consternation.
"The other things are all for the party."
"The--party?"
"Presents!" she said, rubbing her hands. "Oh, dear! I'm so tired!
And I'm so happy! Oh, nobody was ever so happy. The party (that's the
secret) is to be next Thursday a week; that gives me time to make my
dress. I ordered the cake in Mercer. All pink-and-white
icing--perfectly lovely! And I have a present for everybody. Here's a
work-basket for Martha King. And I have a bird-cage and a canary for
dear Willy (that is to come down to-morrow; I really couldn't carry
everything). And I've got a knitted shawl for Maria Welwood, and a
cloak for her dear Rose--that was rather expensive, but it's always
cheap to get the best. And a cornelian breast-pin for Alice Gray. And
a Roman sash for poor little Mary Gordon; she seems to me such a
forlorn child--no mother, and that rough Alex for a brother.
And--well; oh, dear! I'm so excited I can hardly remember--a book for
Mr. Ezra; a book for Mrs. Dale. Books are safe presents, don't you
think?"
Dr. Lavendar groaned.
"And a picture for Rachel King--that's it; that square bundle. So
pretty!--a little girl saying her prayers; sweet!--it's like her Anna.
And a box of candy for Sally Smith's little brothers; and a pair of
agate cuff-buttons for Sally--" She was moving her packages about as
she checked them off, and she looked round at Dr. Lavendar with a sigh
of pure joy. He could not speak his distracted thought.
"Oh, you mustn't see that," she cried, suddenly pushing a certain
package under the others with great show of secrecy; and Dr. Lavendar
groaned again. "I think a party with presents for everybody will be
very unusual, don't you?" she asked, heaping her bundles up carefully;
two more fingers had burst through her cotton gloves, and as she leaned
forward a button snapped off her jacket. "I don't want to brag," she
said, "but I think it will be as nice a party as we have ever had in
Old Chester."
"But, Lydia, my dear," Dr. Lavendar said, gently, "I am afraid it is
extravagant, isn't it, to try to give us all so much pleasure? And is
a blue silk frock very--well, serviceable, I believe, you ladies call
it?"
"No, indeed it isn't," she said, with sudden, pathetic passion.
"_That's why I got it_. I never, since I was a girl, have had anything
that wasn't serviceable."
"But," Dr. Lavendar said, "I rather hoped you would see your way clear
to making your house a little more comfortable?"
"Why, but I'm perfectly comfortable," she assured him; "and even if I
was not, I'd rather, just for once in my life, have my party and give
my presents. Oh, just once in a lifetime! I'd rather," she said, and
her eyes snapped with joy--"I'd rather have next Thursday night, and my
house as it is, than just comfort all the rest of my days. Comfort!
What's comfort?"
"Well, Lydia, it's a good deal to some of us," Dr. Lavendar said. And
then his eyes narrowed. "Lydia, my dear--does Mr. Rives know about
this?"
Miss Lydia, counting her packages over, said, absently, "No; it is to
be a surprise to William."
"If I am not mistaken," said Dr. Lavendar, "it will be a very great
surprise to William."
And then he fell into troubled thought; but as he thought his face
brightened. It brightened so much that by the time they reached Old
Chester he was as joyously excited about the party as was Miss Lydia
herself, who made him a thousand confidences about her dress and her
presents and the food which would be offered to her guests. His
joyousness had not abated when, the next morning, Mrs. Barkley
presented herself, breathless, at the Rectory.
"I think," said she, in an awful bass, sitting up very straight and
glaring at Dr. Lavendar, "that this is the most terrible thing that
ever happened."
"There are worse things," said Dr. Lavendar.
"_I_ know of nothing worse," Mrs. Barkley said, with dreadful
composure. "You may. You know what the unregenerate human heart may
do. I do not. This is the worst. What will people say? What will
Mrs. Dale say? It must be stopped! She ran in this morning and told
me in confidence. She came, she said, to know if she could borrow my
teaspoons next Thursday week. I said she could, of course; but I
suppose I looked puzzled; I couldn't imagine--then she confessed. She
said you knew, but no one else. Then, before I came to my senses, she
ran out. I came here at once to say that you must stop it."
"In the first place," said Dr. Lavendar, thrusting his hands down into
his dressing-gown pockets, "I couldn't stop it. In the second place, I
haven't the right to stop it. And in the third place, I wouldn't stop
it if I could."
"Dr. Lavendar!"
"I am delighted with the plan. We need gayety in Old Chester; _I think
we'll get it_. I hope she'll have Uncle Davy in, with his fiddle, and
we'll have a reel. Mrs. Barkley, will you do me the honor?"
It came over Mrs. Barkley, with a sudden chill, that there was
something the matter with Dr. Lavendar.
"I have calculated," said the old minister, chuckling, "that Miss Lydia
has in hand, at present, about $1.75 of our $100. This sum I trust she
will give to Foreign Missions. The need is great. I shall bring it to
her attention."
"Dr. Lavendar," said Mrs. Barkley; and paused.
"Ma'am?"
"I don't understand you, sir."
Dr. Lavendar looked at her and smiled.
IV
And so the night of Old Chester's festivity approached. Miss Lydia's
invitations were delivered the morning of the day, but a rumor of the
party was already in the air. There had been some shakings of the head
and one or two frowns. "It will cost her at least $3," said Martha
King, "and she could get a new bonnet with that."
"It's her way of thanking us for her present," said the doctor, "and a
mighty nice way, too. I'm going. I'll wear my white waistcoat."
Mrs. Drayton said, calmly, that it was dishonest. "The money was given
to her for one purpose. To ask people to tea, and have even only cake
and lemonade, is spending it for another purpose. It will cost her at
least $4.50. Not a large sum, compared with the whole amount donated
in charity. But the principle is the same. I always look for the
principle--it is a Christian's duty. And I could not face my Maker if
I ever failed in duty."
Then Mrs. Dale's comment ran from lip to lip: "Miss Lydia has a right
to do as she pleases with her own; if she invites me to tea, I shall go
with pleasure."
When the rumor reached William Rives's ears he turned pale, but he made
no comment. "But I came to ask you about it, Lydy," he said. This was
Wednesday evening, and William stood at the front door; Miss Lydia was
on the step above him. "I won't ask you to come in, William," she
said, "I'm so busy--if you'll excuse me."
"I am always gratified," said William, "when a female busies herself in
household affairs, so I will not interrupt you. I came for two
purposes: first, to inquire when you intend to begin the improvements
upon your house; and, secondly, to say that I hope I am in error in
regard to this project of a supper that I hear you are to give."
"Why?" said Lydia.
"Because," William said, with his sharp, neat smile, "a supper is not
given without expense. Though I approve of hospitality, and make a
point of accepting it, yet I am always conscious that it costs money.
I cannot but calculate, as I see persons eating and drinking, the
amount of money thus consumed, and I often wonder at my hosts. I say
to myself, as I observe a guest drink a cup of tea, 'Two cents.' Such
thoughts (which must present themselves to every practical man) are
painful. And such a supper as I hear you mean to give would involve
many cups of tea."
"Twenty-seven," said Miss Lydia.
"And is there to be cake also?" said William, breathlessly.
"There is," said Miss Lydia; "a big one, with a castle in
pink-and-white icing on it--beautiful!"
William was stricken into silence; then he said, shaking his head, "Do
you really mean it, Lydy?"
"I do, William."
Mr. Rives sighed.
"Well," he said--"well, I regret it. But, Lydy, we might utilize the
occasion? Refreshment is always considered genteel at a marriage. Why
not combine your supper with our wedding? We can be married to-morrow
night. Dr. Lavendar is coming, I presume? I can get the license in
the morning."
Miss Lydia was silent; the color came into her face, and she put her
hand up to her lips in a frightened way. "Oh, I--don't know," she
faltered. "I--I am not--not ready--"
"Oh," William urged, "never mind about being ready; I should be the
last to wish you to go to any of the foolish expense of dress customary
on such occasions. Yes, Lydy, it is an opportunity. Do agree, my
dear; we will save money by it."
Miss Lydia drew in her breath; she was very pale; then suddenly she
nodded. "Well, yes," she said. "I will, if you want to, William.
Yes, I will."
"I will communicate with Dr. Lavendar," said Mr. Rives, joyfully, "and
ask him to hold himself in readiness, but not to speak of it outside."
Miss Lydia nodded, and, closing the door, went back to her engrossing
affairs. Presents and a party and a wedding--no wonder the poor little
soul was white and dizzy with excitement!
Long will Old Chester remember that occasion: The little house, lighted
from garret to cellar; candles in every possible spot; flowers all
about; the mantel-piece heaped with bundles; William King's bird-cage
hanging in the window; Uncle Davy's fiddle twanging in the kitchen; and
Miss Lydia in front of the smoky fireplace, banked now with larkspurs
and peonies--Miss Lydia in a light, bright blue silk dress trimmed with
lace; Miss Lydia in white kid gloves, buttoned with one button at the
wrist, and so tight that the right glove split across the back when she
began to shake hands. Oh, it was a great moment.... No wonder she was
pale with excitement! ... She was very pale when William Rives
arrived--arrived, and stood dumfounded!--staring at Miss Lydia; staring
at the packages which were now finding their way into astonished hands;
staring at the refreshment-table between the windows, at the great,
frosted cake, at the bottles of Catawba, at Mrs. Barkley's spoons stuck
into tall glasses of wine jelly. Mr. Rives stood staring at these
things, his small eyes starting out upon his purpling cheeks, and as he
stared, Miss Lydia, watching him, grew paler and paler.
[Illustration: "MISS LYDIA, WATCHING HIM, GREW PALER AND PALER"]
Then, suddenly, William, stealthily, step by step, began to back out of
the room. In the doorway he shouldered Mrs. Barkley, and, wheeling,
turned upon her a ferocious face:
"_And I contributed $1.50--_"
But as he retreated and retreated, the color returned to Miss Lydia's
cheek. She had almost stopped breathing as he stood there; but when he
finally disappeared, she broke out into the full joyousness of the
occasion. The opening of each present was like a draught of wine to
her, the astounded or angry thanks went to her head; she rubbed her
hands until the left glove split also; and then Uncle Davy's fiddle
began in good earnest, and she bustled about, running and laughing, and
arranging partners for the reel.
Yes, it was a great occasion. Old Chester talked of it for months; not
even William Rives's most unexpected and unexplainable departure the
next day on the morning stage could divert the appalled, excited,
disapproving interest that lasted the year out. Not even Miss Lydia's
continued faithfulness to the portrait, which had condoned so many
offences in the past, could soften Old Chester's very righteous
indignation. There were, it must be admitted, one or two who professed
that they did not share the disapproval of all right-thinking persons;
one was, if you please, Mr. Smith! (He was one of the new Smiths, so
one might expect anything from him.) He had not been invited to the
party, but when he heard of it he roared with most improper mirth.
"Well done!" he said. "By Jove! what a game old party. Well done!
The money was champagne on an empty stomach; of course, she got drunk.
It would have been cheaper to have bought a bottle of the genuine
article and shut herself up for twenty-four hours. Well, it's worth
the cost of a new chimney. I'll put her repairs through, Dr.
Lavendar--unless you want to get up another present?" And then he
roared again. Very ill-bred man he was.
Dr. Lavendar said that there would not be another present. He said
Miss Lydia had a right, in his opinion, to spend her money as she
chose; but there would not be another present.
And then he walked home, blinking and smiling. "Smith's a good
fellow," he said to himself, "if he is one of the new folks. But what
I'd like to know is: _did Lydia think $100 a low price?_"
AMELIA
I
The exception that proved Old Chester's rule as to the subjection of
Youth was found in the household of Mr. Thomas Dilworth.
When the Dilworth children (at least the two girls) hung about their
father when he came home at night or teased and scolded and laughed at
him at their friendly breakfast-table, an observer might have thought
himself miles away from Old Chester and its well-brought-up Youth. The
way those girls talk to Thomas Dilworth! "Where will it end?" said Old
Chester, solemnly. For instance, the annual joke in the Dilworth
family was that father had been in love with mother for as many years
as she was old, less so many minutes.
Now, imagine Old Chester children indulging in such familiarities!
Yet on Mrs. Dilworth's birthday this family witticism was always in
order:
"Father, how long have you been mother's beau?"
And Thomas, rosy, handsome, looking at least ten years younger than his
Amelia, would say: "Well, let's see: forty-one years" (or two or three,
as the case might be), "eleven months, twenty-nine days, twenty-three
hours, and forty minutes; she was twenty minutes old when I first laid
eyes on her, and during those twenty minutes I was heart-whole."
But Mrs. Dilworth, smiling vaguely behind her coffee-cups, would
protest: "I never heard anything about it, Tom, until you were sixteen."
And then the girls would declare that they must be told just what
father said when he was sixteen and mother was twelve. But Thomas drew
the line at that. "Come! come! you mustn't talk about love-making. As
for marrying, I don't mean to let you girls get married at all. And
Ned here had better not let me catch him thinking of such nonsense
until he's twenty-five. He can get married (if I like the girl) when
he is twenty-eight."
"You got married at twenty-two, sir," Edwin demurred.
"If you can find a woman like your mother, you can get married at
twenty-two. But you can't. They don't make 'em any more. So you've
got to wait. And remember, I've decided not to let Mary and Nancy get
married, ever. I don't propose to bring up a brace of long-legged
girls, and clothe 'em and feed 'em and pay their doctors' bills, and
then, just as they get old enough to amount to anything and quit being
nuisances, hand 'em over to another fellow. No, sir! You've got to
stay at home with me. Do you understand?"
The girls screamed at this, and flung themselves upon him to kiss him
and pull his hair.
No wonder Old Chester was shocked.
Yet, in spite of such happenings, Thomas and Amelia Dilworth were of
the real Old Chester. They were not tainted with _newness_--that sad
dispensation of Providence which had to be borne by such people as the
Macks or the Hayeses, or those very rich (but really worthy) Smiths.
The Dilworths were not new; yet their three children had the
training--or the lack of training--that made the Hayes children and
their kind a subject for Old Chester's prayers.
"Who can say what the result of Milly Dilworth's negligence will be?"
Mrs. Drayton said, sighing, to Dr. Lavendar; who only reminded her that
folks didn't gather thistles of figs--generally speaking.
But in spite of Dr. Lavendar's optimism, it was a queer household,
according to Old Chester lights.... In the first place, the father and
mother were more unlike than is generally considered to be
matrimonially safe. Amelia was a dear, good soul, but, as Miss Helen
Hayes said once, "with absolutely no mind"; while Thomas Dilworth was
eminently level-headed, although very fond (so Mrs. Drayton said) of
female society. And it must be admitted that Thomas had more than once
caused his Milly a slight pang by such fondness. But at least he was
never conscious that he had done so--and Milly never told him. (But
Mrs. Drayton said that that was something she could not forgive in a
married gentleman. "My dear husband," said Mrs. Drayton, "has never
wandered from me, even in imagination.") Added to conjugal incongruity
was this indifference on the part of Thomas and his wife to the
training of the children. The three young Dilworths were allowed to
grow up exactly as they pleased. It had worked well enough with Mary
and Nancy, who were good girls, affectionate and sensible--so sensible
that Nancy, when she was eighteen, had practically taken the
housekeeping out of her mother's hands; and Mary, at sixteen, looked
out for herself and her affairs most successfully. With Edwin the
Dilworth system had not been so satisfactory. He was conceited (though
that is only to be expected of the male creature at nineteen) and
rather selfish; and he had an unlovely reserve, in which he was
strikingly unlike his father, who overflowed with confidences. This,
and other unlikeness, was, no doubt, the reason that there were
constant small differences between them. And Mrs. Dilworth--vague,
gentle soul!--was somehow unable to smooth the differences over as
successfully as most mothers do.
Now, smoothing things over is practically a profession to mothers of
families. But Milly Dilworth had never succeeded in it. In the first
place, she had no gift of words; the more she felt, the more
inexpressive she became; but, worst of all, she had, poor woman, not
the slightest sense of humor. Now, in dealing with husbands and
children (especially with husbands), though you have the tongues of
men--which are thought to be more restrained than those of women--and
though you have the gift of prophecy (a common gift of wives) and
understand all mysteries--say, of housekeeping--and though you give
your body to be used up and worn out for their sakes, yet all these
things profit you nothing if you have no sense of humor. And Milly
Dilworth had none.
That was why she could not understand.
She loved, in her tender, undemonstrative way, her shy, unpractical,
secretive Edwin and her two capable girls; she loved, with the single,
silent passion of her soul, her generous, selfish, light-hearted Tom,
who took her wordless worship as unconsciously and simply as he took
the air he breathed; she loved them all. But she did not pretend to
understand them. Thus she stood always a little aside, watching and
loving, and wondering sometimes in her simple way; but often suffering,
as people with no sense of humor are apt to suffer. Dear, dull, gentle
Milly! No one could remember a harsh word of hers, or mean deed, or a
little judgment. No wonder Dr. Lavendar felt confident that there
would be no thistles in her household.
Thomas Dilworth had the same comfortable conviction, especially in
regard to his girls. "Now, Milly, honestly," he used to say, "apart
from the fact that they are ours, don't you really think they are the
nicest girls in Old Chester?"
Milly would admit, in her brief way, that they were good children.
"And Edwin means all right," the father would assure himself; and then
add that he couldn't understand their boy--"at least, I suppose he's
ours? Willy King says so. I have thought perhaps he was a changeling,
put into the cradle the first day."
"But, Tom," Milly would protest, anxiously, "Neddy couldn't be a
changeling. He was never out of my sight for the first week--not even
to be taken out of the room to be shown to people. Besides, he has
your chin and my eyes."
"Well, if you really think so?" Thomas would demur. And Mrs. Dilworth
always said, earnestly, that she was sure of it.
Still, in spite of eyes and chin, Ned's unpracticalness was an anxiety
to his father, and his uncommunicativeness a constant irritation.
Thomas himself was ready to share anything he possessed, money or
opinions or hopes, with any friend, almost with any acquaintance. "I
don't want to know anybody's business," he used to say; "I'm not
inquisitive, Milly; you know I'm not. But I hate hiding things! Why
shouldn't he say where he's going when he goes out in the evening?
Sneaking off, as if he were ashamed."
"He just doesn't think of it," the mother would say, trying to smooth
it over.
"Well, he ought to think of it," the father would grumble, eager to be
smoothed.
But Milly found it harder to reconcile her husband to their boy's
indifference to business than to his reserves.
"He sees fit to look down on the hardware trade," Tom told his wife,
angrily. "'Well, sir,' I said to him the other day, 'it's given you
your bread-and-butter for nineteen years; yes--and your fiddle, too,
and your everlasting music lessons.' And I'll tell you what, Milly, a
man who looks down on his business will find his business looking down
on him. And it's a good business--it's a darned good business. If Ned
doesn't have the sense to see it, he had better go and play his fiddle
and hold out his hat for pennies."
Milly looked anxiously sympathetic.
"I don't know what is going to become of him," Thomas went on. "When
you come to provide for three out of the hardware business, nobody gets
very much."
Mrs. Dilworth was silent.
"I was talking about him to Dr. Lavendar yesterday, and he said: 'Oh,
he'll fall in love one of these days, and he'll see that fiddling won't
buy his wife her shoe-strings; then he'll take to the hardware
business,' Dr. Lavendar said. It's all very well to talk about his
falling in love and taking to business; but if he falls in love, I'll
have another mouth to fill. And maybe more," he added, grimly.
"Not for a year, anyway," his wife said, hopefully. "And, besides, I
don't think Neddy's thinking of such a thing."
"I hope not, at his age."
"You were engaged when you were nineteen."
"My dear, I wasn't Ned."
Mrs. Dilworth was silent.
"The Packards telegraphed to-day that they wouldn't take that reaper,"
Tom Dilworth said.
Milly seemed to search for words of sympathy, but before she found them
Tom began to talk of something else; he never waited for his wife's
replies, or, indeed, expected them. He was so constituted that he had
to have a listener; and during all their married life she had listened.
When she replied, she was a sounding-board, echoing back his own
opinions; when she was silent, he took her silence to mean agreement.
Tom used to say that his Milly wasn't one of the smart kind; he didn't
like smartness in a woman, anyway; but she had darned good sense;--for,
like the rest of us, Thomas Dilworth had a deep belief in the
intelligence of the people who agreed with him....
"I have a great mind," he rambled on, "to go up to the Hayeses'. You
know that note is due on the 15th, and I believe I'll have to ask him
to extend it. I hate to do it, but Packard has upset my calculations,
and I'll have to get an extension, or else sell something out; and just
now I don't like to do that."
"Very well," she said. It was her birthday--the one day in the year
that her Thomas remembered that he had been in love with her for so
many years, months, days, hours, minutes--a fact she never for one day
in the year forgot. But she could no more have reminded him of the day
than she could have flown. She was constitutionally inexpressive.
Tom began to whistle:
[Illustration: music fragment]
but broke off to say, "Well, since you advise it, I'll see Hayes"; then
he gave her a kiss, and immediately forgot her--as completely as he had
forgotten his supper or any other comfortable and absolutely necessary
thing. Then he lighted his cigar and started for the Hayeses'.
II
"And who do you suppose I found there?" he said, when he got home, well
on towards eleven o'clock, an hour so dissipated for Old Chester that
Milly was broad awake in silent anxiety. "Why, Ned, if you please! He
was talking to Hayes's daughter Helen. She seems a mighty nice girl,
Milly. I packed young Edwin off at nine; he was boring Miss Helen to
death. Boys have no sense about such things. Can't you give him a
hint that women of twenty-five don't care for little boys' talk?
By-the-way, she talks mighty well herself. After I settled my business
with Hayes, we got to discussing the President's letter; she had just
read it."
"Do you mean to say _that the President has written to Helen Hayes_?"
cried Mrs. Dilworth, sitting up in bed in her astonishment.
Thomas roared, and began to pull his boots. "Why, they are regular
correspondents! Didn't you know it?"
"No! I hadn't the slightest idea--Tom, you're joking?"
"My dear, you can't think I am capable of joking? But, Milly, look
here, I'll tell you one thing: she was mighty sensible about Ned. She
thinks there's a good deal to him--"
"I don't need Helen Hayes to tell me that," said Ned's mother.
Tom, who never paused for his wife's reply, was whistling joyfully:
[Illustration: music fragment]
Helen Hayes had been very comforting to him; he had protested, when Ned
reluctantly departed, that a boy never knew when to clear out; and Miss
Helen had pouted, and said Ned shouldn't be scolded; "I wouldn't let
him 'clear out'--so there!" Few women of thirty-two can be cunning
successfully, but Tom thought Miss Helen very cunning. "I just
perfectly love to hear him talk about his music," she said.
"He can't talk about anything else," Ned's father said. "That's the
trouble with him."
"The trouble with him? Why, that's the beauty of him," said Miss
Hayes, with enthusiasm; and Thomas said to himself that she was a
mighty good-looking girl. The rose- lamp-shade cast a soft
light on a face that was not quite so young as was the frock she
wore--rose- also, with much yellowish lace down the front. It
was very unlike Milly's dresses--dark, good woollens, made rather
tight, for Milly, short and stout and forty-three, aspired (for her
Thomas's sake) to a figure,--which is always a pity at forty-three.
Furthermore, Helen Hayes's hands, very white and heavy with shining
rings, lay in lovely idleness in her lap; and that is so much more
restful in a woman's hands than to be fussing with sewing "or
everlasting darning," Thomas thought. In fact, what with her lovely
idleness and her praise of his boy, Tom Dilworth thought he had rarely
seen so pleasing a young woman. "Though she's not so very young, after
all; she must be twenty-five," he told his wife.
"She'll never see thirty again."
"Well, she's a mighty nice girl," Thomas said.
Except to look pretty, Miss Helen Hayes had done nothing to produce
this impression, for she had contradicted Mr. Dilworth up and down
about Ned.
"He has genius, you know."
"You mean his fiddle?" Tom said, incredulously.
"I mean his music. We'll hear of him one of these days."
"I don't care much whether we ever hear from his music," he said, "but
I wish I could hear that he was applying himself to business."
"Business!" cried Helen Hayes. "What is business compared to Art?"
Thomas looked over at Mr. Hayes in astonishment, for in those days, in
Old Chester, this particular sort of talk had not been heard; the older
man sneered and changed his cigar from one corner of his mouth to the
other. Miss Hayes did not get much sympathy from her family. But she
went on with pretty dogmatism:
"You see, in a man like your son--"
"A man! He's only twenty, my dear young lady."
"In a _man_, sir! like your son--genius is the thing to consider; and
you owe it to the world to let genius have its fullest play. Don't
bring Pegasus down to plough Old Chester cornfields. Why, it seems to
me," said Helen Hayes, "that he ought to be allowed to just soar. We
common folk ought to do the ploughing."
"Thunder an' guns!" said Tom Dilworth.
"I don't care if he can't be sure that two and two make four," cried
Miss Helen (Thomas, bubbling into aggrieved confidence on this sore
subject, had alleged this against his son); "he can put four notes
together that open the gates of heaven. And he'll distinguish himself
in music, because his father's son is bound to have tremendous
perseverance and energy."
Old Mr. Hayes snorted and spat into the fire; but Miss Helen's look
when she said "his father's son" made Mr. Thomas Dilworth simper.
"That girl has sense," he said to himself as he walked home at a
quarter to eleven. But he only told Mrs. Dilworth that she had better
hint to Ned to be a little more backward in coming forward. "That
Hayes girl is nice to him on our account," said Tom, "but he needn't
bore her to death. Milly, why don't you have one of those pink
wrappers? She had one on to-night. Loose, you know, and trimmed down
the front."
"A wrapper isn't very suitable for company," Mrs. Dilworth said,
briefly. "It didn't matter with you, because you're an old married
man; but she oughtn't to go round in wrappers when Neddy's there."
"Why, it was a sort of party dress--all lace and stuff. I wish you had
one like it. As for Ned, he's a babe; and her wrapper thing was
perfectly proper, of course. Can't you ask her for the pattern?"
And then Thomas went to sleep and dreamed of a large order for
galvanized buckets; but his Milly lay awake a long time, wondering how
she could get a pink dress; pleased, in her silent way, that Tom should
be thinking about her clothes; but with a slow resentment gathering in
her heart that Helen Hayes's clothes should have suggested his thought.
"And pink isn't my color," she thought, a vision of her own mild, red
face rising in her mind. Still, a fresh pink lawn--"that's always
pretty," Milly Dilworth said to herself, earnestly.
III
Tom Dilworth's boy was a curious _sport_ from the family stock. He
did, indeed, look down on the hardware business, but not much more than
on any business, although galvanized utensils were perhaps a little
more hideous than most things. Business in itself did not interest
him. Money-making was sordid folly, he said; because, "What do you
want money for? Isn't it to buy food and clothes and shelter? Well,
you can't eat more food than enough; you can only wear one suit of
clothes at a time; and an eight-foot cell is all the shelter that is
necessary."
"Eight-foot--_grandmother!_" his father would retort; "you'll inventory
that lot of spades, young man, and dry up."
And Ned, with shrinking hands and ears that shuddered at the hideous
screech of scraping shovels, would make out his inventory with
loathing. His mother was not impatient or contemptuous with him--she
could not have been that to any one; she simply could not understand
what he meant when he spouted upon the folly of wealth (for, like most
shy people, he occasionally burst into orations upon his theories), or
when he set off some fireworks of scepticism borrowed from Mr. Ezra
Barkley, or undertook (when Thomas was not present) to prove his
father's politics entirely wrong. On such occasions Nancy would say,
"Oh, Ned, _do_ be quiet!" and Mary would yawn openly. As for his
music, nobody cared about it, except, perhaps, his mother. "But I must
say, Neddy, I like a tune," she would say, mildly, after Edwin had
tucked his violin under his chin and poured out all his young soul in
what was a true and simple passion.
"A tune!" poor Ned said, and groaned. "Mother, I wish you wouldn't
call me that ridiculous name."
"I'll try not to, Neddy, dear," she would promise, anxiously; and Ned
would groan again.
With such a family circle, one can fancy what it was to the lad when
quite by accident he found a friend. It was the summer that he was
twenty, that once, coming back in the stage with him from Mercer, Miss
Helen Hayes showed a keen interest in something he said; then she asked
a question or two; and when, hesitating, waiting for the laugh which
did not come, he began to talk, she listened. Oh, the joy of finding a
listener! She looked at him, as they sat on the slippery leather seat
of the old stage, with soft, intelligent eyes, her slightly faded
prettiness giving a touch of charm to the high and flattering gravity
of her manner. When she asked him to bring his violin sometime and
play to her, the boy could almost have wept with joy. He made haste to
work off several of his dearest and most shocking phrases, which she
took with deep seriousness: A whale's throat is not large enough to
swallow a man--therefore the Biblical account is false, etc., etc. "In
fact," said Ned, "if I could have a half-hour's straight conversation
with Dr. Lavendar, I could prove to him the falsity of most of the Old
Testament."
Helen Hayes was shocked; she regretted Mr. Dilworth's scepticism with
almost tearful warmth; yet she realized that a powerful mind must
search for truth, above all. She wished, however, that he would read
such and such a book. "I can't argue with you myself," she said--"you
are far too clever for my poor little reasoning powers."
It was in April that Edwin entered into this experience of feminine
sympathy; and by mid-summer, at the time when Mr. Thomas Dilworth also
found Miss Helen Hayes so remarkably intelligent, the boy was absorbed
in his new emotion of friendship. He never spoke of it at home, hence
his father's astonishment at finding him at the Hayeses'. And when, a
week later, he found him a second time, Tom Dilworth was much perplexed.
"I dropped in on my way back from the store," he told his wife, "and
there was that boy. I said to Miss Helen that she really must not let
him bother her. I told her he was a blatherskite, and she must just
tell him to dry up if he talked too much."
"Tom, I don't think you ought to talk that way about Neddy," Mrs.
Dilworth said. "He's a dear boy."
"He may be a dear boy, but he is a great donkey," Ned's father said,
dryly; "and I think it is very good in Helen Hayes to put up with him.
I can see she does it on my account. Milly, why don't you ask her to
come to supper, sometime? I like to talk to her; she's got brains,
that girl. And she's good-looking, too. Ask her to tea, and have
waffles and fried chicken, and some of that fluffy pink stuff the
children are so fond of, for dessert."
"She's not much of a child," said Mrs. Dilworth, her face growing
slowly red. "She's thirty-two if she's a day."
"My dear, she has aged rapidly; you said thirty a month ago. I like
the pink stuff myself, and I'm nearly fifty. I bet the Hayeses don't
have anything better at their house."
Milly softened at that. Where is the middle-aged housekeeper who does
not soften at being told that her pink stuff is better than anything
the Hayeses can produce? Yet Tom's talk of Miss Helen's brains pierced
through her vagueness and bit into her heart and mind; and she could
not forget that he had called the girl good-looking. "Girl!" said Mrs.
Dilworth. She was standing before the small swinging glass on her high
bureau, looking at herself critically; then she slipped back and locked
her door; then took a hand-glass and stood sidewise to look again. Her
hair was drawn tightly from her temples and twisted into a hard knot at
the back of her head; she remembered that the Hayes girl wore high
rats, which were very fashionable, and had a large curl at one side of
her waterfall. "But it's pinned on," Milly said to herself; "anyway,
mine's my own." Then she pulled her cap farther forward (in those days
mothers of families began to wear caps when they were thirty) and
looked in the glass again: Helen Hayes did not have a double chin.
"She's a skinny thing," Milly said to herself. Yet she knew, bitterly,
that she would rather be skinny than see those cruel lines, like
gathers on a drawing-string, puckering the once round neck below the
chin. And her forehead: she wondered whether if, every day, she
stroked it forty-two times, she could smooth out the wrinkles?--those
wrinkles that stood for the tender and anxious thought of all her
married life! She had heard of getting rid of wrinkles in that way.
"It would take a good deal of time," she thought, doubtfully. Still,
she might try it--with the door locked. These reflections did not,
however, interfere with the invitation which Thomas had suggested.
Milly had her opinion of a middle-aged woman who wore wrappers in
public; but if Tom wanted her and her wrappers, he should have them.
He should have anything in the world he desired, if she could procure
it. Had he desired Miss Hayes hashed on toast, Milly would have done
her best to set the dainty dish before her king. And no doubt poor
Miss Helen in this form would have given Mrs. Dilworth more personal
satisfaction than did her presence at Tom's side (for the invitation
was promptly accepted) in some trailing white thing, her eyes fixed on
her host's face, intent, apparently, upon any word he might utter.
Watching that absorbed and flattering gaze, Milly grew more and more
silent. She heard their eager talk, and her mild eyes grew round and
full of pain with the sense of being left out; for Miss Hayes, though
patient with her hostess, and even kind in a condescending way, hardly
spoke to her. Once when, her heart up in her throat, Mrs. Dilworth
ventured a comment, it seemed only to amuse Thomas and his guest--and
she did not know why.
"This morning," Tom said, "I was h'isting up a big bunch of galvanized
buckets to our loft with a fall and tackle, and all of a sudden the
strap slipped, and the whole caboodle just whanged down on the
pavement--"
"O-o-o-o!" said Helen Hayes, putting her hands over her ears with
dramatic girlishness.
"It was terrific, and just at that moment up came Dr. Lavendar. Well,
of course I couldn't express my feelings--"
"Poor Mr. Dilworth!"
"--he came up, and gave me a rap with his stick. 'Thomas,' he said
(you know how his eyes twinkle!)--'Thomas, this is the most profane
silence I ever heard.'"
Everybody laughed, except Milly and Edwin, the latter remarking that he
didn't see anything funny in that. At which Miss Hayes said to him,
under her breath, "Oh, you superior people are so contemptuous of our
frivolity!" And Ned blushed with satisfaction, and murmured, "Why, no;
I'm not superior, I'm sure."
As for Milly, with obvious effort and getting very red, she said that
she didn't see how silence could be profane. "As long as you didn't
say anything, you conquered your spirit," she added, faintly.
And then they all (except Edwin) laughed again. After that she made no
attempt to be taken into the gayety about her, but her heart burned
within her. The next morning at breakfast some words struggled out:
"You'd think she was a young thing, she laughs so. And she's nearly
thirty-five."
"How time flies!" said Tom, chuckling. And then, to everybody's
astonishment, the mute Edwin spoke up, and said that as for age it was
a matter of the soul and not of the body. "Some people are always
young," said Edwin. "Dr. Lavendar is, and you are, father--"
"Thank you, grave and reverend seignior."
"--and mother," continued the candid youth, "has always been old.
Haven't you, mother?"
"True, for you, my boy," said the father; "your mother has the wisdom
of the family."
Milly Dilworth's face grew dully red to the roots of her hair; a wave
of anger rose up in her inarticulate heart. They called her old, these
two. She could hardly see her plate for tears.
Edwin, however, was so thrilled by the elegance of his sentiment that
he was eager to repeat it to Miss Hayes; but, somehow, he always had
difficulty in introducing the subject of age. When he did succeed in
getting in his little speech, she said that he impressed her very much
when he said things like that. "Your insight is wonderful," she
murmured, looking at him with something like awe in her eyes. (Miss
Helen was never cunning with Ned.)
"I guess you're the only person that thinks so," Ned said; "at home
they're always making fun of me."
"My friend," she said, gravely, "what else can you expect? You are an
eagle in a pigeon's nest. I don't mean to criticise your family, but
you know as well as I that you are--different. You are an inspiration
to me," she ended. And Ned blushed with joy.
It certainly is inspiring to be told you are an inspiration.... Mr.
Thomas Dilworth did not blush when he learned that mentally he was the
most stimulating person that Miss Hayes had ever met; but he had an
agreeable consciousness of his superiority, which he made no effort to
conceal from his wife. He never made any effort to conceal anything
from Milly, not even that fondness for female society which Mrs.
Drayton had deplored.
And by-and-by Milly's tears began to lie very near the surface. They
never gathered and fell, but perhaps they dropped one by one on her
heart, leaving their imprint of patiently accepted pain. At this time
she thought of her own mental deficiencies very constantly. Her mind
had no flexibility, and she reached conclusions only by toilsome
processes; but once reached, they were apt to be permanent. Her slow
reasoning at this time led her to conclude that her Thomas was not to
blame because he admired some one who was cleverer than she. "Why,
he'd be foolish not to," she thought, sadly.
But this eminently reasonable conclusion did not save Mrs. Dilworth
from turning white and red with misery, when, for instance, her husband
observed that he had had to take down two bars of the Gordon fence, so
that Miss Hayes could go home across lots. Then Thomas chuckled, and
added that Helen Hayes was the brightest woman he knew.
He did not go on to tell of his walk in the October dusk, and Miss
Helen's arch appeal to him for instruction on a certain political point
on which she was ignorant. Thomas had instructed her so fully and
volubly, while she looked at him with her reverent gaze, that it had
grown dark; and that was why he had to take her home across lots.
Thomas had not mentioned these details; he merely said he thought Miss
Helen Hayes a bright woman--the brightest, to be exact, that he knew.
And yet his Milly went into the kitchen pantry and hid her face in the
roller behind the door and sobbed.
Well, of course! It's very absurd. A fat, wordless woman, who ought
to be darning her children's stockings, it's very absurd for her to be
weeping into a roller because her man, who has loved her for
forty-three years, eleven months, twenty-nine days, twenty-three hours,
and forty minutes--her man, to whom she is as absolutely necessary as
his old slippers or his shabby old easy-chair--because this man does
not think her the brightest woman he knows. But absurd as it is, it is
suffering.
The woman of faithful heart who has been left behind mentally by her
husband is a tragic figure, even if she is at the same time a little
ridiculous--poor soul! Her futile, panting efforts to catch up; her
brave, pitiful blunders; her antics of imitation; her foolish pink lawn
frocks--of course they are funny; but the midnight tears are not funny,
nor the prinking (behind locked doors), nor the tightened dresses, nor
the stealthy reading to "improve the mind"--that poor, anxious, limited
mind which knows only its duty to its dearest and best. These things
mean the pain--a hopeless pain--of the recognition of limitations.
What did it matter that once a year Tom announced that he had loved his
Amelia for so many years, months, days, hours, and minutes?--He did not
talk to her about the President's letter! But he talked to Helen Hayes
about it. And yet she was a pale thing. "She never had my color,"
poor Milly thought; "and they say she doesn't get along well at home.
And she's no housekeeper. Mrs. Hayes herself told me she was just real
useless about the house. I can't understand it."
Of course she could not understand it. What feminine mind ever
understood why uselessness attracts a sensible man? It is so foolish
that even the most foolish woman cannot explain it.
As the autumn closed in on Old Chester, nobody in the family noticed
Milly Dilworth's heavier look and deeper silence. Tom himself was more
talkative than usual; business had been good, and he was going to get
something handsome out of a deal he had gone into with Hayes. This
took him often to the Hayeses' house; and after the two men had had
their talk, Miss Helen was to be found at the parlor fireside, very
arch and eager with questions, but most of all so respectful of Tom's
opinions. His Amelia was respectful of his opinions, too, but in such
a different way. Perhaps just at this time Thomas Dilworth pitied
himself a little--the middle-aged husband does pity himself once in a
while. Perhaps he sighed--certainly he whistled. There is no doubt
that Mrs. Drayton would have felt he was wandering from his Amelia--at
least in imagination. And yet Tom was as settled and grounded in love
for his middle-aged wife as he ever had been.
This, however, cannot be understood by those who do not know that the
male creature, good and honest and faithful as he may be, is at heart a
Mormon.
"I declare," Tom said, coming home at twelve o'clock at night--"I
declare I feel younger."
Milly was silent.
Then Tom began to whistle:
[Illustration: music fragment]
Then he broke off to say that he didn't think that Helen Hayes was
over-happy at home. "The Hayeses are commonplace people, and she is
very superior. I guess they don't get along well."
Milly thought to herself that when a girl didn't get along with her own
mother it didn't speak well for the girl; but she did not say so.
But Thomas went on to declare that he didn't know what to make of Ned.
"Hanging round the Hayeses till I'm ashamed of him! Why doesn't he
know better? I never bored a woman to death when I was his age." And
his wife thought, in heavy silence, that there were other people who
hung round the Hayeses.
However, Thomas made his feeling so clear to his son that during the
winter Ned was never seen at the Hayeses' on the same evening that his
father was there. But there was an hour in the afternoon, from five to
six, when the boy was free and Thomas was busy with his spades and
buckets;--but you can't look after a boy every minute.
IV
Poor Amelia, in her bedroom, in the chilly December dusk, sopped her
eyes with cold water and looked in the glass. "I _mustn't_ cry any
more," she said to herself, despairingly--"they're so red now!"
A door opened down-stairs, and there was a burst of laughter; and Mrs.
Dilworth, in the cold twilight, went on sopping her eyes. Tom and the
girls evidently didn't need her. "They could get along just as well
without me. And if the Lord would take me, Tom could--could--so he
could--"
Her soul was dumb, even to itself; but she knew what it was that Tom
"could" do.
And she knew it without bitterness. Like every other woman whose love
for her husband has in it the maternal element (and most good women's
love has this element), she had always felt that if she died Thomas
ought to marry again; but this simple creature went one ahead of that
rather elementary feeling, and specified: she was willing to have him
marry _her_.
"If the Lord would only remove me," said poor Milly, looking miserably
in the glass at her plump figure, which showed no indications of
removal. Her eyes were hopelessly red; she didn't see how she could
possibly go down to supper. But of course she had to go down. The
mother of a family and the mistress of one servant must go down to
supper, no matter what the condition of her eyes may be. She slunk
into her seat behind her teacups, and scarcely dared to look about her
noisy, hungry circle, still less at her Thomas, who was smiling to
himself, but who did not share his amusement with his family. Still,
when he suddenly said something about the refreshment of talking to
intelligent people, it was not hard to guess the direction of his
thoughts. "It sharpens your brains up," said Thomas. "I was going to
suggest, Milly, that you should ask Helen Hayes to tea again; but she's
got company; and when they leave she's going off to make a visit to
some of her relations, she tells me."
Amelia's mild lips tightened silently. So they had been together
again. Her hand shook as she poured out another cup of tea for her
Thomas, who took that moment to say, with all a husband's candor, that
she was getting fatter than ever. "I thought you were starving
yourself to get thin, Milly?" he said, smiling. Milly smiled, too,
faintly; but she was saying to herself: "What did they talk about? How
long were they together? Oh, if I could only be taken away!"
It would be interesting to follow the processes of a mind like Mrs.
Dilworth's: how did a wife and mother of children reach the point of
feeling that her family would be better off without her? Anybody in
Old Chester could have told her such a belief was folly, and wicked
folly at that. But it seemed just plain reason to Milly Dilworth: "I'm
not necessary to anybody. Thomas likes somebody younger. He can't
marry her because I'm alive; he could marry her (and she would be good
to the children) if I were not here. But I _am_!" she would end,
hopelessly.
Morning after morning, as she went about her household duties, or when
before tea she sat in her little, old rocking-chair, mending the family
stockings, she used to break herself against the hopelessness of the
situation: She was there; and unless the Lord would remove her (any
other sort of removal was impossible to her devout imagination) Tom
could not have what he wanted--yes, and needed, too. For it was at
this period that Mrs. Dilworth recognized, what most wives of men do
recognize at one time or another, that although being a wife and mother
is the only vocation of a married woman, being a husband and father is
only one of many vocations of a married man. Hence the companionship
of an eminently worthy wife is almost never enough for the male
creature. When this harsh truth burst upon Milly, she wiped her eyes
on the stocking she was mending and groaned aloud. But she did not
rail against the fact, nor did she attempt to deny it; wherein she
showed a superfeminine intelligence. She only said to herself that
Thomas could not have what he wanted while she was alive; yet she
couldn't, it seemed, die, although she was so miserable that she didn't
know how she lived! It was at this point that she began to make wild
schemes to relieve the situation: Suppose she asked that Hayes girl to
come and make them a visit? But no--a man wants more than to just look
at a pretty girl across the table. Suppose she went away herself and
made a visit, and asked Miss Helen Hayes to come and keep house for
her? (Like all good wives, Milly had no hesitation in offering up
another woman to the pleasure of her lord.) No; people would talk
about Tom if she did that.... The amount of it was, poor Milly,
although she did not know it, was really planning that Thomas should
have two wives at the same time--and, dear me! how that would simplify
things! There would be the old, sensible, matter-of-fact wife to mend
his stockings and order his good dinner and nurse him through the
indigestion consequent upon the dinner--the old, anxious wife, who has
had the children and reared them, who has planned and economized and
toiled with him, who has borne the burden and heat of the day at his
side--the prosaic wife, who gives, unasked, such good advice. Every
one will admit that this elderly person has been, and (to a limited
degree) still is, a necessity to every Thomas. But sometimes Thomas
thinks, in his simple way, that it would be pleasant to have the
luxuries as well as the necessities of life; to have, for instance, a
young wife--a pretty wife, clever and light-hearted and gayly
tyrannical; a wife who never knew enough to advise anybody, who should
be a relaxation and a refreshment, _and just a little bit of a fool_;
for, as every intelligent (unmarried) woman knows, men like fools;
feminine fools. Of course the trouble is that if you supply a wife for
two sides of a man's character--for utility, so to speak, and for
diversion--he may, not unreasonably, demand that every side and angle
and facet of his jewel-like nature have its own feminine setting. That
was probably Solomon's idea. Well, well! the time is not yet for this
reasonable arrangement; and it is possible that trade in galvanized
buckets will never warrant its extensive existence.
But all this is very frivolous compared to the reality of this poor
woman's pain, a pain that finally evolved a plan which, although less
picturesque than the harem, was of the same grade in the eye of the
law, though, curiously enough, not in her own eye. She could not, as
she expressed it to herself, be dead, so that her Thomas might have his
wish; _but he could think she was dead_.
When this extraordinary idea came into Milly Dilworth's head, she felt
as one imprisoned in darkness who sees, far off, the glimmer of
daylight. He "could think she was dead!" And if he thought so, of
course there could be nothing wrong in his marrying "_her_." (Miss
Hayes's moral status did not enter into Milly's calculations.)
The light in her darkness dazzled poor Milly at first, and the way was
not clear. It took two weeks of further thought to decide upon the
step, and then to evolve its details; but one need not go into them as
Milly did.... As she sat at her work, day after day, she thought her
plan out slowly and toilsomely. At first she kept balking at the
enormity of it. Then some chance word would betray Tom's admiration
for brains, and she would beat and spur her mind up to her project
again.... And at last she accepted it.... Once accepted, the thing
was settled. Her mind had about as much flexibility as a bar of lead,
and there was no changing it. It only remained to decide upon the
details. This she did slowly and painfully. Each step was planned,
each contingency arranged for.
And by-and-by the day came to act.
The night before, at supper, Mrs. Dilworth, her hands stumbling among
her teacups, said, faintly, "I'm going over to the other side of the
river to-morrow to order some chickens from Mrs. Kensy."
"That Kensy house is right by the railroad station," Ned said,
scowling; "I don't believe she has any hens."
"Yes, she has, Neddy," said Mrs. Dilworth.
Edwin frowned blackly. "I do wish you wouldn't call me by that absurd
name, mother."
"I keep forgetting, Neddy dear."
Edwin held up his hands despairingly.
"What are you two people talking about?" demanded Thomas.
"I'm going to walk over, across the ice, to the Bend, to-morrow," said
Milly.
"Walk!" her husband protested. "What do you walk for? It's cold as
Greenland on the ice, and, besides, they were cutting at the pool by
the Bend; you don't want to go that way, Milly. Take the stage round."
Mrs. Dilworth crumbled a piece of bread with shaking fingers, and said
nothing.
"What time are you going, mother?" inquired Edwin.
"In the afternoon, about four."
"Why, you went there only two days ago," Edwin said, irritably. "I saw
you on the back road carting a big bundle."
"It would have been more to the point if you'd done the carting for
your mother," Tom Dilworth said, sharply.
His wife paled suddenly at that word about a bundle, but the subject
was not pursued. Edwin said, grumbling, that he didn't see what
possessed his mother to choose such an hour. "It's too dark for a lady
to be out," Edwin protested.
"Too dark for a--_grandmother_!" his father said. "Don't you criticise
your mother, young man." And then he added: "Look out for the places
where the men were cutting, Milly. It hasn't frozen over yet."
And Mrs. Dilworth said, after a pause, "I know."
That night was a misery of dreams that the deed was done, broken by
wakings desperate with the knowledge that it was yet to do. In the
morning she seemed to have lost all power of words; she bore her
husband's reproaches that Ned was late for breakfast; she went about
her household duties; she watched the girls start for school (she did
not kiss them; demonstrations of affection had never been possible to
this dumb breast; but she stared after them with haggard eyes); and
through it all she hardly uttered a word; when she did speak, it seemed
as though she had to break, by agonizing effort, some actual lock upon
her lips. When the girls had gone she looked about for her eldest; but
Ned was not to be found. "I never knew him to go to the store before
breakfast," she thought, miserably. His father, pulling on his coat in
the hall, said that Ned was getting industrious to go to his work so
early! His wife was silent.
When he started, whistling cheerfully,
[Illustration: music fragment]
she watched him from the window, straining her eyes until he was out of
sight. Then she went up-stairs to her bedroom, and, opening his closet
door, leaned her head against one of his coats, trembling very much.
Afterwards she wandered about the house in aimless, restless waiting
for Ned.
In the course of the morning Tom sent over to inquire why the boy had
not come to the store. Milly told the messenger to tell Mr. Dilworth
that Mr. Edwin was not at home. "Say I thought he was at the store,"
she said. "I'll give him his father's message when he comes in to
dinner." But he did not come in to dinner; and minute by minute the
afternoon ticked itself away. She had said to herself that she must
start about four, before Nancy and Mary got home from school. "It must
be so that it would be dark when I was coming back," she reminded
herself. "If I leave here at four, and get my bundle from Mrs. Kensy
at five, it would be pretty dark by the time I would be going home.
Mrs. Kensy will tell them that it was dark."
At four Edwin had not appeared; Milly, having no imagination, had no
anxiety; she merely gave up, patiently, the hope of a wordless
good-bye. But she kept looking for him; and when she finally put on
her things, she paused and turned back to the window, to look once more
towards Old Chester; but there was no sign of Ned. It did not occur to
her to postpone her plan; her mind, run into the mould of sacrifice,
had hardened into rigidity. So at last, miserably, the tears running
down her face, she stepped out into the cold and went down through the
garden to the river. There she turned and looked back, with dumb
passion in her eyes; the firelight was winking from the parlor windows
and all the warm commonplace of life seemed to beckon her. She put her
muff up to wipe her eyes, but she made no prayer or farewell; her
silence had reached her soul by that time.
[Illustration: "THERE SHE TURNED AND LOOKED BACK"]
It was very cold; the ice was rough, and the wind had blown the dry
snow about in light drifts and ripples, so that walking was not
difficult. She trudged out, up towards the Bend, skirting the place
where the men had been cutting. They had gone home now, and the ice
about the black, open space of water was quite deserted. The wind came
keenly down the river, blowing an eddy of snow before it; the bleak sky
lay like lead over the woods along the shore. There was not a house in
sight. Amelia Dilworth looked furtively about her; then she bent down
and scraped at the snow on the edge of the ice, as one might do who, in
the water, was struggling for a hold upon it. After that, for a long
time, she stood there, looking dumbly at the current running, black and
silent, between the edges of the ice. At last, her hand over her mouth
to check some inarticulate lament, she stooped again, and put her
little black muff on the broken snow close to the water.
When she reached Mrs. Kensy's she was quite calm. She said briefly
that she had come to order some chickens; "--and I'll take that bundle
I asked you to keep for me."
The woman brought it, and Milly tucked her fingers through the stout
strings she had tied so carefully a few days before. When she would
open it in the woods, and put on the new dress and shawl and the heavy
veil that it held, and then, in the dark, take the half-past-five
train, no one would know that Thomas Dilworth's wife had fled away into
another State. They would find the muff, and they would think--there
would be only one thing to think.
"I want the chickens for Sunday," she said; "please send them over on
Saturday." Then it came into her mind with a little gush of happiness
that she would pay for them on the spot, instead of having the bill
sent to Tom, as was her custom; she had drawn a sum of money from the
bank a fortnight ago--a small sum, but her own; now it was all in her
purse; she would buy Tom's Sunday dinner out of her little fund.
Except to leave him, it was the last thing she would ever do for him.
She put her hand into her pocket--and chilled all over. Then stood
blankly looking at the woman; then plunged her hand down again into her
pocket; then exclaimed under her breath; then tore her bag open and
fumbled distractedly among brushes and night-gown and slippers; then
pulled her pocket wrong side out with trembling fingers.
"_My purse!_" she said, breathlessly. Then she searched everything
again.
"It ain't any difference," Mrs. Kensy protested.
"I must have left it at home. I can't go back for it. It is too late."
"What for?" said Mrs. Kensy.
"The--the train."
"Oh, you was going on, was you?" Mrs. Kensy said. "Well, I can let you
have the price of a ticket a little ways."
But Mrs. Dilworth, with shaking hands, pulled everything out of her
bag, shook her skirts, fumbled in the bosom of her dress, ran out and
searched the garden-path, strained her eyes across the snow on the
river--all in vain. "Oh, my!" she said, faintly.
"But I can lend you the price of a ticket, ma'am," Mrs. Kensy said
again.
"No matter," Mrs. Dilworth said, dully. "I'll go home."
Even as she spoke she heard the train tooting faintly far up the
valley. She sat down, feeling suddenly sick.
V
There was nothing to do but to go home. She remembered now how in her
agitated watching for her son she had put her purse down on the corner
of her bureau--and left it there. Yes; there was nothing to do but go
back. "I can start to-morrow," she said to herself. But in the sick
reaction of the moment she knew that she could never start again; her
purpose had been shattered by the blow. She took her bundle--the
bundle that meant flight and disguise and self-sacrifice, and that
stood for the shrewdness which is so characteristic of the kind of
stupidity which forgets the purse--and went stumbling down in the
darkness to the river. She said to herself that she must get her muff;
and she thought heavily that it would be pretty hard to carry so many
things across the ice. She was numb with the shock of interrupted
ecstasy. She could not feel even mortification--only fatigue. She was
so tired that, seeing in the darkness a hurrying figure approaching
her, she did not recognize her husband until he was almost upon her.
"_Milly_? My God! Milly!"
He had her muff in his hand, and as he reached her he caught at her
shoulder and shook her roughly. "Milly--I thought--I thought--" He
stammered with agitation. "I found this muff, and I thought it was
yours; and Neddy's gone, too, and I thought--both of you--"
"Neddy _gone_?" she repeated, dully.
She stood still on the ice, trying to get her wits together.
"He's disappeared. He isn't in town. He went out early this morning.
To skate, I suppose. Nora saw him from her window; at about six, she
says. And this open water"--she felt him quiver at her side--"and then
this muff--"
"No!" she said. "I--I made a mistake." She did not take in the words
about Ned.
"But where is he? Nobody's seen him. I suppose I'm a fool, but I'm
uneasy. I came to meet you because I thought you might know. But when
I saw this muff--it is yours, Milly, isn't it?--I got into a panic
about you, too."
"Why," she said--"it's mine; yes. I--I left it--I suppose. Neddy
wasn't with me. Did you think he was with me? I don't understand,"
she ended, bewildered.
"He hasn't been at home all day," her husband said, "nor in town,
either." And then he repeated the story, while she looked at him, slow
understanding dawning in her eyes.
"Neddy--gone! Where?"
"But that's what I don't know," the father said.
And his wife, dazed still, but awake to the trouble in his voice, began
to comfort him, alarm rising slowly in her own heart like an icy wave.
"Maybe he went to see somebody in Upper Chester?"
"But he doesn't know anybody at Upper Chester. Of course it's
possible. Only--you gave me such a fright, Milly!" Mrs. Dilworth put
her hand over her mouth and trembled. "However, I guess he's all
right, as you say. I guess we'll find him at home when we get back.
It's lucky I came to meet you, because I can lug your things for you.
How did you drop your muff, dear? Here, take it; your hands must be
cold. Oh, Milly, you gave me an awful fright--it was right on the very
edge of the ice; those confounded cutters hadn't put up any ropes. You
do really think there's no reason to be uneasy about Ned?"
"No," she said. Her knees shook; she had to pause to swallow before
she spoke. Oh, what if he should find her out? As she trudged along
at his side in the cold darkness she said to herself, with a sickening
sense of apprehension, that if he found her out she should die. Then
as her mind cleared she tried in her brief way to encourage him about
their boy; yet, as they drew nearer home and she saw again the firelit
windows, she began to awaken to the situation: Neddy had gone out to
skate; at six, did Nora say? Of course he might have stopped to see
somebody in Upper Chester; only Neddy never went to see anybody
anywhere--except (Amelia Dilworth had forgotten her!)--except that
Hayes girl--and she wasn't at home. Yes, it was strange; and worrying,
perhaps. But she only repeated, as they went hurrying up to the back
door, that she was sure Neddy was all right. But she held her breath
to listen for his voice haranguing his sisters in the sitting-room.
Instead, the two girls came running out to meet them.
"Oh, father, did you find Ned? Oh, here's mother; she'll know where he
is."
"Mother, I'm sort of scared about him," Mary whispered.
"He's gone to see some friend," the mother said, and her brevity, so
agonizing to her, seemed to reassure the others.
"He hasn't any friend except Miss Helen Hayes," Nancy said, "and she
went away last week."
"Maybe he's gone to hunt her up," Mary said, giggling, and her father
told her to be quiet.
"It's thoughtless in him to be so late. But your mother isn't worried,
so I guess we needn't be. Your mother says there is not the slightest
cause for anxiety, and she knows."
"Come to supper," Amelia said, her heart sinking; and the commonplace
suggestion cheered them all, although Tom Dilworth did not like to lose
the assurance of his wife's presence, even to have her go up-stairs to
take off her bonnet, and went with her, saying again, decidedly, that
there was, as she said, no possible reason for uneasiness, and that he
himself hadn't a particle of anxiety. "But I'll give that boy a piece
of my mind for worrying you so. Why, Milly, what a fat pocket-book!
Where did you get so much money, my dear? I didn't know the hardware
trade was so prosperous. Look here, Milly--it is pretty late,
honestly?"
She took her purse out of his hands, her own trembling. For a moment
she could not speak, and leaned forward to look into the swinging glass
and make pretence of untying a knot in her bonnet-strings. "Oh, he'll
come home soon," she said.
In spite of assurances, the tea-table was not very cheerful--the girls
stopped short in the middle of a sentence to listen for a step on the
porch. Tom got up twice to look out of the window. Mrs. Dilworth
thought she heard the gate slam, and held her breath; but no Ned
appeared. The evening was endlessly long. Tom pretended to read his
newspaper, and kept his eye on one spot for five minutes at a time. At
ten he packed the girls off to bed; at eleven he was walking up and
down the room; at twelve he told his wife to go to bed; but somehow or
other he went himself, while she sat up, "to let the boy in."
You can make excuses for this sort of lateness up to a certain point;
but it is curious that at about 2.30 in the morning the excuses all
give out. Tom Dilworth got up and dressed. "Something has happened,
Milly," he said, brokenly. His wife put her arms around him, trying to
comfort him.
"If Miss Hayes was only at home," she said, "maybe she would have some
idea of his plans. He might have told her. And she could tell us what
to do."
"Who?" said Tom--"that Hayes girl? Maybe so. I hadn't thought of her.
No, I don't believe she'd be any help. She hasn't got much sense in
that kind of way."
Such ages and ages was Milly away from her great experience of jealousy
that she felt no relief at this bald betrayal. Together they went out
onto the porch, listening, and straining their eyes. The moon was just
going down; it was very cold; far off a dog barked. But there was no
human sound. The two haggard people went shivering back into the hall,
where a candle burned dimly in the glass bell hanging at the foot of
the stairs.
"Something has certainly happened," Tom said again. "Oh, Milly, you
are always so calm and I go all to pieces." He leaned his elbow
against the wall and hid his face in his arm. His wife heard him groan.
"And--I've been hard on him sometimes," he said.
She took his hand and kissed it silently.
Poor Tom went to pieces more than once in the days that
followed--dreadful days of panic and despair. Old Chester, aroused at
daybreak by the terrified father, decided at once that the boy was
drowned; but everybody stood ready to help the stricken parents with
hopeful words to the contrary, words which rang as hollow to Thomas and
his wife as to the well-meaning liars.
It was on Wednesday that he had disappeared. On Friday they dragged
the river through the open holes; on Saturday, blew up the ice and
dragged all the way down to the second bend. That night Nancy and Mary
crept away to cry in their own room; Tom sat with his head buried in
his arms; his wife knelt beside him, touching him sometimes with a
quiet hand, but never speaking. Dr. Lavendar came in and put his hand
on Tom's shoulder for a minute, and then went away. The firelight
slipped flickering about the room; sometimes the coal in the grate
snapped and chuckled, and a spurt of flame shone on the two suddenly
aged faces. And then into the silent room came, with hurried,
shamefaced triumph--Edwin.
"I--I'm afraid you've been anxious--"
"He ought to have written," said another voice, breathless and
uncertain, and breaking into nervous laughter. "It is naughty in him
to have forgotten. I--I told him so."
Thomas Dilworth lifted his head and stared, silently; but his wife
broke out into wild laughter and streaming tears; she ran and threw
herself on Edwin's breast, her throat strangling with sobs.
"Oh--she's found Neddy! She has brought him back to us!--she has found
him! Oh, Miss Hayes, God bless you--God bless you! Oh, where did you
find him?"
Miss Hayes opened her lips--then bit the lower one, and stood, scarlet.
"I meant to write," Edwin began to explain--"of course I meant to
write, but--"
"Oh, dear Mrs. Dilworth," Helen's fluttering voice took up the excuse,
"you must forgive him"--she came as though to put her arms about Ned's
mother. "After all, a bridegroom, you know--"
Milly lifted her head from Edwin's shoulder and gaped at her.
"Bridegroom?"
Thomas Dilworth got on his feet and swore. Miss Helen Hayes--or, no;
Mrs. Edwin Dilworth--came and hung upon his arm.
[Illustration: "THOMAS DILWORTH GOT ON HIS FEET AND SWORE"]
"You won't mind very much? You'll forgive him? We couldn't tell,
because--because papa would have interfered; but I knew your dear, kind
heart. Mrs. Dilworth, I have so revered Mr. Dilworth!--that was one
reason I said _yes_. You'll let me be your little girl, Mr. Dilworth?"
"Little--_grandmother_!" said Tom Dilworth; and burst into a roar of
laughter; then stopped, and said through his set teeth to his son, "You
scoundrel!"
"Thomas--don't!" the mother entreated. "He has come back."
"He'd better have stayed away!" Thomas said, furiously, in all the
anger of suddenly relieved pain.
"Oh, dear Mrs. Dilworth," Helen murmured, "forgive us! He ought to
have written--I ought to have reminded him. But--_you_ understand? I
know you do. Just these first beautiful days, one forgets everything."
"Well, I tell you I meant to write," Ned persisted, doggedly. "But
mother put me all out by going over to the Bend in the afternoon. I
was going to take that train, and of course I couldn't; Kensy's house
is right there by the station. And I had to take the morning train
instead; and it put me all out. I had to get up so early I forgot to
take any clothes," he added, resentfully. "It wasn't my fault."
"Not your fault?" his father said, and then turned to his wife, almost
with a sob. "Milly, can he be our boy, this sneak?"
"Yes; yes, he is, Tom; indeed he is, dear. And he just forgot; he
didn't mean anything wrong." Milly was almost voluble, and she was
crying hard. And then she looked at the woman who had brought him
back--the faded, anxious, simpering woman, who for once had no words
ready. Milly looked at her, and suddenly opened her arms and took her
son's elderly wife to her heart. "Oh, you poor woman," she said, "how
unhappy you must have been at home!"
Helen looked at her blankly, then dropped her head down on the kind
shoulder, and Milly felt her quiver.
"She's fifty!" Tom said, trembling with anger. "How the devil a son of
mine can be such a jack--"
"Tom, dear! there now, _don't_," the mother said; "he's at home. Just
think; he's at home! and we thought--we thought--" Her voice broke.
"We'll all love you, Miss Hayes--I mean Helen," she whispered to the
sobbing woman.
Then, with a sort of gasp, she put her daughter-in-law's arms aside
gently, and went over and kissed her husband.
As for Thomas Dilworth, after the first shock of anger and
mortification had passed, and the young couple had finally settled
themselves upon the disgusted bounty of the respective fathers, he used
to whistle incessantly a certain song much in vogue at the time:
"I hanker
To spank her,
Now I'm her papa!"
"AN EXCEEDING HIGH MOUNTAIN"
I
Robert Gray's first wife, Alys (Old Chester had hard work to swallow
her name; "but it's better than any of your silly 'ie's,'" said Old
Chester)--this first Mrs. Gray was a good deal of a trial to everybody.
She was not only "new," but foreign; not only foreign, but indifferent
to Old Chester. Indeed, it took all Old Chester's politeness and
Christian forbearance to invite Mrs. Robert Gray to tea--with the
certainty that the invitation would be declined. She was an English
girl whom Robert met somewhere in Switzerland--a heavy-eyed, silent
creature, certainly a very beautiful woman, but most inefficient and
sickly; and there were so many nice, sensible girls in Old Chester!
(However, there is no use saying things like that: as if a man ever
married a girl because she was sensible!)
Yet young Gray certainly needed a sensible wife; his wealth was limited
to character and good manners, plus a slender income as tutor in the
Female Academy in Upper Chester. Excellent things, all; but a wife
with sense (and money) would have been an agreeable addition to his
circumstances. Whereas, this very beautiful English girl was a
penniless governess, left stranded in Germany by an employer, who had,
apparently, got tired of her. Robert Gray had met the poor, frightened
creature, who was taking her wandering way back to England, and married
her, frantic with rage at the way she had been treated. When he
brought her home, he was so madly in love that he probably did not half
appreciate Old Chester's patience with her queer ways. But the fact
was, that for the few months she lived, she was so miserable that Old
Chester could not help being patient, and forgiving her her half-sullen
indifference, and her silence, and her distaste for life--even in Old
Chester!
For in spite of Robert's adoration, in spite of all the ready
friendliness about her, in spite of the birth of a baby girl, she
seemed, as it were, to turn her face to the wall. She died when the
child was about a week old. Died, the doctor said, only because, so
far as he could see, she did not care to live.
"You ought to try to get better for the baby's sake," said Miss Rebecca
Jones, who had come in to help nurse her. And the poor girl frowned
and shook her head, the heavy, white lids falling over her dark eyes.
"I don't like it."
And Rebecca (who had too much good sense to be shocked by the vagaries
of a sick woman) said, decidedly: "Oh, you'll learn to like her. Come,
now, just try!"
But she did not seem to try; even though Robert, kneeling with his arm
under her pillow, holding her languid hand to his lips, said, sobbing,
"Oh, Alys, Alys--for God's sake--don't leave me--"
Then she opened her beautiful eyes and looked at him solemnly.
"Robert," she said, "I am sorry. I am--sorry. I--am--"
"What for, precious?" he entreated; "sorry for what? to leave me? Oh,
Alys, then live, live, dear!"
"I--am--" she began; and then her voice trailed into eternity.
Miss Rebecca Jones hung about the house for a few days, to make the
poor gentleman comfortable; then he was left alone with the child
(purchased at so dreadful a cost) and one servant, and his daily work
of teaching the polite languages at the Female Academy. Miss Rebecca's
hard face softened whenever she thought of him; but all she could do
for him was to go often to see the poor seven-months baby--which seemed
for a time inclined to follow its mother.
Now it must be understood at once that Rebecca Jones was not a schemer,
or a mean or vulgar woman. She was merely a hard-headed,
honest-hearted product of years of public-school teaching, with a
passion for truth and no grace in telling it. She was sorry for Mr.
Gray, and sorry for the poor baby, who was being allowed, she said to
herself, to grow up every which way; and sorry for the comfortless
house left to the care of what she called "an uneducated servant-girl."
So, after school, and on Saturday mornings, she used to go over to Mr.
Gray's house and bustle about to the bettering of several things.
Indeed, old Mr. Jones told her more than once that he didn't know what
that there widower would do without her. And Rebecca said, truthfully
enough, that she didn't know, either. And when she said it her heart
warmed with something more than pity.
As for Robert Gray, dazed and absent, trying to do his duty at the
Academy during the day, and coming home at night to look blankly at his
child, he, too, did not know what he would have done that first year
without Miss Rebecca's efficient kindness. He was so centred in his
grief, and also of so gentle a nature, that he took the kindness as
simply as a child might have done. Like many another sweet-minded man,
he had not the dimmest idea of the possible effect of his rather
courtly manner and his very delicate courtesy upon a woman of slightly
different class, whose life had been starved of everything romantic or
beautiful. He became to sharp-tongued Miss Rebecca Jones a vision of
romance; and, somehow, quite suddenly, about eighteen months after his
wife's death, he discovered that he was going to marry her. In his
startled astonishment, he realized that he had himself led up to her
avowal of willingness by some talk about her kindness. Perhaps she had
misunderstood his words; if she had, Robert Gray was not the man to
offer an explanation.... However, after the first shock of being
accepted, he was gently explicit:
"I realize that the child ought to have the care of a good woman, and
therefore I--"
"I'll do my duty by her," Rebecca said.
"I want her brought up to love and reverence her mother. I want her
brought up to be like her. It is for the child's sake that I--I marry
again. I speak thus frankly, Miss Rebecca, because I so entirely
respect you that I could not be anything but frank."
Rebecca's square face flushed over the high cheek-bones to the gaunt
forehead and the sparse hair; then her eyes looked passionately into
his. "I understand. Yes; I understand. And I will be good to your
child, Mr. Gray."
And so he married her; and, when you come to think of it, it was a very
sensible thing to do. Even Old Chester said he was very sensible. A
man of thirty, with a baby--of course he ought to marry again! "But
why on earth," said Old Chester, "when there are so many girls of his
own class!--not but what Rebecca Jones is a very worthy person."
Meanwhile, Rebecca, with hard conscientiousness, set herself to bring
the child up. She trained her, and disciplined her, and made a painful
point of talking to her about the first Mrs. Gray, according to her
promise to teach her to "love and reverence her mother." The
discipline sometimes made Robert Gray wince; but it was wise, and never
unkind; so he never interfered;--but he left the room when it was going
on. Once he said, nervously:
"I scarcely think, Mrs. Gray, that it is necessary to be quite so
severe?"
"She must be made a good child," Rebecca answered.
"I am not afraid that she will not be a good child," Robert Gray said;
"she is her mother's daughter."
"Well, she is her father's daughter, too," Rebecca declared, briefly.
And her husband, shrinking, said:
"Light is stronger than darkness; Alice's mother was a creature of
light. I am not afraid of her inheritance of darkness."
As for Rebecca, she went away and shut herself up in the garret.
"'Creature of light!'" she said, sitting on the floor under the
rafters, and leaning her head on an old horsehair-covered trunk wherein
were packed away Mr. Gray's winter flannels--"well, I am a good wife to
him, if I ain't a 'creature of light.'"
Yes, she was a good wife.... How carefully she put his flannels away
in May; how prudently she planned his food; how she managed to make the
two ends of his little income meet--yes, and lap over, so that every
summer he could go away from her for a two months' vacation in the
woods! Not once did he find a button lacking; not once had he put on a
clean pair of stockings and then pulled them off because of a hole in
the heel. Can our lords say as much, my mistresses? I trow not! Yes,
a good wife: that lovely being who left the world with a faint,
unfinished regret upon her pitiful lips could never have made him so
comfortable.
Indeed, the whole household revolved upon Robert's comfort. Every
domestic arrangement had reference to his well-being. That he did not
become intolerably selfish was not Rebecca's fault, for, like many good
wives, she was absolutely without conscience in the matter of
self-sacrifice; but Robert escaped spiritual corruption, thanks to his
own very gentle nature and his absolute unconsciousness of the
situation. Perhaps, too, Rebecca's tongue mitigated the spoiling
process. She never spared him what she considered to be the truth
about himself or Alice. But her truthfulness stopped here; she spared
the dead, perforce. For what could she say ill of that beautiful
creature whose only wrong-doing lay in dying? But she knew, with
shame, that she would have liked to speak ill of her--in which
reprehensible impulse to remove a fellow-being from a pedestal, Rebecca
showed herself singularly like the rest of us.
In this bleak air of unselfishness and truth-telling, Robert Gray
became more and more aloof. Gradually he retreated quite into his
past, doing his daily work at the Academy--where successive classes of
young ladies adored him for his gentle manners and his mild, brown
eyes--and living very harmlessly with his memories, which he kept fresh
and fragrant by sharing them with Alys's daughter, who, it must be
admitted, being young and human, was not always intensely interested;
but Rebecca had trained her too well for Alice ever to show any
weariness. Robert kept his little collection of pictures and
photographs of his first wife shut behind the curtained doors of an old
secretary. If his second wife found him standing, his hands clasped
behind him, his eyes wandering from one lovely presentment to another,
he never displayed an embarrassed consciousness, but he shut the doors.
He accepted Rebecca's devotion respectfully; he was never impolite,
still less unkind; in fact, in all their married life he had never, she
used to tell herself, spoken unkindly save once; and then his words
were nothing more dreadful than, "We will not discuss it, if you
please, Mrs. Gray." At first he had, very gently, made some
grammatical suggestions; and she had profited by them, though, being a
true Pennsylvanian, she never mastered "shall" and "will," nor did she
lose the Pennsylvania love for the word 'just'; to the end of her days,
Rebecca was 'just tired out'; or 'just real glad'; or 'just as busy as
could be.' Grammar, however, was as far as Robert Gray went in any
personal relation. He addressed her, in his courteous voice (always a
little timidly), as "Mrs. Gray"; and he kept as much as possible out of
her way. Meantime, Rebecca (remembering why he had married her) did
her duty by the child, and never failed to mention, in her hard voice,
that Alice must try to grow up like her mother.
"Make me a good girl," Alice used to say in her sleepy prayers every
night--"make me a good girl, like my dear mother." Once, of her own
accord, the child added, "And make me pretty like her, too." Rebecca,
listening to the little figure at her knee, said, sternly, when Alice
got up and began to climb into the big four-poster:
"Don't be vain. Don't ask God for foolish things. Beauty is foolish
and favor is deceitful. Just ask Him to make you as good as your
mother was."
And, indeed, it must be admitted that the child did not inherit her
mother's wonderful beauty. At first her father had expected it; he
used to take liberties with his Horace, and say:
"O filia pulchra matre pulchriore."
But as Alice grew older, Robert Gray had to admit that the dead woman
had taken her beauty away with her. The child had just a pleasant
face; eyes that were gray or blue, as it happened; a commonplace nose,
and uncompromisingly red hair. In those days red hair was thought to
be a mortifying affliction, and poor, plain Alice shed many tears over
the rough, handsome shock of hair that broke into curls about her
forehead and all around the nape of her pretty, white neck.
II
But in spite of red hair, and what Old Chester religiously believed to
be its accompanying temper, Alice Gray was a lovable girl, and at
twenty, behold, she had a lover; indeed, she had more than one (not
counting Dr. Lavendar); but Alice never gave a thought to anybody but
Luther Metcalf. Luther was a good boy, Old Chester said; but added
that he would never set the river on fire.
Certainly he did not use his incendiary opportunity; he had a small
printing-office, and he owned and edited Old Chester's weekly
newspaper, the _Globe_; but neither the news nor the editorial page
ever startled or displeased the oldest or the youngest inhabitant. The
_Globe_ confined itself to carefully accredited cuttings from
exchanges; it had a Poet's Corner, and it gave, politely, any Old
Chester news that could be found; besides this, it devoted the inner
sheet to discreet advertisements, widely spaced to take up room. All
Old Chester subscribed for it, and spoke of it respectfully, because it
was a newspaper; and snubbed its editor, because he was one of its own
boys--and without snubbing boys are so apt to put on airs! Poor Luther
was never tempted to put on airs; he was too hard-worked and too
anxious about his prospects. He and Alice were to get married when he
and the _Globe_ were out of debt; for his father had left him a
mortgage on the office building, as well as an unpaid-for press. When
Luther was particularly low-spirited, he used to tell Alice it would
take him five years to pay his debts; and, to tell the truth, that was
an optimistic estimate, for the _Globe_ and the printing-office
together did very little more than pay the interest on the notes and
Luther's board.
So, when they became engaged, waiting was what they looked forward to,
for, of course, Robert Gray could not help them; it was all Rebecca
could do to stretch his salary to cover the expenses of their own
household. But the two young people were happy enough, except when
Luther talked about five years of waiting.
"We've been engaged two years already," he said, moodily; "I don't want
to be another case of Andrew Steele."
"I'm not afraid," Alice said. "Why, if you get the new job press, and
get that Mercer work, think how much that will help!"
"Well," Luther said, "yes; but if I get the press, there's another
debt. And if I don't get it, I can't get the work; so there it is. A
vicious circle."
This question of the purchase of a new press, before the old press had
been paid for, was a very serious and anxious one. "I wish father
could help," Alice said--they were walking home from Wednesday-evening
lecture, loitering in the moonlight, and wishing the way were twice as
long.
"Oh, I wouldn't think of such a thing," the young man declared; "we'll
pull out somehow. He's gone off to the woods, hasn't he?"
"Yes, he went this morning; he's so pleased to get away! He won't be
back till the Academy opens."
"I suppose he hates to leave you, though," Lute said.
"Yes, but I can see that the getting away is a great relief. I keep
his pictures dusted, and take the flowers up to the cemetery for him;
so he knows things are not neglected."
"But," Luther said, thoughtfully, "I think she's sorry to have him go?"
"Oh yes; sorry, I suppose," Alice admitted. "She's fond of him--in her
way."
"Then why--" Luther began.
"My dear, she's _jealous_ of my mother."
"Oh, Alice!"
"Well, you know," Alice explained, "my mother was so beautiful--and
poor Mrs. Gray! But I must say, Lute, she's the justest person I know.
She's always told me that my mother was perfect. And of course she
was; but when you're jealous, it isn't so easy to acknowledge things
like that."
"But I don't see how you can be jealous of the dead," Luther ruminated.
"Oh, _I_ do! I could be jealous of some girl who was dead, if you'd
loved her, Lute." And then the boy put his arm round her, and they
kissed each other there in the shadows of the locust-trees overhanging
a garden wall. "I'm so glad there isn't anybody, dead or alive," Alice
said, happily; "though I'd rather have her alive than dead. If she
were alive, you'd have quarrelled with her, and stopped loving her.
But if she were dead, she would keep on being perfect. Yes; I'd rather
marry a man who had been--been _divorced_," said Alice, lowering her
voice, because the word was hardly considered proper in Old Chester,
"than a man whose wife was dead, because he would always be thinking
what an angel she was and what a sinner I was."
"He would think you were an angel," the boy told her, blushing at his
own fervency.
But the fervency died on his ardent young lips when they got into the
house and sat decorously in the parlor with Mrs. Gray. Rebecca was
sewing, her hard, square face a little harder than usual. Mr. Gray had
gone away on that annual fishing-trip--gone, with a look of relief
growing in his eyes even as he stepped into the stage and pulled the
door to behind him; pulled it hurriedly, as though he feared she would
follow. Then, baring his head politely, he had looked out of the
window and said:
"Good-bye. You will send for me should you, by any chance, need me. I
trust you will be very well."
"I don't know that I have ever had to interrupt your fishing-trip with
any of my needs," Rebecca had answered, briefly. She spoke only the
truth; she never had interfered with any pleasure of his; and yet
Robert Gray had winced, as if he had not liked her words. Now, alone,
in the parlor, darning his stockings, she wondered why. She never said
anything but the simple truth; but he looked at her sometimes as a dog
looks who expects a blow. He was truthful himself, but he never seemed
to care much to hear the truth, she thought, heavily. Once he told her
that truth was something more than a statement of fact. The statement
of a fact may be a lie, he had said, smiling whimsically; and Rebecca
used to wonder how a fact could be a lie? She recalled the time when,
with brief accuracy, she had mentioned to him in what condition of
ragged neglect she had found his wardrobe after the "creature of light"
had left him; and how he had seemed to shrink not from the shiftless
dead, but from her. And she remembered painfully that one unkindness:
She had told him that, to her mind, not even the weakness of death was
quite an excuse for saying you didn't like your own baby; and he had
said, with a terrible look, "We will not discuss it, if you please,
Mrs. Gray." She had never spoken of it again; but his look had burned
into her poor, narrow, sore mind; she thought of it now, moodily, as
she sat alone, her heart following him on his journey. If his first
wife had only not been so perfect, she said to herself, she could have
borne it better; if she had had a bad temper, even, it would have been
something. But she had often heard Robert tell Alice that her mother
had an "angelic temper." Rebecca wished humbly she herself could be
pleasanter. "I don't feel unpleasant inside; but I seem to talk so,"
she thought, helplessly. She was thinking of this when the two young
people came in; and looking up over her spectacles, she said, coldly:
"Did you remember to wipe your feet, Luther? You are careless about
that. Alice, I found a flower on my daphne; you can carry the pot up
to the cemetery when you go."
"Yes, ma'am," Alice said. She took up her sewing (for Rebecca would
not have idle hands about); sometimes she glanced at Luther, sitting
primly in the corner of the sofa, and once caught his eye and smiled;
but there were no sheep's-eyes or sweet speeches. They were Old
Chester young people, and such things would have been considered
improper; just as sitting by themselves would have been thought not
only indecorous, but selfish.
"Oh, Alice," Luther said, suddenly, "I meant to ask you; wasn't your
mother's name spelled 'Alys'?"
"Yes. Why?"
"Well, it's such an unusual name that it struck my attention when I saw
it in the paper."
"What about it?" Alice asked. "Oh, dear, why didn't father spell me
'Alys' instead of 'Alice'? It's so much prettier!"
"Prettiness isn't everything; and 'Alice' is a sensible name," Rebecca
said. "Don't criticise your father."
"It was an advertisement in one of the _Globe's_ exchanges," Luther
explained. "I was scissoring things, and the name caught my eye. It
was information wanted. Of course it's just a coincidence, but it's
queer, because--here it is," said the editor of the _Globe_, fumbling
in his pocket. "I cut it out and meant to show it to you, but I
forgot." Then he read, slowly, "_Information wanted of one Alys
Winton--_"
"Why, but Winton was my mother's name!" cried Alice.
"_--one Alys Winton, who married sometime in 1845; husband thought to
be an American, name unknown. She (or a child of hers, born in 1846)
is requested to communicate with Amos Hughes, Attorney at Law,_" etc.
Alice stared, open-mouthed. "Why, Lute!" she said--"why, but that must
be my mother!"
Lute shook his head. "I don't think there's anything in it. Do you,
Mrs. Gray?"
"Might be," she said, briefly.
Alice took the crumpled cutting, and holding it under the lamp, read it
through to herself. "But, Lute, really and truly," she said, "it is
queer. Perhaps some of my mother's rich relations have left her a
fortune! Then we could pay off the mortgage. Only I'm afraid my
mother hadn't any rich relations--or poor ones, either. I never heard
of any. Did you, Mrs. Gray?"
"No," Rebecca said.
"She was a governess, you know, Lute, in some horrid English family;
the wife didn't like her, and she discharged my poor little mother;
then the family went off and left her all alone in Germany. Perfectly
abominable!"
"Don't be unjust, Alice; you don't know anything about it," Mrs. Gray
said. "She was very young. Perhaps she couldn't teach the children to
suit their parents. Though it was unkind to leave her unprovided for,"
she added, with painful fairness.
"I guess it was!" cried Alice. "Oh, how angry father gets when he
talks about it! He says she was in such terror, poor little thing,
when he met her. And yet she was very forgiving, father says. He says
she wrote and told the gentleman that she was married. _I_ wouldn't
have. I'd have let him think I'd starved, so he would have suffered
remorse--the wretch!"
"I hope you would not have been so foolish or so selfish," her
step-mother said.
"You see, she had no relations to turn to," Alice explained to Luther;
"if father hadn't come, dear knows what would have become of her."
"I suppose she could have earned an honest living, like anybody else,"
Mrs. Gray said.
"Well, anyway," Alice said, thoughtfully, "this advertisement is queer.
She had no relations that father ever heard of; but there might be some
one. What do you think, Mrs. Gray?"
"There might be," Rebecca said. She thought to herself that it was
very probable; that first wife had brought Robert Gray beauty and love;
it only needed that she should bring him money to make it all perfect.
In her bleak mind a window of imagination suddenly opened, and she had
a vision of what wealth would mean to her husband, coming as a gift
from those dead hands. She set her lips, and said: "Better find out
about it, Luther. Write to the man and say that a person of that name
before her marriage, died here in Old Chester, leaving a child--and
don't keep your hands in your pockets; it's bad manners."
"Do you really think it is worth while, ma'am?" Luther said,
incredulously.
"Of course it is," said Alice. "Suppose it should be some inheritance?
Such things do happen."
"In story-books," Lute said.
"Well, then I'd like to be in a story-book," Alice said, sighing.
"Just think, Lute, we might pay for the press and pay off the mortgage!"
"Golly!" said Lute.
Then they fell to making all sorts of plans, gayly, each tripping the
other up with the prosaic reminder of improbability.
"Or, if it _should_ be anything," Luther said, "it won't be more than
$100."
"Well, that's something; it will meet two monthly payments on the
press."
"It will pay for a diamond-ring for you," Lute said.
"Nonsense! We'll buy father a horse."
"And who will buy the oats?" Rebecca said.
"I could give you a big oleander, Mrs. Gray," Alice told her, smiling.
"You could put the money in the bank, like a sensible girl," Rebecca
said, severely. "Don't speak of this outside, either of you. Mr. Gray
wouldn't wish his wife's name talked about."
"And don't let's write anything about it to him," Alice said; "let's
have it a surprise!--if there is anything in it; only, of course, there
isn't anything," she ended, sighing. "But you might write to the man,
Lute."
"Of course there isn't anything," Lute agreed, sensibly. "I'll write
if you want me to; but I wouldn't build on it, Ally," he said, as he
got up to go. And when he paused a minute in the darkness on the
porch, he added, softly, "If you get rich, maybe you won't want a poor
printer?"
And she laughed, and said, "Maybe I won't!"
Then he kissed her just under her left ear, and said, "Money isn't
everything, Ally."
III
Money isn't everything, but it has so much to do with most things that
even a dim, story-book vision of it stirred Alice's imagination.
Luther, having no imagination, dismissed the vision from his mind after
writing a letter to "Amos Hughes, Attorney at Law." Indeed, Luther had
more practical things to think of than possible legacies, poor fellow.
His balance-sheet for that month of June was very dark. More than
once, after the office was closed for the day, he sat at his desk in
his shirt-sleeves, hot and tired and grimy, poring over his ledger by
the light of a swinging lamp. Alice grew worried about his pallor and
the hollows in his cheeks; but there was nothing she could do, though
she chafed against her helplessness to help, and revolved all sorts of
schemes in her impractical girl-mind. Indeed, she went so far as to
pour out her heart to Dr. Lavendar, in the hope that he could make some
suggestion. She found the old man sitting in the wistaria arbor near
his beehives, smoking peacefully, and throwing sticks to Danny, who
needed exercise and scrambled after them into the tall grass, bringing
them back with fatiguing alacrity.
"Look here, sir," said Dr. Lavendar, "don't find 'em so quick. I'm
worn out pitching them."
Then Alice Gray came down between the box borders and said she wanted
his advice; and Dr. Lavendar, glancing up at her, saw an uncertain lip
and heard a catch in her voice; whereupon he told her to give Danny a
run. "The scoundrel has kept me working for the last half-hour," he
complained.
When she came back, flushed and laughing, and sat down on the arbor
step, her voice was quite steady; so he listened placidly to her story.
"You want to get some work to help Lute, do you, good-for-nothing?"
"Yes," Alice said, eagerly. "Oh, Dr. Lavendar, _can_ you think of
anything? I wanted to go into the office and learn to set type, but
Mrs. Gray--"
"Well?"
"Mrs. Gray said I had better learn to keep house economically. She
said father wouldn't like it."
"Mrs. Gray would always think first of what your father would like."
Alice scratched lines in the gravel with one of Danny's sticks. "I
suppose she would," she admitted.
"And what did Lute say?"
"Oh, he wouldn't listen to it. But I thought maybe you could make him,
Dr. Lavendar?"
"I?" said Dr. Lavendar. "No, thank you. Do you think I'd rob the boy?"
"Rob him?"
"Of his self-respect; a boy wants to stand on his own legs; he doesn't
want a girl propping him up. You let Lute alone. He'll manage. And
you're young yet, anyhow. It won't hurt ye to wait. Mrs. Gray is
right. You learn to be as good a housekeeper as she is; and though you
mayn't put money into Lute's pocket before you're married, you'll not
be taking it out after you're married."
Alice sighed. "Oh, I wish I could help Lute; I wish I had a lot of
money."
"A lot of sense is better," Dr. Lavendar said, chuckling. "Oh, you
women! You steal a man's unselfishness and self-respect, and you put
it down to love. Love? You're a pack of thieves, the lot of you. You
ought to be prosecuted. I'd do it, if I had time. Hey, Danny! bite
her; she's like all the rest of 'em."
Alice hugged him, and defended herself. "You're just an old bachelor;
you don't appreciate us."
"Appreciate ye? I appreciate you. Maybe that's why I'm an old
bachelor."
But though he discouraged Alice's projects for assisting Luther, Dr.
Lavendar went plodding up the printing-office stairs the next morning.
Luther, emerging from behind a press, brightened at the sight of his
caller, and ushered him into a small closet which he called his private
office; and when Dr. Lavendar asked him to print some more
missionary-meeting notices, he said he would put them in at cost price.
"Don't you do it!" said Dr. Lavendar, thumping the floor with his
umbrella. "Look here; I'll have to teach you the first principles of
business: make your profit--and don't go to 'pauperizing the Church,'
sir. There's too much of that sort of thing," he added, with
reminiscent crossness. "Some scalawag of a bookseller wrote and
offered to sell me books at thirty-three per cent. discount because I
was a parson. There's no more reason why a parson should get a
discount than a policeman. I told him so. I tell you so. Print those
slips, and _print 'em better than you did the last lot_! Do you hear
that? You forgot a comma on the second line. How's business, Lute?"
Lute's face fell. Then they talked things over, to the boy's great
comfort; and at the end of the talk Lute straightened his shoulders and
drew a good breath.
"By George! sir, if hanging on does it, I'll hang on--" he stopped,
and looked round, in answer to a knock. "Well?" he said, impatiently.
But the gentleman who stood in the doorway was not rebuffed.
"Are you Mr. Metcalf, the editor of the _Globe_?"
"Yes, sir," said Luther.
"I called in relation to an advertisement"--Luther was instantly alert,
and Dr. Lavendar, scenting a customer, was about to withdraw--"an
advertisement in a New York paper, requesting information of a certain
person--"
"What!" cried Luther. "I had forgotten all about it."
"My name is Carter. I am from the office of Mr. Amos Hughes. Messrs.
Pritchett, Carver, and Pritchett, Solicitors at Law, of London, are our
principals. The advertisement was in relation to a person called Alys
Winton."
Luther, stumbling in his astonishment over his words, began to explain.
"Mrs. Gray is dead," he ended. "And Alice is her daughter; isn't she,
Dr. Lavendar? She asked me to write to you."
"Well, well; this is very interesting," said Dr. Lavendar. "I hope
your object in seeking to obtain information is to benefit this young
lady? She's one of my children."
Mr. Carter, still standing in the doorway, smiled, and said, "Do I
understand that this Miss Alice is the daughter of the person named
Alys Winton?"
"Yes," said Dr. Lavendar. "You can easily satisfy yourself on that
point by consulting my parish records."
"And her mother is the lady you advertised for!" cried Luther. The boy
was red with excitement. It was just as Alice said--a story-book. And
they could get married right away! For it would be a lot of
money--perhaps $5000; people in England didn't advertise for
information of a person dead for twenty-two years for any small amount;
well, even if it were $4000, they could get married; even if it were
$3000. "How m--" he began, and stopped; of course that was not a
proper question. "Alice's mother is the lady you advertised about," he
said, lamely.
"Well, that does not follow, young gentleman; but the coincidence of
the name was of sufficient interest for our firm to feel that I might,
perhaps, just look into it. There may be dozens of Alys Wintons, you
know."
"Oh," said Luther, so blankly that Dr. Lavendar laughed.
"Perhaps before beginning at the beginning you might save time by
looking at the end," he said to the lawyer. "If you will step over to
my church, you will see that our little Alice here is the daughter of
Mr. Robert Gray and a lady named Alys Winton."
"A very good idea, sir. You, I infer, are a clergyman in this place?
Ah, yes; just so. Lavendar? Ah, yes. I shall be pleased to look at
the records, as you suggest, sir."
Luther, rather abashed, longing to accompany them, stood waiting for an
invitation. But none came. Dr. Lavendar went pounding down the
stairs, followed by Mr. Carter, and Lute heard them talking about the
roughness of the road from Mercer over which Mr. Carter had come on the
morning stage.
"Confound the road!" said Lute to himself. "Hi! Davidson! I'm going
out. The first page is all made up; you can close up the fourth."
Then he dashed down the creaking stairs and out into the hot sunshine.
He had a glimpse up the street of the church, and Dr. Lavendar bending
down fumbling with the key of the vestry door; it was evident that
Luther's presence was not considered necessary. "I don't care," the
boy said to himself, joyously, and started at a swinging pace out over
the hill. "I'll be the one to tell her, anyhow!" His face was all
aglow. As he hurried along he made calculations as to the rent of the
little house. To be sure, he was reckoning on Alice's money; but the
boy was so honest, and so in love, that he had no mean
self-consciousness of that kind. "_We can get married!_" He had no
room for any other thought.
Mrs. Gray was sitting on the back porch shelling pease; there was a
grape trellis running out from the porch roof, and under it the shadows
lay cool and pleasant on the damp flagstones. Rebecca, absorbed in the
lulling snap of pods, looked up, frowning, at the noisy interruption,
for the young man burst in, breathless, swinging his cap, his eyes
shining.
"Oh, Mrs. Gray, where's Alice? Oh, my, such news! I never was so
excited in my life!"
"That is not saying much," Rebecca told him; "you've not had a very
exciting life. Alice is in the dining-room. Alice! come out here.
Here's Luther. He says he never was so excited in his life; and I hope
he won't be again, for he has upset my bucket of pods."
Luther, full of apologies, began to pick them up. "I'm so sorry, but I
was so dreadfully excited--"
"Dreadful is a large word," Rebecca said. "I doubt whether either you
or I have ever seen anything 'dreadful' in our lives. Don't
exaggerate, Luther."
"Yes, ma'am," Lute said. "Oh, there's Alice! _Alice!_" He stood up,
his hands full of pods, his face red. "Oh, Alice, what do you suppose
has happened? You'll never guess!"
"The advertisement man!" cried Alice. Luther's face fell a little, and
he laughed.
"Well, you're pretty smart. Yes, it is--"
"_What?_" said Rebecca Gray. As for Alice, she whirled out on the cool
flags and jumped up and down.
"Oh, Lute, tell us--tell us! What does he say? Has he sent some
money? Oh, how much is it? Oh, Lute, we'll pay for the press. Lute,
is it--is it $1000? Tell us; hurry, hurry!"
Upon which Lute began to subside. "Well, it isn't quite--I mean, he
didn't--he hasn't said just exactly how much. I mean, of course, I
suppose, it isn't certain; but I'm sure there isn't a particle of
doubt; only--"
"Now, Lute, begin at the beginning and tell us." Alice sat down
breathlessly beside her step-mother, and began mechanically to shell
the pease.
"Don't," Rebecca said; "I will do my own work. You'd better get your
table-cloth and finish that darning." Her face had grown quite pale;
she saw the fabric of her life crumbling at the base; if, through that
first wife, money should come into the family, what use for her patient
economies? What use for her existence? That first wife, yet more
perfect, would crowd her further from her husband's life. In her
heart, used to the long, dull ache of unloved years, rose up a
murderous hatred of the dead woman. At first she hardly heard Luther's
story, but as it went on she began to listen and the pain in her
tightened throat of unshed tears lessened. It might not be. As this
Mr. Carter said, there might be dozens of Alys Wintons. Her hands,
motionless after the first shock, went at their work again.
"You're the daughter of a lady of that name," she said, coldly; "but
she may not be the lady they want. Better not count on it." Alice
looked rather blank for a moment; and then she burst into even more
than Luther's confidence.
"Do you suppose it will be $2000? Oh, Lute, just think, we'll pay for
the new press right down!"
"No, we won't, either," Lute said, stoutly. "I'm not going to let you
spend your money on printing-presses."
"Nonsense!" Alice cried, laughing and stamping her foot.
Rebecca frowned and looked at her over her glasses. "Don't be
unlady-like, Alice."
"No, 'm," Alice said; and then she laughed at her own excitement; "it
may be only $100."
"It may be nothing at all," Rebecca Gray said, and got up and took her
pan and bucket and went into the house. It seemed to her that if she
had to hear any more of Alys Winton she would speak out and say some
dreadful thing about her. But what could she say with any kind of
truth? What could she say ill of that poor creature, so beloved and so
harmless? For, after all, though a woman ought to see that a man's
buttons are sewed on, you can't say that mere shiftlessness is a sin.
Besides, she was sick for those few months. "Perhaps if my health
hadn't been good, I would have been careless myself," Rebecca thought,
with painful justice. But she went up-stairs to her own room and
locked the door. She felt sure that it was as Alice and Luther said:
there would be money, and she would be of still less consequence to her
husband; for what did Robert Gray, nervously polite, really care for
her economies and her good housekeeping?
"Not _that_!" she said to herself, bitterly.
IV
"You will stay and have dinner with me," Dr. Lavendar had told the
lawyer, hospitably, "and then Goliath and I will take you up the hill
to Mr. Gray's house."
And so, in the early afternoon, Goliath brought Mr. Carter to the
Grays' door. Alice, who was on the porch, insisted that Dr. Lavendar
should come in, too; she leaned into the buggy to whisper, joyously,
"If it is anything nice, I want you to hear it."
But for once Dr. Lavendar did not laugh and give her a kiss and call
her his good-for-nothing; he got out silently, and followed Mr. Carter
into the parlor, where Luther and Mrs. Gray were awaiting them. There
was a tense feeling of expectation in the air. The two young people
were together on the sofa, smiling and laughing, with small, whispered
jokes of presses and diamond-rings and mortgages. Rebecca sat by the
table, her worn hands in a trembling grip in her lap; she sat very
upright, and was briefer and curter than ever, and she looked most of
the time at the floor.
"You have been informed of my errand, madam?" said Mr. Carter. "It is
unfortunate that Mr. Gray is not at home, but perhaps you may be able
to give us some information on certain points, which will at least
instruct me as to whether the facts in the case warrant further
reference to him for confirmation. I will ask a few questions, if you
please?"
"Go on," Rebecca said.
"The late Mrs. Gray, the mother of this young lady," said Mr.
Carter--"do you happen to know her nationality?"
"English."
"Ah, yes. Just so. And do you know the date of her marriage to Mr.
Gray?"
Rebecca gave it.
"If any facts in regard to her occur to you--" the lawyer began.
"I've heard Mr. Gray say that she was a governess in the family of a
Mr. Urquhart," Rebecca said; and added, "They discharged her in Berlin."
Mr. Carter, glancing at a memorandum, his face keen with interest,
said, eagerly, "Pray proceed, madam."
"I don't know much more; Mr. Gray met her in Interlaken. They were
married three weeks afterwards."
"Ah, Switzerland? That explains; there was no record of a marriage at
the Embassy. Can you tell me anything of the parentage of the lady?"
"Her father's name was George Winton," Alice broke in, "and they lived
in a place called Medfield. He was a clergyman. Her mother's name was
Alys, too. Father has a prayer-book belonging to my grandmother; it
has her name in it, and my mother's. Would you like to see it, sir?",
"Exceedingly," Mr. Carter said; and while Alice ran to get the book, he
studied his memorandum so closely that no one dared to ask him a
question, if, indeed, any one wanted to. Rebecca had answered him
dully, looking out of the window part of the time, part of the time at
the floor. Dr. Lavendar, on the other side of the room, his hands on
the head of his cane, sat silently staring down at the carpet, his face
heavy and rather stern. Lute, radiant, twirled his cap in his hands,
and resolutely held his tongue.
Alice, as she handed the prayer-book to Mr. Carter, stopped on her way
back to Luther and squeezed Dr. Lavendar's hand. "Isn't it wonderful?"
she whispered; and he shook his head a little impatiently.
"Go and sit down, my dear," he said.
Mr. Carter, glancing at the name on the flyleaf, looked at his notes
again and then at Alice, "And this young lady--can she give me the date
of her birth?"
There was a little laugh, and Luther and Alice gave it together,
eagerly.
There were two or three more questions, and then Mr. Carter folded his
memorandum and slipped it within its rubber band with a snap; then he
smiled. Rebecca looked at him drearily. "Of course," he said,
addressing himself to her, "a question of identity cannot be decided
offhand; it is necessary to have certain affidavits which the surviving
husband of the deceased (who is asserted to be the person in question)
would be obliged, legally, to furnish. I think, however, that I am not
going beyond the line of discretion and propriety if I say that _if_
Mr. Robert Gray can produce such proofs (which I think I am not
unwarranted in saying I believe he can)--_if_ he can, then this young
lady is the heir to a very considerable fortune. I think, in point of
fact, I have the right to say that, if (as I have said before) these
proofs are forthcoming, the amount to be paid to the daughter of Alys
Winton is L5000."
Rebecca Gray put her hand to her mouth and stared blindly at the floor.
Dr. Lavendar thrust out his lower lip and frowned. As for Alice, she
laughed aloud, then burst out crying.
"Oh, _Lute_!" she said, tremulously; and, somehow, the two children
found themselves holding hands. "It's--it's so much!" she faltered.
"Five thousand pounds is--is $25,000!" the boy said, turning pale.
There was a pause; no one seemed to know just what to say. Then Lute,
suddenly: "Is it your mother's father that left it to you, Alice?"
She turned to Mr. Carter, drawing in her breath like a child. "Is it?"
"Ah--no," he answered, briefly.
"But I didn't know my mother had any relations?" Alice said, in a dazed
way; "I thought father said--I'm sure he said--she hadn't any
relations? Perhaps--perhaps it is a mistake, after all?"
"The testator was not a relative of the Alys Winton in question," Mr.
Carter said. He glanced uneasily at Dr. Lavendar, who lifted his head
and looked at him searchingly. "It will be best to make further
explanations to Mr. Gray," Mr. Carter said, hurriedly.
"But who has left the money to me--if it is to me?" Alice said,
bewildered. "Can't I ask that? What is the name of the kind person?
I think I might ask that."
[Illustration: "'WHAT IS THE NAME OF THE KIND PERSON?'"]
"The name of the testator was Urquhart," Mr. Carter said, "but--but,
you know, my dear young lady, the identity is not yet legally
authenticated; so--therefore--perhaps--I think, Dr. Lavendar, I had
best go now? I think you mentioned that the stage leaves at four?"
"Urquhart?" Alice said; "the man who was so unkind? Oh, Lute, I
suppose he repented. Oh, how astonished father will be! He'll have to
forgive him now."
"It's a pretty late repentance," Luther said, with a chuckle; "and how
did he know about you, Alice? I don't see why he should leave you
money, even if he was a brute to your mother. Still," said the boy,
gayly, "I guess we won't complain?"
"Gracious!" cried Alice, "that is queer. Well, he _was_ a kind person!"
Rebecca Gray stared, frowning, at the lawyer. "He knew--this
Urquhart--that she had a child?" she said, slowly.
Mr. Carter was gathering up his papers. "Yes," he said--"yes; he--knew
it."
"What?" said Rebecca, in a very low voice--"_what?_"
"In view of the fact that, legally, the matter is still undecided," Mr.
Carter said, hurriedly, "perhaps we need not take this point up? At
all events, not here."
"Sir," said Rebecca, "why does Mr. Urquhart leave L5000 to Robert
Gray's daughter?"
"He was sorry he was unkind to my mother," Alice said, her voice
quivering. ("Oh, Lute, $25,000!")
"Alice," her step-mother said, in a loud, harsh voice, "you had better
leave the room. Luther, go with Alice, please."
The two young people, bewildered, got up with blank faces, and with
obvious reluctance obeyed. "But why should I be sent out, Lute?" Alice
said, hotly, when they were in the hall. "It's my money--if I'm the
person."
Luther stopped, and stood, frowning. On the boy's open, honest face
came a perplexed look. But Alice said again, in injured tones, that
she didn't know what Mrs. Gray meant. In the parlor the three elders
looked at each other in silence. Mrs. Gray had risen, and stood
leaning forward, her trembling hands flat on the table.
"I don't--understand," she said.
"Mr. Carter," said Dr. Lavendar, "certain remarks of yours on our way
up here made me apprehensive. I see that my friend, Mrs. Gray, is
also--apprehensive. I would suggest that you have a few words with her
alone. I will leave you."
"No," Rebecca said; "hear the end of it." Her hard face was red and
hot. "Why does Mr. Urquhart leave the child of Robert Gray L5000?
Why?"
"It is as I think you surmise, madam," John Carter said, gravely.
Rebecca recoiled, with a broken exclamation of horror.
Dr. Lavendar drew in his breath. "Oh, my poor Robert!" he said.
"It is so stated in the will," the lawyer went on; "there is no
disguising it; nor, as far as I can see, can it be hidden from the
legatee. The directions for finding this heir make the thing explicit.
The testator states that he received information of the expected birth
of his child _after_ the marriage of the person in question, who did
not mention her married name--hence our difficulty in tracing her."
Rebecca, her eyes narrowing into a cruel smile, sat down and rocked
backward and forward in her chair.
"Dreadful--dreadful--dreadful!" she said, aloud, exultantly.
V.
The last quarter of an hour, packed with tragic revelation, lost Mr.
Carter the stage.
"I hope you will put up at the Rectory, sir," Dr. Lavendar said, as
they drove away from Robert Gray's door.
"I thank you, sir," said Mr. Carter.
Then they fell into silence--Mr. Carter from politeness, Dr. Lavendar
from horror. He was going back in his memory with painful effort; but
it was all very vague.... He had hardly known her; she had been ill
for those months that she had been in Old Chester, and she had made it
very clear that she did not care to see people. He thought of her
beautiful, sullen face; of Robert Gray's passionate devotion; of Old
Chester's silent disapproval.... He groaned to himself, and John
Carter looked at him sidewise.
After supper at the Rectory, they sat down to smoke in heavy silence;
Mr. Carter respected the old man's distress, but wondered if he should
not have been more comfortable with Van Horn at the Tavern. The
glowing July day had darkened into rainy night, with a grumble of
thunder back among the hills; but in the midst of a sudden downpour
they heard footsteps on the path, and then some one pushed open the
hall door, and flapped a wet umbrella on the steps before entering. A
minute later Luther Metcalf stood, hesitating, on the study threshold.
"Dr. Lavendar--"
The old man got up hurriedly. "Yes, Lute. Come into the dining-room.
You will excuse me, sir?" he said to Mr. Carter. He put his hand on
Lute's arm, in a friendly grip, for there was a break in the boy's
voice.
"I know about it," Lute said. They sat down at the dining-room table;
Lute swallowed hard, and pulled with trembling fingers at his hatband;
he did not lift his eyes. "And--and I want you to tell her not to take
it."
"How is she, Lute?"
"I haven't seen her. She wouldn't come down-stairs. She sent me a
little note," Luther said, taking it out of his breast-pocket, and then
putting it back again tenderly. "'Course I won't pay any attention to
it."
"Saying she'd release you, I suppose?"
"Yes; but that's nothing. I'll make her understand the minute I see
her. But, Dr. Lavendar, I don't want that--that money!" the boy ended,
almost with a sob. "I want you to tell her not to take it."
Dr. Lavendar was silent.
"At first I thought--I couldn't help thinking--we could get married
right off. We could get married and have a home of our own; you know,
we'd be rich people with all that money. And I suppose, honestly, that
as things are now, there's no chance of our getting married for a good
while. But I--I tell you what, sir. I'd rather never get married
than--than touch that money!"
Dr. Lavendar nodded.
"You won't let her, sir? You'll make her give it back?"
"My dear boy, I can't 'make' Alice do anything. The money is hers."
"Oh, but Dr. Lavendar, won't you go and talk to her? It may be a
temptation to her, just as it was to me, for a minute. We could just
make the office hum, sir. We could put it right on its feet; we could
have a real Daily. I know she'll think of that. _I_ just thought we
could get married. But Alice will think about helping the office, and
me."
"Of course the money would bring ease to her father--" Dr. Lavendar
stopped abruptly.
"Oh, my _God_!" Lute said, and dropped his head on his arms.
"Bring ease to--to the family," Dr. Lavendar ended lamely.
"You know Mr. Gray won't touch it," Lute burst out; "and I can't let
Alice, either. Dr. Lavendar, I thought maybe you'd let me hitch
Goliath up and drive you out to the house?"
"Not to-night, Lute. Alice has got to be alone. Poor child, poor
child! Yes; we've all of us got to meet the devil alone. Temptation
is a lonely business, Lute. To-morrow I'll go, of course. Did you
answer her note?"
"Oh yes; right off. I just said, 'Don't be foolish,' and--and some
other things. I didn't tell her we mustn't take the money, because I
hadn't thought of it then. Mrs. Gray said she wouldn't come out of her
room. Oh, just think of her, all by herself!" Luther bent over and
fumbled with his shoelace; when he looked up, Dr. Lavendar pretended
not to see his eyes.
When the boy went away, Dr. Lavendar went back to the study and asked
John Carter some legal questions: Suppose he had not found this child,
what would have become of the money? Suppose the child should now
decline to take it, what then?
"Well," said Mr. Carter, smiling, "as a remote contingency, I suppose I
might reply that it would revert to the residuary estate. But did you
ever know anybody decline L5000, Dr. Lavendar?"
"Never knew anybody who had the chance," Dr. Lavendar said; "but
there's no telling what human critters will do."
"They won't do that," said John Carter.
What a long night it was, of rain and wind and dreadful thought! ...
Rebecca had told Alice, with kindness, but with such a grip upon
herself lest exultation should tremble in her voice, that she seemed
harsher than ever. Then she told Lute. He pleaded that Alice would
speak to him, and Mrs. Gray had gone to the girl's room and bidden her
come down-stairs.
"Come, Alice. You must control yourself. Come down and talk to
Luther."
Alice shook her head. "I'll--write him a note."
Mrs. Gray carried the note back to Lute, and brought up the answer,
which Alice read silently. Rebecca watched her; and then, with an
effort, she said:
"Alice, remember we are not to judge. We don't understand. We must
not judge. Good-night." She opened the door, and then looked at the
child, seated, speechless, with blank eyes, on the edge of the bed.
"Good-night, Alice. I--I'm sorry for you, poor girl!" and she came
back hastily and kissed her.
At that, even in her daze of horror, a glimmer of astonishment came
into Alice's face. But she did not look up or speak. When it grew
dark, she began mechanically to get ready for bed; she knelt down, as
usual, at the big chintz-covered winged chair and began to say her
prayers, her mind blind as to her own words: "Bless dear father--"
Then she cried out, suddenly and dreadfully, and covered her poor,
shamed head with her arms, and prayed no more. Then came a long fit of
crying, and then a dreary calm. Afterwards, as the night shut in with
rain and rumble of thunder, the shame lightened a little, for, though
she could not read it in the darkness, she held Lute's little note
against her lips and kissed it, and cried over it, and said his words
over to herself, and felt that at any rate there was one bright spot in
it all: Lute would never have any more anxieties. Of Robert Gray she
thought pitifully, but with not much understanding. Oh, dreadful,
dreadful! But he had loved his wife so much (so the child thought) he
would surely forgive her. Not knowing how little forgiveness counts
for when a star goes out. Sometimes, sitting there on the floor,
listening to the rain, she slept; then woke, with a numb wonder, which
darkened into cruel understanding. _Shame; shame_--but Lute wouldn't
be worried any more; Lute would be rich.
[Illustration: "SHE KNELT DOWN, AS USUAL, AT THE BIG CHINTZ-COVERED
WINGED CHAIR"]
So the night passed....
Rebecca Gray did not sleep. When the house was still she went
up-stairs, eager to be alone. She shut her bedroom door softly; then
she put her brass candlestick on the high bureau and looked about
her.... Everything seemed strange. Here was her old-fashioned bed
with its four mahogany posts like four slender obelisks; there was the
fine darn in the valance of the tester; the worn strip of carpet on
which she had knelt every night for all these twenty years; it was all
the same, but it was all different, all unfamiliar. The room was
suddenly the room where that woman had died; the old four-poster was
the bed of that heartbreaking night, with sheets rumpling under a
wandering hand and pillows piled beneath a beautiful, dying head; not
her own bed, smooth and decorous and neat, with her own fine darn in
the tester valance. She did not know the room as it was now; she did
not know herself; nor Robert; nor that--that--_that woman_. She sat
down, suddenly a little faint with the effort of readjusting a belief
of twenty-two years. "She was a wicked woman," she said, out loud; and
her astounded face stared back at her from the dim mirror over the
mantel-piece. After a while she got up and began to walk back and
forth; sometimes she drew a deep breath; once she laughed. "A wicked
woman!" ... Now he would know. Now he would see. And he would loathe
her. He would hate her. He would--her lip drooped suddenly from its
fierce, unconscious smile; he would--suffer. Yes; suffer, of course.
But that couldn't be helped. Just at first he would suffer. Then he
would hate her so much that he would not suffer. Not suffer? It came
over her with a pang that there is no suffering so dreadful as that
which comes with hating. However, she could not help that. Truth was
truth! All the years of her hungry wifehood rose up, eager for
revenge; her mind went hurriedly, with ecstasy, over the contrast; her
painful, patient, conscientious endeavor to do her best for him. Her
self-sacrifice, her actual deprivations--"I haven't had a new bonnet
for--for four years!" she thought; and her lip quivered at the
pitifulness of so slight a thing. But it was the whole tenor of her
life. _She_ had no vacations in the mountains; she would have liked
new valances, but she spent hours in darning her old ones to save his
money; she had turned her black silk twice; she had only had two black
silks in twenty years. All the great things she had done, all the
petty things she had suffered, rose up in a great wave of merit before
her; and against it--what? Hideous deceit! Oh, how he would despise
the creature! Then she winced; he would--suffer? Well, she couldn't
help that. It was the truth, and he had got to face it. She was
walking up and down, whispering to herself, a sobbing laugh on her
lips, when suddenly, as she passed the mirror, she had a dim, crazy
vision of herself that struck her motionless. A moment later she took
the candle, and with one hand clutching for support at the high
mantel-shelf--for her knees were shaking under her--held it close to
the glass and peered into the black depths. Her pale, quivering face,
ravaged with tears, stared back at her, like some poor ghost more ugly
even than in life. "_A wicked woman._" Yes--yes--yes; and he would
have to know it. But when he knew it, what then? If his eyes opened
to sin, would they open to--
"I have tried to make him comfortable," she said, faintly.
Suddenly she put the candle down and sank into a chair, covering her
face with her poor, gaunt hands....
And so the night passed.... The dawn was dim and rainy. It was about
four o'clock that Alice, sitting on the floor, sleeping heavily, her
head on the cushion of the chair, started, bewildered, at the noise of
the opening door. Rebecca, in her gray dressing-gown, one hand
shielding the flare of her candle, came abruptly into the room.
"Alice," she said, harshly, and stopped by the empty bed; then her eyes
found the figure on the floor ("you ought to be in bed"), she said, in
a brief aside; then: "Alice, I've been thinking it over. You can't
take that money."
"I don't understand," Alice said, confused with sleep and tears.
"You can't take that money. If you do, your father would have to know.
And he never must--he never must."
Alice pulled herself up from the floor and sat down in her big chair.
"Not take the money?" she said, in a dazed way; "but it's mine."
"That's why you needn't take it. Thank God it was left to you, not
just to 'her heirs.' Alice, I've gone all over it. I--I wanted you to
take it"--Rebecca's voice broke; "yes, I--did."
"Well, it's mine," Alice repeated, bewildered.
Rebecca struck her hands together. "Yours not to take! Don't you see?
You can save your father."
Alice, cringing, dropped her head on her breast with a broken word.
"Don't be a fool," the older woman said, trembling. "He's been your
father ever since you were born. And it would be a pretty return for
his love to tell him--"
Alice burst out crying; her step-mother softened.
"I am sorry for you, you poor girl. But, oh, Alice, think, _think_ of
your father!" She clasped her hands and stood, trembling; she took a
step forward, almost as if she would kneel.
"If he would feel so dreadfully," Alice said, at last, "why--we needn't
tell him where the money comes from."
"Now, Alice, that is absurd. Of course he would know. He would have
to know. A girl doesn't inherit L5000 without her father's knowing
where it comes from. And, anyway, Mr. Carter said that Mr. Gray would
have to make a statement and swear to it. Of course he would--know."
"Do you mean you don't want me to have it at all?" Alice said, blankly.
"I've just explained it to you," Rebecca said, her voice harsh with
anxiety. "You _can't_ have it."
"But it's my money; I have a right to it. And it would make all the
difference in the world to Lute. If he is going to take a girl--like
me, he ought to have the money, anyhow."
"And kill your father?" Rebecca said. "Alice! Don't you see, he must
go on believing that she is"--her voice grew suddenly tender--"that she
is 'a creature of light?'"
"I want Lute to have the money," Alice said.
"Alice!" the other exclaimed, with dismay, "don't you think of your
father at all? And--for your mother's sake."
Alice was silent; then, in a hard voice, "I don't like her."
"Oh!" Rebecca cried, and shivered. There was a pause; then she said,
faintly, "For your own sake?"
Alice looked up sullenly. "Nobody need know; we would only say it had
been left to--her. Nobody would know."
Suddenly, as she spoke, despite the plain face and the red hair, Alice
looked like her mother. Rebecca stepped back with a sort of shock.
Alice, crying a little, got up and began to pull down her hair and
braid it, with unsteady fingers. Her step-mother watched her silently;
then she turned to go away; then came back swiftly, the tears running
down her face.
"Oh, Alice, it is my fault! I've had you twenty-two years, and yet you
are like-- See, Alice, child; give her a chance to be kind to him, in
you. Oh, I--I don't know how to say it; I mean, let her have a chance!
Oh--don't you see what I mean? She said she was sorry!" All the
harshness had melted out of Rebecca's face; she was nothing but
gentleness, the tears falling down her cheeks, her voice broken with
love. "Alice, be good, dear. Be good. Be good. And I--I _will_ be
pleasanter, Alice; I'll try, indeed; I'll try--"
VI
"Well," said Mr. Amos Hughes, a week later, in the cool dusk of Dr.
Lavendar's study, just before tea, "this is a most extraordinary
situation, sir!"
"Will ye have a pipe?" said Dr. Lavendar, hospitably.
John Carter, his feet well apart, his back to the fireless grate, his
hands thrust down into his pockets, said, looking over at his partner:
"Amos, Dr. Lavendar once remarked to me that there was no telling what
human critters would do."
Dr. Lavendar chuckled.
"Very true," Amos Hughes admitted, putting one fat knee over the other;
"but I must say that I never before knew a human critter throw away
L5000."
"I'm sorry you haven't had better acquaintances," said Dr. Lavendar.
"I have. I'm not in the least surprised at this child's behavior. Mr.
Carter, are you looking for anything? You'll find a decanter on the
sideboard in the next room, sir. This is a pretty good world, Mr. Amos
Hughes; I've lived in it longer than you have, so you'll take my word
for it. It's a pretty good old world, and Miss Alice Gray has simply
decided to do the natural and proper thing. Why, what else could she
do?"
"I could mention at least one other thing," said Mr. Carter.
"Extraordinary situation! but I suppose the residuary legatees won't
make any objection," murmured Amos Hughes.
Dr. Lavendar rapped on the table with the bowl of his pipe. "My dear
sir, would you have a girl, for a paltry L5000, break her father's
heart?"
"Her father?"
"Mr. Gray would not, in my judgment, survive such a revelation," said
Dr. Lavendar, stiffly.
"May I ask one question?" John Carter said.
"G'on," said Dr. Lavendar.
"What I would like to know is: How did you bring Miss Gray to look at
the thing in this way?"
"I didn't bring her," said Dr. Lavendar, indignantly; "her Heavenly
Father brought her. Look here, sir; this business of the law is all
very well, and necessary, I suppose, in its way, but let me tell you,
it's a dangerous business. You see so much of the sin of human nature
that you get to thinking human nature has got to sin. You are
mistaken, sir; it has got to be decent. We are the children of God,
sir. I beg that you'll remember that--and then you won't be surprised
when a child like our Alice does the right thing. Surprise is
confession, Mr. Carter."
Mr. Carter laughed, and apologized as best he could for his view of
human nature; and Dr. Lavendar was instantly amicable and forgiving.
He took Mr. Amos Hughes's warning, that he should, as a matter of duty,
lay very clearly before the young lady the seriousness of what she
proposed to do, and not until he had exhausted every argument would he
permit her to sign the papers of release which (as a matter of
precaution) he had prepared. "She's of age," said Amos Hughes, "and
nobody can say that she has not a right to refuse to proceed further in
the matter. But I shall warn her."
"'Course, man," said Dr. Lavendar; "that's your trade."
And so the evening came, and the three men went up to Robert Gray's
house.
It was a long evening. More than once Dr. Lavendar trembled as he saw
the kingdoms of the world and the glory of them spread before his
child's eyes. But he said no word, and once, sternly, he laid his hand
on Rebecca's arm to check some word of hers.
"Let her alone," he said.
It was eleven o'clock before there came a moment of solemn silence.
Alice bent over a paper, which John Carter had read aloud to her, and
signed her name. Luther and Rebecca and Dr. Lavendar witnessed the
signature. Then Rebecca Gray took the girl in her arms.
"That young man has got something to him," Mr. Amos Hughes said, as
they went back to the Rectory.
"If you could put some printing in his way, it would be a favor to me,"
said Dr. Lavendar.
"I shouldn't wonder if I could," the lawyer said.
"The girl is a fine creature, poor child," said Mr. Carter.
"Gentlemen," said Dr. Lavendar, "they are both good children, and they
have behaved well; but there's somebody else, let me tell you!"
However, he did not tell them. Perhaps he kept his opinion for Robert
Gray's ears, for once he said, smiling, in Rebecca's presence:
"Robert, this wife of yours is a noble woman."
Mr. Gray, a little surprised, said, politely, looking with kind eyes at
Rebecca, "Mrs. Gray is a very good wife, sir."
And Rebecca went up and hid herself in the garret and cried with joy.
AT THE STUFFED-ANIMAL HOUSE
I
Willy King's buggy, splashed to the top of the hood with mud and
sagging sidewise on its worn old springs, came pulling up the hill past
the burial-ground. The doctor himself, curled in one corner, rested a
leg on the dash-board and hung his reins on the hook over his head. He
was very sleepy, for he had been up until three with an old woman who
thought she was sick, and he had been routed out of bed again at five
because she told her family that she was going to die. William King
was not given to sarcasm, but he longed to say to the waiting
relatives, "There is no hope!--she'll live." Instead, he looked
seriously sympathetic and kept his thoughts to himself. When he got
home to breakfast, his wife told him how foolish he was to take so much
trouble. "There's nothing the matter with Mrs. Drayton," said Mrs.
King; "and I should tell her so, flatly and frankly. It would do her
good."
William said that he would like another cup of coffee.
"It wouldn't be good for you," said his Martha; "you are drinking too
much coffee. You can have shells if you want to. Shall I have some
shells warmed up?"
William said "No," and went trudging off to his office; and then, at
ten, started on his round of calls, his old buggy still unwashed from
the morning jaunt to the hypochondriac's death-bed. The day was still
and sunny, the road quite deserted and full of pleasant shadows under
the May foliage. But the sleepy doctor saw it all through half-closed
eyes, and yawned, and rested one plump leg on the dash-board, and let
the reins hang swaying from the hook in the roof of the bug-pry. Then,
suddenly, his mare stopped and William opened his eyes.
"Caught you napping, Willy!" said a loud, hearty voice. And the doctor
sat up and drew his leg in and laughed.
"Well, Miss Harriet, how do you know but what I was worrying over a
case?"
"Much worrying you do, young man!" She sat down on a log on the
road-bank and smiled at him. She was a big, vigorous woman with a
fresh, brown face and a keen, kind eye. She had a gun in her hand, and
a rabbit's white tail stuck out of the hunting-wallet slung over her
shoulder. She had broken through the underbrush on the hill-side just
as Willy's buggy jogged into the shadow of a sycamore that stretched
its mottled arms over the deserted road.
"Willy," she went on, in her loud, cheerful voice, "do you doctor-men
smile at one another when you meet, like the Augurs, because you fool
us so easily with your big words? You call a scratched finger an
'abrasion of the epidermis'--and then you send a bill. And, bless me!
what a serious air you put on at a minute's notice!--I saw you pull
your leg in, Willy. Come, now; you were in my Sunday-school class--why
don't you just admit to me that that piercing look over your
eye-glasses is one of the tricks of the trade? I won't tell."
William King chuckled. "You just get a touch of lumbago, Miss Harriet,
and you'll believe in my tricks."
"Lumbago!" said his reviler. "Not I; a day's shooting would cure it
quicker than a barrel of your pills."
"Been shooting this morning?"
"No; I set a trap in Dawson's hollow." She pulled out the rabbit and
held it up. "Not a bone broken. Handsome, isn't he? Poor little
thing!"
William looked at the soft, furry creature, limp in the big brown hand,
with critical appreciation. "Yes, beautiful. Miss Annie didn't find
him, to let him out?"
The hunter's face changed to amused impatience. "Willy, she opened
three traps last week. And she was so shrewd about it; you would never
believe how clever she is. Of course it's no use to scold."
"Of course not. What excuse does she make?"
"Oh, just the same thing: 'Sister, it hurts me to think they can't get
out.'"
"Poor thing!" said the doctor.
"I have tried to make her promise not to interfere with the traps. You
know, if I could once get a promise out of her I would be all right;
Annie never broke a promise in her life. But she is too shrewd to be
led into it. She always says, 'I'm the oldest, and you mustn't order
me round.' It would be funny if it weren't so provoking."
"Poor thing!" said the doctor again.
"She follows me and takes the bait out of the traps once in a while;
but she prefers to let things go. And she is certainly wonderfully
bright about it," Miss Harriet said. "Now, why can't she be sensible
in other things?"
"Well, you know she has always been about twelve; it's the young head
on old shoulders."
"I must tell you her last performance," Miss Harriet said. "You know
that picture of Aunt Gordon that hung in the dining-room? Dreadful
thing! I never saw the poor woman, but I believe she wasn't quite as
ugly as that portrait, though Alex looks just like her, Dr. Lavendar
says; and Alex is dreadfully ugly, with those pale eyes of his. Well,
I happened to say--it was last Tuesday, at tea, and Matty Barkley was
there: 'That picture of Aunt Gordon is awful! I can't bear it.' Of
course I never thought of it again, until I came home the next day--and
what do you suppose?"
Willy began to grin.
"Yes! she had got up on a chair, if you please, and cut it out of the
frame and slashed it all to pieces."
"Well done!" said Willy King, slapping his thigh.
"No such thing. It was ugly, but it was a family portrait."
"What did she say?"
"Oh, she had her excuse.... Willy, I can't understand her mind; it is
so unreasonably reasonable: 'Sister, you said you couldn't bear it, so
what was the use of having it?' After all, that was sense, William."
"So it was," said the doctor, and unhooked his reins and nodded.
"Well," he said--
But Miss Harriet laughed awkwardly. "Wait a minute, can't you? It
won't kill anybody to do without a pill for five minutes."
"Well, no, I suppose it won't," William admitted; "but with a view to
getting home in time for dinner--"
"Oh, let Martha wait. Willy, you are the meekest being--let her wait.
Tell her you'll have your dinner when you're good and ready."
"Martha is only concerned on my own account," the loyal William
protested.
"Well, I'm not going to keep you long," his old friend said, roughly;
"I--I just want to ask you a question." Her face grew suddenly a dull
red. "Not that I believe in your pills and potions--just please
remember that. But I suppose you do know a little something."
"I could diagnose a scratched finger," said the doctor, meekly.
"Well--" she said, and looked at the lock of her rifle; "there's
nothing in the world the matter with me, but--"
"You don't look like a confirmed invalid," the doctor assured her.
"No!--do I?" she said, eagerly. "I really am very well, William--very
well. Dear me, when I get home after a round of my traps (when Annie
hasn't teased me by letting things out) and eat a good dinner, and sit
down with a taxidermy magazine, I--I wouldn't thank King George to be
my uncle. Yes, I am _very_ well."
Her emphasis had in it a certain agitation that caught the doctor's
eye. "Your out-of-door life is calculated to keep you well," he said.
Miss Harriet got up and thrust the rabbit back into the pouch at her
side. "Of course; and, anyhow, I'm not the sick kind. Imagine me shut
up between four walls! I should be like Sterne's starling. Do you
remember?--'I want to get out, I want to get out.' No, there's nothing
the matter with me. Absolutely nothing."
She did look very well, the big, brown woman, towering up at the
road-side, with her rifle in her hand and the good color in her cheeks
and lips. Yet her eyes had a worn look, William thought. "Pain
somewhere," said the doctor to himself.
"You know, I don't believe in your pills and truck," she insisted,
frowning.
"Of course not," he assured her easily. "Come, now, Miss Harriet,
what's wrong?"
"Nothing, I tell you," she said, sharply; and then, with impatient
brevity, she spoke of some special discomfort which had annoyed her.
"It began about six months ago."
"Probably you've taken cold," William King said, and then he asked a
question or two. She answered with irritable flippancy:
"Now don't put on airs, Willy. There's no use trying to impress me; I
know you. Remember, you were in my Sunday-school class."
"Why didn't you make a better boy of me, then? You had your chance.
Miss Harriet, would you mind coming into my office and just letting me
look you over? Come, now, why shouldn't I get a job out of you for
once? Here you tackle me on the road-side and get an opinion for
nothing."
She chuckled, but retorted that she hated doctors and their offices.
"I'm not that Drayton cat," she said, "always wanting a doctor to fuss
over me. No, you can give me a pill right here--though I haven't a bit
of faith in it."
"I wouldn't waste a good pill on you," the doctor defended himself.
"You've got to come and see me."
But when she had promised to come, and William, slapping a rein down on
the mare's flank, was jogging along under the sycamore branches, he did
not fall into his pleasant drowse again. "She looks so well," he said
to himself, "she must be all right--"
II
Miss Harriet's house, called by Old Chester children "The
Stuffed-Animal House," was on the hill-road a stone's-throw beyond the
burial-ground. It was of weather-worn brick, and its white lintels,
carved in thin festoons of fruit and flowers, were nearly hidden by ivy
that stretched dark figures over the marble, and, thickening with the
years across the tops of the windows, made the rooms within dim with
wavering leaf shadows. A brick path, damp and faintly green with moss,
ran down to a green gate set in a ragged privet hedge that was always
dusty and choked with dead twigs. The house itself was so shaded by
horse-chestnuts that grass refused to grow in the door-yard. A porch
shadowed the front door, which opened into a dark, square hall, full of
dim figures that hung from the ceiling and stood in cases against the
walls. A dusty crocodile stretched overhead, almost the width of the
hall; a shark, with varnished belly splitting a little under one fin
and showing a burst of cotton, lurked in a dim corner; over the parlor
door a great snake, coiled about a branch, looked down with glittering,
yellow eyes; and along the walls were cases of very beautiful birds,
their plumage dulled now, for it was forty years since Miss Harriet's
father had made his collection. But all around the hall were
glistening eyes that stared and stared, until sometimes an Old Chester
child, clinging to a mother's protecting hand, felt sure they moved,
and that in another moment the crocodile's jaws would snap together, or
the eagle's wings would flap horribly in the darkness.
Yet there was an awful joy to Old Chester youth in being allowed to
accompany a mother when she made a polite call on Miss Harriet. This
hall, that was dark and still and full of the smell of dead fur and
feathers and some acrid preservative, had all the fascination of
horror. If we were very good we were allowed to walk from case to case
with old Miss Annie, while our mothers sat in the parlor and talked to
Miss Harriet. Miss Annie could not tell us much of the creatures in
the cases, and for all she used to laugh and giggle just as we did, she
never really knew how to play that the hall was a desert island and the
wild beasts were lurking in the forest to fall upon us.
"It isn't a forest, it's our front hall," Miss Annie would say; "and
you must do what I tell you, because I'm the oldest, and I don't want
to play desert island. But I'll show you my chickens," she would add,
with eager politeness.
Sometimes, if Miss Annie were not in the room, we would hear Miss
Harriet tell some story about her mischievousness, and our mothers
would sigh and smile and say, "Poor dear!" Our mothers never said
"poor dear!" about us when we did such things. If one of us Old
Chester children had spoken out in church as Miss Harriet said Miss
Annie did once, and told Dr. Lavendar that he was telling a story when
he read in the morning lesson that the serpent talked to
Eve--"because," said Miss Annie, "snakes can't talk"--if we had done
such a dreadful thing, we should have been taken home and whipped and
sent to bed without any supper, and probably the whole of the third
chapter of Genesis to learn by heart. We should not have been "poor
things!" This was very confusing to Old Chester youth until we grew
older and understood. Then, instead of being puzzled, we shrunk a
little and stayed close to our mothers, listening to Miss Harriet's
stories of Miss Annie with strange interest and repulsion, or staring
furtively at the little old woman, who laughed often and had a way of
running about like a girl, and of smoothing back her gray hair from her
temples with a fluttering gesture, and of putting up her lip and crying
when she was angry or frightened or when she saw anything being hurt.
Miss Annie could never bear to see anything hurt; she would not let us
kill spiders, and she made us walk in the grass instead of on the brick
path, because the ants came up between the bricks, and she was afraid
we would step on them.
"Annie is very kind-hearted," Miss Harriet used to tell our mothers.
"She can't bear my traps."
Miss Harriet's traps were her passion; her interest in taxidermy had
come to her from her father, and though she had not been able to add
anything of real value to Mr. Hutchinson's collection, her work was
thoroughly well done; and she even made a fair sum of money each year
by sending her squirrels and doves to town for the Christmas trade.
But more important than the money was the wholesome out-of-door life
her little business entailed, which had given her her vigorous body and
sane mind. She needed both to live with this gray-haired woman, whose
mind was eleven or twelve years old. It was not a bad mind for eleven
or twelve, Willy King used to say. Old Miss Annie had a sort of crude
common-sense; she could reason and determine as well as any other
twelve-year-old child--indeed, with an added shrewdness of experience
that sixty years of bodily age made inevitable. She knew, innocently,
much of life that other children were guarded from knowing; she knew
death, too, but with no horror--perhaps as we were meant to know
it--something as natural as life itself, and most of all as a release
from pain. For old Annie knew pain and feared it as only the body in
which the soul is not awake can fear it. She wept at the sight of
blood and moaned when she heard a squirrel squeak in the trap; she
shivered with passionate expectation of relief when Miss Harriet's
kindly chloroform brought peace to fluttering wings or beating claws.
When some soft, furry creature, hurt in the trap, relaxed into happy
sleep in the thick, sweet smell that came out of Miss Harriet's big
bottle, Miss Annie would laugh for joy, the tears of misery still wet
upon her wrinkled cheeks.
"Don't come into my shop," Miss Harriet used to say, laughing and
impatient, when Miss Annie would follow her into the room in the barn
where she did her work--"don't come in here, and then you won't see
things that hurt your feelings."
But Annie, smoothing her hair back from her temples with a curious,
girlish gesture, would only shake her head and sidle closer to her
sister, the young, guileless eyes in the withered face full of protest
and appeal. Her horror of pain lost Miss Harriet many a fine specimen;
for, in her pity for the trapped creatures, Annie, noiselessly, like
some Indian hunter, used to follow on her sister's footsteps through
the woods, lifting the baits out of the traps, or if she found a snared
creature unhurt, letting it go, and then creeping home, frightened at
Miss Harriet's anger, which, if she discovered the old child's
naughtiness, fell like a thunderbolt, and then cleared into patient
amusement, as a black shower brightens into sunshine. The big, kind
woman with a man's mind could not be angry at this poor creature; so
she did her duty by her and tried not to think about her. She went her
way, and set her traps, and prepared her few specimens, brushing Annie
or any other annoyance aside with careless good-nature.
"Don't think about unpleasant things," she used to say, in her loud,
cheerful voice. "The trouble with you doctors and ministers," she told
Dr. Lavendar, "is that you make people think about their insides. It's
stomachs with Willy and souls with you. Nobody ought to know that they
have a stomach or a soul. I don't. A tree don't. And there isn't an
oak in Old Chester that isn't pleasanter than Mrs. Drayton. Yet she's
always fussing about her insides--spiritual and material."
"It's when you don't have 'em that you fuss," Dr. Lavendar said; "the
trouble isn't too much soul, it's too little. And I guess it's the
same with stomachs."
"Then you say Mrs. Drayton has no soul?" Miss Harriet said, pleasantly.
"I never said anything of the sort," said Dr. Lavendar.
As for Miss Harriet, she went on to Willy King's office, prepared, as
usual, to make him as uncomfortable as she could. But she never put
Willy out. Her flings at his profession tickled him immensely, and if
now and then the good, honest William practised, as Miss Harriet said,
a few of the tricks of his trade, he was not averse to sharing their
humor with some one who could appreciate it.
"So you have that Drayton cat on your hands again?" Miss Harriet said,
plumping herself down in William's own chair in front of his office
table so that she could pick up and examine what she called his
"riffraff." ("Do open your windows, William. I don't see how you can
be so shut up. Po-o-o! how can people live so much in-doors?")
"Well," said William, doing as he was bid, "she enjoys my visits and I
enjoy her checks. I don't complain."
"That's like the profession," said Miss Harriet; "you put your hands in
our pockets whenever you get a chance. Well, you'll get nothing out of
my pocket, William, for there's nothing in it."
"Miss Harriet," said William, chuckling--"you won't tell anybody, will
you? But Mrs.--well, I won't name names; that's not professional--"
"Call her a 'Female,'" said Miss Harriet.
"Well, a Female sent for me on Tuesday, in a dreadful hurry; I must
come, 'right off! quick!' I was just sitting down to breakfast, but of
course I ran--"
"Martha must have been pleased?"
"I ran; and arrived, winded. There was--the Female, at _her_
breakfast. 'Oh,' she said, 'doctor, the baby has slept right through
from six last night, and he hasn't wakened up yet. I am afraid there
is something the matter with his little brain.'"
"William, if you didn't say that there was something the matter with
_her_ little brain--"
"I didn't," William said, grimly, "because she hasn't any. Now, Miss
Harriet, let's talk about yourself; it's pleasanter."
"Oh, there was not the slightest occasion to come to see you. But I
said I would, and here I am. I suppose you'll send me a bill as long
as my arm. Do you have a system of charges, Willy? So much for a look
over your glasses? So much for that solemn cough? I suppose you grade
all your tricks. Now work off the most expensive ones on me; I propose
to get the worth of my money, young man."
"Thought you said you weren't going to pay any bills?" William reminded
her; and then refused to be side-tracked any longer, but asked question
after question, bringing her up once or twice with a sharp turn.
"Don't joke now, please, Miss Harriet. Be as exact as you can. Is
this condition thus, or so--?" And when he got through with his
questions, he took up the joking rather heavily.
"You're so faithless about pills," he said, "that I'm not going to give
you any."
"What! no pills?" said Miss Harriet.
William King laughed awkwardly. "Not a pill! I don't see any
condition which warrants them: but--"
"What did I tell you? There's nothing the matter, and you just dragged
me here to give your office a busy look."
"I didn't suppose you'd see through it," said Willy King. "But, Miss
Harriet, I--I don't feel _quite_ satisfied. I--do you know I've a
great mind to get a man in Mercer to look you over? I want you to go
up with me to-morrow and see him."
"Nonsense!"
"No, truly," he said; "I am not satisfied, Miss Harriet."
"But what do you mean?" she insisted, sharply. "There's nothing the
matter with me. You said yourself I didn't need any medicine. Give me
some opiate to stop this--this discomfort when it comes on, and I'll be
all right."
"You can't bear opiates," he said, bluntly; "your heart won't stand
them. Don't you remember the time you broke your ankle and I tried
morphine--a baby dose--to give you some relief? You gave me a scare, I
can tell you."
Miss Harriet was silent. Then: "I've known my heart wasn't right for
two years. But--"
"Oh, your heart doesn't give me any concern--if you don't take
liberties with it. Perhaps it isn't quite as good as it was thirty
years ago, but--"
"Ah, I lost it to you then, Willy. You were a sweet little fellow when
you came into my class. Do you remember once when--"
"Miss Harriet, you've got to go to Mercer with me to-morrow," William
King interrupted, quietly. "I hope there's nothing much out of the
way. I hope not. I--I believe not. But I'm not sure. We'll go up
and see Greylord and find out. He'll give you some pills, maybe," he
ended, and laughed and got up. "Now I'm off to the cat, Miss Harriet."
And Miss Harriet, to her astonishment, found herself dismissed before
she had made the boy tell her what he was afraid of. "He _is_ a boy,"
she said to herself. "Of course he wouldn't be apt to know what was
the matter. I ought to have gone to see some Mercer man to begin with.
I remember when Willy was born."
III
When they came out of the Mercer doctor's door William King's fresh
face had gone white, but Miss Harriet walked smiling. At the foot of
the steps the doctor paused and stood an instant leaning on the
hand-rail, as though for support and to get his breath. Miss Harriet
looked at him with concern. "Why, Willy!" she said.
"Miss Harriet," William said, hoarsely, "he may be mistaken. It's
perfectly possible that he is mistaken."
"I guess not, Willy," she said, simply. "Come, now, don't be such a
wet string." She struck him a friendly blow on the shoulder that made
the doctor take a quick step forward to keep his balance; but it gave
him the grip upon himself that for a single instant he had lost.
"And, anyhow," he said, "even if he is right, it may not develop. I've
known a case where it was checked for two years; and then the patient
died of small-pox."
"Pleasant alternative," said Miss Harriet; she was smiling, her face
full of color, her shoulders back, her head up. "Come, Willy, let's
have a spree. Here we are for a day, and Martha's at home. We'll have
a good dinner, and we'll do something interesting. _Hurrah!_" said
Harriet Hutchinson.
And the doctor could do no less than fall into step at that martial
note and march at her side proudly. And by some spiritual contagion
his courage met hers like the clash of swords. They went to get their
good dinner, and Miss Harriet ate it with appetite. Afterwards she
declared they would go to the circus. "It's in town; I saw the tents.
I haven't been to a circus for forty years," she said; "but I know just
how the pink lemonade tastes. You've got to treat, Willy."
"I'll throw in pea-nuts," said William King; and with that they left
the restaurant and went sauntering along the hot, grimy street in the
direction of the open lots beyond the blast-furnaces, where, under a
deep June sky, dazzling even though it was smudged by coils of smoke,
were stretched the circus tents, brave with flags and slapping and
billowing in a joyous wind. William King held on to his hat and looked
at the great, white clouds, domed and shining, piled all along the
west. "We'll get a shower, I'm afraid, Miss Harriet."
"Well, take a pill, Willy, and then it won't hurt you," she told him,
with a laugh that belonged to the sun and wind, to the flags whipping
out on their halyards and the signs of the side-shows bellying from
their guy-ropes, to the blare of music and the eager circus crowd--that
crowd that never changes with changing generations. Still there is the
old man gaping with excited eyes; still the lanky female in spectacles;
the cross elder sister afraid of crushing her fresh skirts; the little
boy absorbed in thought; the little girl who would like to ride on the
Shetland pony when the clown offers any miss in the audience an
opportunity. We know them all, and doubtless they know us, the
patronizing, amused on-lookers, who suddenly become as eager and
absorbed as any graybeard or child in the crowd. We know the red
boxes, too, where men with hard faces and wearied eyes shout
mechanically the same words of vociferous invitation to the side-shows.
Children, pulled along by their elders, would stop, open-mouthed,
before these men; but somehow they never see the wild man or the fat
lady. Ah, the regret for the unseen side-shows!--the lady with the
snakes; the skeleton man; the duel between the educated hyena and his
trainer--that hyena of whom the man in the red box speaks with such
convincing enthusiasm. "_I have been_," cries the strident voice--"_I
have been connected with circuses all my life--all my life, ladies and
gentlemen!--and I give you my sacred word of honor that this is the
most magnificent specimen of the terrible grave-robbing hyena that I
have ever seen!_" Why did we never see that hyena? Why, why did we
always hurry on to the main tent? It is the pang that even paradise
must know, of the lost experience of earth--or perhaps of hell.
"We ought to see the fat lady," said Dr. King.
"I'm afraid we'll be late," Miss Harriet objected, eagerly.
So they pushed on with the impatient, good-natured crowd. The smell of
tan-bark and matted pelts and stale pea-nut shells came in a gust as
they jostled under the flap of the outer tent and found themselves
inside the circle of gilded cages. "Shall we go right in and get our
seats?" William said.
"What! and not look at the animals? Willy, you're crazy. I want to
feed the elephants. Why, there are a lot of them, six or seven."
So they trudged around the ring, their feet sinking deep into the
loose, trampled earth. Miss Harriet poked the monkeys clinging to the
grating of their car, with her big umbrella, and examined the
elephant's hide with professional interest. "Imagine curing that
proboscis," she said. And then they stopped in front of a miserable,
magnificent lion, turning, turning, turning in a cage hardly more than
his own length. Miss Harriet drew in her breath. "It's being trapped
that is so awful, Willy. The consciousness that _you can't get out_.
It isn't the--the pain of it; it's being trapped."
William King, looking at the poor tawny creature of the desert and free
winds and life that dealt death with passion, blinked suddenly behind
his glasses. "But you trap things yourself," he protested, a moment
afterwards.
"Oh, but I don't keep 'em trapped; I kill 'em," she defended herself.
"I couldn't keep things shut up. I'd be as bad as Annie if I saw any
living creature that wasn't free to get out-of-doors." And then she
pushed on to the next cage, and the next; then suddenly feared that
they would not get good seats if they wasted any more time among the
animals. "For we won't have any reserved doings," she said. "I want
to sit on those boards that I sat on forty years ago."
She was as excited as she might have been forty years ago; and pushed
ahead into the big tent, dragging William by the hand, and climbing up
tier after tier, to get a good view of the ring. When they sat down,
she made haste to spread open the flimsy pink sheet of the programme
with its pale type, and read to William, in a loud, ecstatic voice,
just what was going to happen:
"_Display No. 1. Gigantic Pageantric Prelude--presenting Equitational
Exercises, Hippo-dramatical Revivals, Pachydermical Aggregations--the
only terpsichorean Pachyderms ever taught to tread the mazes of the
Quadrille._
"_Display No. 2. Claire St. Jeal and her company--the loveliest
daughters of Italy, and world-famous bareback equestriennes--_"
"You are sure you are not getting tired?" William King interrupted.
"Tired?" she repeated, scornfully. "William, as Matty Barkley would
say, you are a perfect fool. Why should I be tired? I feel first
rate--never better. I wouldn't thank King George to be my uncle! I've
wanted to come to the circus for years. Willy, what will your wife
say?"
"Nothing," said William, significantly.
At which Miss Harriet laughed until the tears stood in her eyes.
"William, you have more sense than I gave you credit for. But I am not
sure that, as your Sunday-school teacher, I ought not to tell you to
confess. Hullo! look what's coming."
Flare of banners! Prancing horses! Roman soldiers in rumbling
gold-and-crimson chariots! Elephants bearing, throned upon their
backs, goddesses of liberty and queens of beauty! Miss Harriet was
leaning forward, her lips parted with excitement. William King looked
at her and drew in his breath.
[Illustration: "MISS HARRIET WAS LEANING FORWARD"]
"'Not more than six months;' God grant not!--I wish it might not be
more than two."
"Willy, read what comes next," she said, shoving the programme at him;
"I can't stop looking."
The canvas was darkening a little overhead, so that William had to put
on his glasses and hold the printed sheet at arm's-length to decipher
the blurred, smudged text sufficiently to say that "Mademoiselle
Orinda, Queen of the Flying Trapeze, would give her marv--"
"William--what shall I do about Annie?" Miss Harriet said.
"You know we will all take care of Miss Annie," he said, tenderly;
"and--"
"Oh, Willy, there's the red lemonade," she interrupted, standing up and
beckoning with her crumpled programme. "Did you ever see so deadly a
drink? You forgot the pea-nuts," she reminded him, reproachfully. And
when William secured his hot, brown-paper bag, she ate the pea-nuts and
watched the changing wonders of the ring with intent eyes. She laughed
aloud at the clown's endeavors to ride a kicking donkey, and when the
educated dogs carried one another about in a wheelbarrow she applauded
generously. "They are wonderful!" she said.
William King looked at her keenly; it was all real. Miss Harriet was
incapable of pretence.
The brilliant day, that had showed between lacings of the tent like
strings of sapphires, had dimmed and dimmed; and by-and-by, unnoticed
at first, there was the drip of rain. Here and there an umbrella was
raised, and once or twice a bedraggled man or woman led out a reluctant
child--"For I ain't a-goin' to have you catch your death of cold for no
trained elephants," a mother said, decidedly, pulling a whining boy
from beside Miss Harriet.
"Perhaps," ventured the doctor, "we really ought to go. I can't have
you 'catch your death of cold,' Miss Harriet."
"I won't die of a cold, William," she said, her eyes narrowing.
And William swore at himself under his breath, but said, with clumsy
jocularity: "Well, not if I can help it. But I don't know why you
should be so sure; it might give you bronchitis for a year."
"I won't have bronchitis for a year," Miss Harriet said, gazing at the
clowns.
And William King swore at himself again.
The rain increased to a downpour; little streams at first dripped, then
poured, upon the thinning benches. The great centre pole was streaming
wet; the clown stood in a puddle, and the red triangle on his
chalk-white forehead melted into a pink smear.
"Really, Miss Harriet," William said, anxiously, "I'm afraid--"
"If you're afraid for yourself, I'll go," she said; "but we ought to
wait for the grand concert. (Ah! there's the man with the red
balloons. If you had a half-dozen children, Willy, as you ought to
have, I'd buy him out.) Well, are you sugar or salt, to be so scared
of a drop of rain?"
She did not look afraid of rain herself when she got up and pushed past
the scattered spectators, her hair glistening with drops, her cheeks
red, her eyes clear. "William," she said, when they got outside and
were hurrying along to catch the stage for Old Chester--"William, that
has done me good. I feel superbly. Do you know, I haven't had an
instant's pain since I first spoke of the thing to you? That's three
days entirely free. Why, such a thing hasn't happened in--in three
months. Just think of that--entirely free. William, I'll cheat you
doctor-men yet." She looked at him with glowing courage. "I feel so
well," she said.
She held out her hand, there in the rain on the black cinder-path, and
William King struck his into it with a sort of shout.
"Hurrah!" he said, as she had said when they had come out from hearing
the sentence in the Mercer doctor's office.
The long ride home in the stage, in which they were the only
passengers, was perhaps a descending scale.... At first they talked of
the circus. "I liked the man and the bear best," William said.
"Oh, he wasn't as fine as that beautiful lady in pink petticoats who
rode the fat, white horse. Did you ever see a horse with so broad a
back, Willy? Why, I could have ridden him myself."
"He would need a broad back," William said; and Miss Harriet told him
to hold his tongue and not be impudent. The rain was pattering on the
roof and streaming down the windows, and in the dark, damp cavern of
the stage they could not see each other's face very well; but the
stretches of tense silence in the circus talk made William King's heart
beat heavily, although he burst out gayly that the afternoon had
brought back his youth. "Miss Harriet, when you were a child, didn't
you always want to poke around under the seats when it was over and
find things? William Rives once found five cents. But William would
find five cents in the Desert of Sahara. I never had his luck, but I
was confident that watches were dropped freely by the spectators."
"Of course," cried Miss Harriet. "Or diamond-rings. My fancy led me
towards diamond-rings. But I suppose you never knew the envy of the
ladies' clothes? Dear me--those petticoats!"
"The ring-master's boots were very bitter to me; but my greatest desire
was--"
"Willy," Miss Harriet said, hoarsely, "I don't want anybody to know."
"Of course not," William King said. "Why should they? We may hold
this thing at bay for--"
"We will hold it at bay," she said, with passion. "I will! I _will_!
Do you hear me?"
Willy King murmured something inarticulately; his eyes suddenly smarted.
The ride to Old Chester seemed to him interminable; and when, after
wandering snatches of talk about the circus, the stage at last drew up
at the green gate in Miss Harriet's privet hedge, his nerves were tense
and his face haggard with fatigue.
At home, at his belated supper-table, his good Martha was very severe
with him. "You oughtn't to allow yourself to get so tired; it's wrong.
You could just as well as not have ordered your things by mail. I must
say, William, flatly and frankly, that a doctor ought to have more
sense. I hope there was nobody in the stage you knew to talk you to
death?"
"Miss Harriet came down," William said, "but she hadn't much to say."
"I suppose she went to buy some of her horrid supplies?" Martha said.
"I can't understand that woman--catching things in traps. How would
she like to be caught in a trap? I asked her once--because I am always
perfectly frank with people. 'How would you like to be caught in a
trap, Miss Harriet?' I said. And she said, 'Oh, Annie would let me
out.' You never can get a straight answer out of Harriet Hutchinson."
"My dear, I'll take another cup of tea. Stronger, please."
"My dear, strong tea isn't good for you," Martha said.
IV
When Miss Harriet woke the next morning the blue June day was flooding
her room. At first she could not remember.... What was the something
behind her consciousness? It came in an instant. "_Trapped_," she
said, aloud, and turned her head to see Miss Annie at her bedside.
"What is trapped, sister?" said Miss Annie, her little old face
crumpling with distress.
"I am," Harriet said; and laughed at the absurdity of telling Annie in
such a fashion. But of course there was no use in telling Annie. She
couldn't understand, and all that there was for her to know, the
ultimate fact, she would find out soon enough. The younger sister felt
a sick distaste of dealing with this poor mind; she wanted to be kind
to Annie; she had always wanted to be kind to her--but she didn't want
her round, that was all. And so she sent her off, patiently and not
ungently: "Don't bother me, Annie, that's a good girl. No--I don't
want any roses; take them away. No--I don't want to look at pictures.
You go away now, that's a good girl."
And the wrinkled child obeyed meekly. But she told the deaf Augustine
that Harriet was cross. "I'm the oldest, and she oughtn't to order me
round," she whimpered.
Poor Miss Annie was constantly being told to be a good girl and go
away, in the days that followed--days, to Miss Harriet, of that
amazement and self-concentration which belong to such an experience as
hers. There had been no leading up to this knowledge that had come to
her--no gradual preparation of apprehension or suspicion. The full
speed of living had come, _crash!_ against the fact of dying. The
recoil, the pause, the terrible astonishment of that moment when Life,
surging ahead with all his banners flying, flings himself in an instant
against the immovable face of Death--leaves the soul dazed by the
shock--dazed, and unbelieving. "_It cannot be._" That is the first
clear thought. It is impossible; there is a mistake somewhere. A day
ago, an hour ago, Death was lying hidden far, far off in the years.
Sometime, of course, he would arrive--solemn, inevitable, but
beneficent, or at least serene. He would send soft warnings before
him--faint tollings of fatigue, vague mists of sunset shadows. The
soul will be ready for him when he comes then; will even welcome him,
for after a while Life grows a little tired and is ready to grasp that
cool hand and rest. We all know how to meet Death then, with dignity
and patience. But to meet him to-morrow--to-day, even, when we are
full of our own business, of our own urgent affairs--the mere
interruption of it is maddening. Across the solemnity of the thought
comes with grotesque incongruity an irritated consciousness of the
_inconvenience_ of dying.
As for Harriet Hutchinson--"I don't believe it," she said to herself,
that first morning. And then, breathlessly, "Why, I can't--die!"
She was not afraid, as one counts fear, but she was absorbed; for there
is a dreadful and curiously impersonal interest in the situation that
takes possession of the mind in moments like this. No wonder she could
not think about Annie. She could not think about anything except that
that man in Mercer had said that in a very short time--
"Why, but it's perfectly ridiculous!" she told herself; "it _can't_ be.
I'm not sick--"
As she lay there in her bed that morning, after she had sent Miss Annie
away, she lifted her hand--a large hand, with strong, square fingers,
brown with weather and rough with her work, and looked at it curiously.
It was a little thin--she had not noticed that before; but there it
was, eager, vital, quick to grip and hold, life in every line. And it
would be--still? No; she did not believe it. And, besides, it
couldn't be, it mustn't be. She had a hundred things to do. She must
do them; she couldn't suddenly--_stop_. Life surged up in a great wave
of passionate determination. She got up, eager to go on living, and to
deny, deny, deny! It was the old human experience which is repeated
and repeated until Life can learn the fulfilment of Death. Poor Life,
beaten by the whips of pain, it takes so long sometimes to learn its
lesson!
In those weeks that followed--weeks of refusal, and then struggle, and
then acceptance, and last of all adjustment--Miss Harriet found old
Annie's companionship almost intolerable. She was very unreasonable
with her, very harsh even; but all she asked was solitude, and solitude
Annie would not give. She ran at her sister's heels like a dog; sat
looking at her with frightened eyes in the bad hours that came with
relentlessly increasing frequency; came whimpering to her bedside on
those exhausted mornings when Harriet would scourge her poor body onto
its feet and announce that she was going out. "These four walls
smother me," she used to say; "I must get out-of-doors."
Sometimes it seemed as if the big, kind nature that had borne the
pin-pricks so patiently all these years had reached the breaking-point,
and another day or another hour of poor old Annie's foolish love would
cause it to burst out in frantic anger:
"It hurts, sister?"
"Yes, Annie; but never mind. If I could only get out-of-doors I
wouldn't mind."
"Oh, sister, don't let it hurt."
"Can't help it, Annie. Now, don't think about it, that's a good girl.
Maybe I can get out to-morrow a little while."
"But I can't bear it."
"Got to, my dear. Come, now, run away. Go and see your chickens."
"Sister, I can't bear it."
"Annie, you drive me wild. Augustine--oh, she can't hear.
_Augustine!_ you must take Miss Annie away. Annie, if you say another
word--"
"I'm the oldest and I have a right to talk. Why don't you smell your
big bottle? When the squirrels smell it they are not hurt."
"Well, I'm not a squirrel. Annie, if you stay another minute,
I'll--I'll-- Oh, for Heaven's sake, let me alone!"
She could stand it, she told herself, if she was alone. For though she
finally accepted the fact, her own weakness she could not accept. "I
am ashamed," she told William King, angrily.
"But there's nothing to be ashamed of," Willy King protested, in his
kind way. "Dear Miss Harriet--"
"Hold your tongue. Nothing to be ashamed of? I guess if your body had
put your soul in a corner, with its face to the wall--I guess you'd be
ashamed. Yesterday I--I-- Well, never mind. But my body got me down,
I tell you--got my soul down. Isn't that something to be ashamed of?
Don't be an ass, William. I'm ashamed."
It was this consciousness of her own weakness that made her hold
herself aloof from her friends.
In those days people did not have trained nurses; they nursed one
another. It was not skilful nursing; it frequently was not wise, as we
count wisdom to-day; but it was very tender and loving, and it was very
bracing. In these softer times, when we run so easily to relief from
pain, we do not feel the presence of the professional nurse a check
upon our weakness; if we suffer, we are willing that this skilful,
noiseless machine, who will know exactly how to relieve us, shall see
the suffering. We are neither mortified nor humiliated by our lack of
endurance or of courage. But in Old Chester, when we were ill, and
some friend or relative came to sit by our bedside, we had--for their
sakes--to make an effort to control ourselves. If the effort failed,
our souls blushed. Miss Harriet would not run the risk of failure; her
body, as she said, got the better of her soul when she was alone; it
should not have the chance to humiliate her publicly; so, roughly, she
refused the friendly assistance so eagerly offered: "Thank you;
Augustine can look after me. I don't want anybody. And besides, I'm
perfectly comfortable. (William, I won't have anybody. Do you
understand? It's bad enough to disgrace myself in my own eyes; I won't
have Matty Barkley sit and look on.)"
And William King put people off as well as he could: "I go in two or
three times a day, just to say how do you do; and Miss Annie is about
and can bring her anything she needs. And Augustine is very faithful.
Of course, she is deaf as a post, but she seems to know what Miss
Harriet wants."
So the situation was accepted. "Here I am," she told the doctor,
grimly, "dying like a rat in a hole. If I could only get
out-of-doors!--or if I had anything to do!--I think it's the having
nothing to do that is the worst. But I'll tell you one thing, Willy--I
won't be pitied. Don't have people mourning over me, or pretending
that I'm going to get well. They know better, and so do I."
Those who dared to pity her or who ventured some futile friendly lie
about recovery were met by the fiercest impatience. "How do I feel?
Very well, thank you. And if I didn't, I hope I wouldn't say so. I
hope I'm well enough bred not to ask or answer questions about
feelings. There is nothing in the world so vulgar," she said, and
braced herself to one or another imprudence that grieved and worried
all the kind hearts that stood by, eager to show their love.
"It breaks my heart to see her, and there's nothing anybody can do for
her," Mrs. Barkley told Dr. Lavendar, snuffling and wiping her eyes.
"She positively turned Rachel King out of the house; and Maria Welwood
cried her eyes out yesterday because she was so sharp with her when
Maria said she was sorry she had had a bad night and hoped she'd soon
feel better."
The old man nodded silently. "Poor Miss Harriet!" he said.
"Don't say 'poor Miss Harriet!' to her. Dr. Lavendar, Harriet and I
have been friends since we were put into short dresses--and she spoke
to me to-day in a way--! Well, of course, I shall go back; but I was
ready to say I wouldn't. And she treats poor old Annie outrageously."
Dr. Lavendar nodded again. He himself had seen her several times, but
she had never let him be personal: "Was Mrs. Drayton still gossiping
about her soul?" "Wasn't it nearly time to get a new carpet for the
chancel?" etc., etc. It was her way of defending herself--and Dr.
Lavendar understood. So he only brought her his kindly gossip or his
church news, and he never looked at her mournfully; but neither did he
ever once refer to a possible recovery--that poor, friendly pretence
that so tries the soul absorbed in its own solemn knowledge!
But in the afternoon, after his talk with Mrs. Barkley, the old man
went plodding up the hill to the Stuffed-Animal House, with tender and
relentless purpose in his face. It was a serene September day, full of
pulsing light and fragrant with the late mowing. William King's mare
was hitched to a post by the green gate in the hedge, and the doctor
was giving her a handful of grass as Dr. Lavendar came up. "How is
Miss Harriet, Willy?" the old man said.
William climbed into the buggy and flicked with his whip at the
ironweed by the road-side. "Oh--about the same. Dr. Lavendar, it's
cruel--it's cruel!"
"What's cruel, William?"
"I can't give her any opiate--to amount to anything."
"Why?"
"Her heart."
"But you can't let her suffer!"
"If I stopped the suffering," the doctor said, laconically, "it would
be murder."
"You mean--"
"Depressants, to amount to anything, would kill her."
Dr. Lavendar looked up into the sky silently. Willy King gathered up
the reins. "And Annie?" Dr. Lavendar said.
"She is just a poor, frantic child. I can't make her understand why
Miss Harriet shouldn't have two powders, when one 'sugar,' as she calls
it, gives her a little comfort for a little while. She says, 'Harriet
wouldn't let a squirrel stay hurt.' Miss Harriet says she told her the
other day that she wasn't a squirrel; but it didn't seem to make any
difference to Miss Annie. She has a queer elemental reasonableness
about her, hasn't she? Well, I must go. Dr. Lavendar, I--I hope you
won't mind if I say that perhaps--I mean she doesn't want anybody to
refer to--to anything religious."
"William," said the old man, mildly, "if you can mention anything which
is not religious to a woman who is going to die within a very few
weeks, I will consider it."
And William King had the grace to blush and stammer something about
Miss Harriet's hating anything personal. Dr. Lavendar listened
silently; then he went on up the path to the Stuffed-Animal House. Old
Miss Annie let him into the darkened hall, a burst of western sunshine
flooding in behind him and making the grim, dead creatures dart out of
their shadows for a moment, and sink back into them again when the door
was shut. The old child had been crying, for Miss Harriet had turned
her out of her room, and so he had to sit there in the hall, under the
shark, and try to comfort her and bid her go out and see her chickens.
But for once Miss Annie would not be diverted:
"Harriet wants to go out-of-doors, and she can't. And she is hurt; and
Willy King won't give her sugar in a paper to stop the hurting. He is
wicked."
"By-and-by," said Dr. Lavendar, "Harriet will fall asleep and not be
hurt any more."
"Not till she is dead," Miss Annie said; "Augustine told me so."
"I meant that," Dr. Lavendar said, stroking the poor, gray head
grovelling against his knee.
"Then why didn't you say so? It is a story to say sleep when you mean
dead."
"I ought to have said dead," he acknowledged, gently, "so that you
could understand. But I want you to remember that death is a happy
sleep. Will you remember that?"
"A happy sleep," Miss Annie repeated; "yes; I will remember. _A happy
sleep._" She lifted her head from his knee and smiled. "I'll go and
see my chickens," she said.
[Illustration: "'A HAPPY SLEEP,' MISS ANNIE REPEATED"]
And Dr. Lavendar took his way up-stairs, past the cases of birds, to
Miss Harriet's room. She received him with elaborate cheerfulness.
As for Dr. Lavendar, he lost no time in pretence. "Miss Harriet," he
said, "I am not going to stay and talk and tire you. You've seen
people enough to-day--"
"I'm not tired in the least."
"But I have a word to say to you."
She looked at him angrily. "I would rather not talk about myself, Dr.
Lavendar, please."
"I don't want to talk about yourself," he said.
Her face cleared a little. "That's a relief. I was afraid you were
going to talk to me about 'preparing,' and so forth."
A sudden smile twinkled into Dr. Lavendar's old eyes. "My dear Miss
Harriet, you've been 'preparing' for fifty years--or is it fifty-one?
I've lost count, Harriet. No; you haven't got anything to do about
dying; dying is not your business. In fact, I sometimes think it never
is our business. Our business is living. Dying is God's affair."
"I haven't any business, that's the worst of it," Miss Harriet said,
bitterly. "I've nothing to do--nothing to do but just lie here and
wait. I don't mind dying; but to be here in this trap, waiting. And
I've always been so busy, I don't know how to do nothing."
"That's what I wanted to say to you. There is something you can do.
In fact, there's something you must do."
"Something I must do?" Miss Harriet said, puzzled.
"My dear friend, you must meet this affliction; you can't escape; we
can't save you from it. But there is one thing you can do: you can try
to spare the pain of it to other people. Set yourself, Miss Harriet,
to make it as easy as you can for those who stand by."
Harriet Hutchinson looked at him in amazement. No pity? No
condolences? Nothing but the high charge to spare others. "You mean
my temper?" she said at last, slowly.
"Yes," said Dr. Lavendar.
Miss Harriet blushed hotly. "It is bad; I know it's bad. But--"
"Mine would be worse," said Dr. Lavendar, thoughtfully. "But look out
for it, Harriet. It's getting ahead of you."
Miss Harriet nodded. "You're right."
"You see, when you are out of temper it shows you are suffering; and
that's hard for us to bear--not the temper, of course, but the
knowledge. So you've got to spare us, Harriet. Understand?"
"I understand."
"It will be hard work for you," he said, cheerfully; and somehow the
words meant, not pity, but "_Shoulder arms!_"
For an instant they gazed, eye to eye--the woman devoured by pain, the
old man with his calm demand; and then the soul of her rose with a
shout. What! there was something left for her to do? She need not
merely sit still and die? She need not wait idly for the end? It was
a splendid summons to the mind--a challenge to the body that had dogged
and humiliated the soul, that had wrung from her good-humored courage
irritability and unjust anger, that had dragged her pride in the dust
of shame, yes, even--even (alone, and in the dark), but even of tears.
"_Make it as easy as possible for those that stand by._"
Some might say that that austere command was the lash of the whip; but
to Miss Harriet it was the rod and the staff. The Spartan old man had
suddenly revealed to her that as long as the body does not compel the
soul, there is no shame. As long as she could hold her tongue, she
said to herself, she need not be ashamed. Let the body whimper as it
may, if the soul is silent it is master. Miss Harriet saw before her,
not humiliation and idleness and waiting, but fierce struggle.... And
it was a struggle. It was no easy thing to be amiable when good Maria
Welwood wept over her; or when Martha King told her, flatly and
frankly, that she was doing very wrong not to make more effort to eat;
or even when Mrs. Dale hoped that she had made her peace with Heaven.
"Heaven had better try to make its peace with me, considering," said
Miss Harriet, grimly; but when she saw how she had shocked Mrs. Dale,
she made haste to apologize. "I didn't mean it, of course. But I am
nervous, and say things to let off steam." Such an admission meant
much from Miss Harriet, and it certainly soothed Mrs. Dale.
But most of all, Harriet Hutchinson forbade her body to dictate to her
soul when Miss Annie hung whimpering about her with frantic persistence
of pity. Never in all their years together had Miss Harriet shown such
tenderness to Annie as now, when the poor old child's mere presence was
maddening to her. For Annie could think of nothing but the pain which
could not be hidden, and her incessant entreaty was that it should be
stopped. "Wouldn't you rather be dead, sister?"
"Yes, Annie."
"Well, then, be dead."
"I can't, Annie. Now let us talk of something else. Tell me what the
black hen did when the speckled hen stole her nest."
Annie joyously told her story, as she had told it dozens of times
before; while Harriet Hutchinson turned her face to the wall. Annie
sat on her heels on the floor beside the bed, rocking back and forth,
and talking: "And so the speckled hen flew off. Sister, I'll get you
your big bottle?"
No answer.
"Sister, don't you want to smell the bottle?"
"No, Annie. No--no--_no_! Oh, Annie, don't you want to go and see
your chickens?"
"Why not?"
"Because it wouldn't be right, Annie."
"Why wouldn't it be right, sister?"
"Because," said Harriet Hutchinson--"because I suppose that's one of
the things that would 'make it harder for those that stand by.'"
"I don't understand," poor old Annie said, timidly.
"Well, Annie, that's the only reason I know of. Oh, Annie, Annie! it
is the only reason there is; it is the root of its being wrong." ...
And then the long moan. When Miss Annie heard that sound she shivered
all over; it was the elemental protest of the flesh, which cannot
understand the regal and unconquered soul.
Those were hard days for Willy King, what with his affection and his
sympathy and his daily struggles with Miss Annie; "for she is frantic,"
he told Dr. Lavendar. They were walking up the hill together in the
late afternoon. Miss Harriet had sent for the old man, on whom now she
leaned even more than on William King, for Dr. Lavendar gave her
granite words instead of Willy's tenderer sympathy. "She insists that
I shall give Miss Harriet something--'stuff out of Harriet's bottle,'
she says. I suppose she means chloroform. I wish to God I could."
"God will do His own work, William."
"Yes, sir; but it's such a waste--this courage that fairly breaks our
hearts."
"Waste! William, what are you talking about? We are every one of us
richer for it. I told her so yesterday."
"Well," said William King, thoughtfully, "perhaps so; in this case we
are richer, I admit. But suppose it were a baby that was suffering--or
a dog? Only, we wouldn't let the dog suffer. Dr. Lavendar, one of
these days--you and I won't live to see it, but one of these days--"
"There is Miss Annie now," said Dr. Lavendar. "Why--look at her!"
The old woman came fluttering down the path towards the green gate in
the privet hedge; she was smoothing her hair back from her temples,
with her strange, girlish gesture, and she was smiling, but there was a
new and solemn age in her face that made the two men look at each
other, startled and wondering.
"Dr. Lavendar! Willy!" she said, her voice breaking with joy, "Harriet
is dead--oh, Harriet is dead!"
They stopped short in the pathway. "What--what?" stammered William
King.
"Oh, Harriet is dead!" the old woman said; "and I'm so happy." She
came and leaned on the closed gate at the foot of the path, smiling up
into their faces. "She isn't hurt any more. Oh, I can breathe, I can
breathe, now," said Miss Annie, laying her withered hands upon her
throat and drawing a deep breath.
"When?" said the doctor.
"Oh, just a little while ago. As soon as she got dead I opened the
windows and let the air blow in; she likes the wind when she isn't
hurt."
William King said, suddenly, "_My God!_" and turned and ran up the
path, into the house, into the room, where, indeed, there was no more
hurting.
"Annie," Dr. Lavendar said, "were you with her?"
"Yes," Miss Annie said. "Harriet was hurt very much. But when she
smelled her bottle she stopped being hurt."
Dr. Lavendar leaned against the gate, his breath wavering; then he sat
down on the grass, and rested his forehead on his hands clasped on the
top of his stick. He was unable to speak. Miss Annie came out into
the road and looked at him curiously. After a while he said, feebly,
"Annie, tell me about it."
"Willy wouldn't give Harriet sugar in a paper to stop the hurting. And
Harriet said she couldn't get her bottle. She said it would be wrong
for her to get it."
Dr. Lavendar lifted his head with a quick gesture of relief. "What!
Harriet, didn't get it herself?"
"Oh no," Miss Annie said. "I got it. And I went into Harriet's room.
Harriet's eyes were shut, and she was--was moaning," said Miss Annie,
shivering. "So I put some stuff out of the bottle on a towel and held
it for Harriet to smell. And Harriet opened her eyes and looked
frightened, and she said, 'No, no!' And I said, 'Yes; I'm the oldest
and you must do what I say.' And she said, 'Augustine! Augustine!'
But Augustine can't hear. And I held it down and I said, 'You won't be
hurt any more.' And Harriet pushed it away and said 'No.' And then
she shut her eyes. And after a while she didn't say anything more.
And I held it, oh, a long time. And then I looked, and Harriet's eyes
were shut. And now she's dead! And it doesn't hurt any more. You
come and look at her, and you'll see it doesn't hurt any more. Now she
wouldn't thank King George to be her uncle! Oh, she's dead," said Miss
Annie, nodding her head and laughing; "a happy sleep." She was
standing there in the dusty road in front of him, telling the story,
her hands behind her, rocking slightly backward and forward, like a
child repeating a lesson. The long afternoon shadows stretched from
the trees across the road, and, swaying lightly, flecked her gray head
with sunshine.
"Annie," said Dr. Lavendar, "come here and sit beside me."
She came, happily enough, and let him take her hand and hold it,
patting it softly for a moment before he spoke.
"Annie, it was not right to give Harriet the stuff out of the bottle;
our Heavenly Father stops the hurting when He thinks best. So it does
not please Him for us to do it when we think best."
"But Willy gave Harriet one sugar in a paper, and that stopped it a
little," Miss Annie said, puzzled; "and if he stopped it a little, why
shouldn't it all be stopped?" The obvious logic of the poor mind
admitted of no answer--certainly no argument.
Dr. Lavendar said, gravely, stroking the hand, as wrinkled as his own:
"It was not right, my child. You will believe me when I say so? And I
do not want any one to know that you did a thing that was not right.
So I want you to promise me now that you will not tell any one that you
did it. Will you promise me?"
"Willy knows it, I guess," Miss Annie said.
Dr. Lavendar was silent. Just what had William heard her say? Only
that Miss Harriet was "dead." "I am pretty sure that Willy doesn't
know it," he said, slowly. "And I am quite sure he would prefer not to
know it; so you mustn't tell him. But you can't understand about that,
Annie. You'll just have to believe me. Will you promise me?"
"Why, yes," Miss Annie said, indifferently, smiling up at the moving
leaves. "Oh, Harriet isn't hurt now!"
Dr. Lavendar trembled with anxiety. "I want a solemn promise, Annie.
What do the children do when they make a solemn promise?"
Miss Annie was instantly interested. "Why, they cross their breast and
say 'honest and true'; don't you know?" ...
"Well, then," said Dr. Lavendar, slowly, "you will make a promise to me
in that way." He stood up and took her hand, his face very pale.
"Promise me that never, so long as you live, will you tell any one--any
one, Annie--that you made Harriet fall asleep by giving her the big
bottle to smell. Now, make the promise, Annie."
Miss Annie slowly crossed her breast. "I promise," she said, in a low
voice; her eyes, widening with awe, were fixed on his face. "I promise:
"Honest and true,
Black and black and blue,
Lay me down
And cut me in two--
if I do."
"_Amen!_" said Dr. Lavendar; and took off his hat, and stood looking up
into the sky, his lip trembling. "Father," he said, "I don't even say
'forgive her!' She is Thy little child." And then they stood for a
moment hand in hand in the sunny silence.
THE END
BY ONOTO WATANNA
A JAPANESE NIGHTINGALE. A love story of Japan. Full-page Drawings in
Color and unique Decorative Color Borders on every page, by the
well-known Japanese artist GENJIRO YETO. Crown 8vo, Ornamented Cloth,
Deckel Edges, Gilt Top (in a box), $2.00 net.
There could not easily be a more charming volume to look at than this,
nor a more delightfully appealing romance to read.--_New York World_.
An idyl of the author's homeland, delicate in fancy and dainty in
expression.--_Public Opinion_.
The author and the artist together have produced a charming work of
art, as thoroughly imbued with the Japanese spirit as a bit of old
Satsuma.--_Buffalo Express_.
"A Japanese Nightingale" is one of the daintiest and most exquisite of
love stories; ... indeed, so exquisite is her art, and so delightful
the humor of her pages, that more than one critic has spoken of the
story as "A Japanese Kentucky Cardinal."--_New York Journal_.
It is full of poetry and charm.--_Current Literature_.
A delicious vein of humor runs through the story, especially in the
love scenes, and the style is distinct with the lyrical delicacy of
Japanese thought.--_Brooklyn Eagle_.
HARPER & BROTHERS, PUBLISHERS
NEW YORK AND LONDON
_The above work will be sent by mail to any part of the United States,
Canada, or Mexico, on receipt of the price_. (_Postage 15 cents._)
BY ROBERT W. CHAMBERS
THE MAID-AT-ARMS. Illustrated by Howard Chandler Christy. Post 8vo,
Ornamented Cloth, $1.50.
Mr. Chambers has long since won a most enviable position among
contemporary novelists. The great popular success of "Cardigan" makes
this present novel of unusual interest to all readers of fiction. It
is a stirring novel of American life in days just after the Revolution.
It deals with the conspiracy of the great New York land-owners and the
subjugation of New York Province to the British. It is a story with a
fascinating love interest, and is alive with exciting incident and
adventure. Some of the characters of "Cardigan" reappear in this new
novel.
HARPER & BROTHERS, PUBLISHERS
NEW YORK AND LONDON
_The above work will be sent by mail, postage prepaid, to any part of
the United States, Canada, or Mexico, on receipt of the price_.
End of Project Gutenberg's Dr. Lavendar's People, by Margaret Deland
*** | {
"redpajama_set_name": "RedPajamaBook"
} | 3,964 |
Here's a cute set of vintage rococo scroll ornaments. I love the oval frame with the polka dots and rose vine border around it! The shell design is very interesting as well. My inner child wants to get out the crayons and start coloring. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,163 |
{"url":"http:\/\/stackoverflow.com\/questions\/13051749\/php-namespaces-and-classes","text":"# PHP namespaces and classes [closed]\n\nI have two classes, both in different namespaces. When i try to create the object from one namespace, then i get the error\n\nFatal error: Class 'Classes\\DrDatabase' not found in C:\\Path\\to\\workspace\\AplicationName\\app\\models\\Users.model.php on line XY\n\n\nIs there something i did not understand with php namespaces?\n\nthe two files:\n\nFILE 1\nnamespace Classes;\n\nclass DrDatabase extends MssqlDatabase {\n\npublic function __construct() {\nparent::createLog(__CLASS__);\nif ($this->link) {$this->log->info('Connected');\n}\n}\n}\n\nFILE 2\nnamespace Model;\n\nuse \\Classes\\DrDatabase;\n\nclass UsersModel implements MasterModel {\nprivate function get($id,$onlyEnabled) {\n$db = new DrDatabase();$getOnlyEnabled = 1;\nif (!$onlyEnabled)$getOnlyEnabled = 0;\n$SQL = \"SELECT * FROM table\";$resultSet = $db->query($SQL);\n$result = array(); while ($row = mssql_fetch_assoc($resultSet)) {$result[] = self::createUserObject($row['col1'],$row['col2']);\n}\n}\n}\n\n-\n\n## closed as too localized by hakre, rdlowrey, j0k, tere\u0161ko, JocelynOct 24 '12 at 23:47\n\nThis question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.\n\nTry it with a \"\\\" in front, like so: \\$db = new \\DrDatabase(); \u2013\u00a0 w00 Oct 24 '12 at 14:54\nYou can only instantiate classes that are defined. In your case, the definition of \\Classes\\DrDatabase has not been loaded. Load it and you're fine. That's just what the error message tells you and it is not lying. \u2013\u00a0 hakre Oct 24 '12 at 14:57\n\\Classes is a terrible name for a namespace. It doesn't tell you anything about what it contains. \u2013\u00a0 GordonM Oct 24 '12 at 16:30\nyes i know, that is just a working name, but only that one namespace has to be renamed. I'm not happy with \"model\" either, so i will be renaming them to something more meaningful \u2013\u00a0 Gabriel Oct 25 '12 at 6:47\n\nOtherwise - How would your UserModel know where to find the DrDatabase class?","date":"2015-03-28 16:27:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4812365472316742, \"perplexity\": 3894.109161884721}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-14\/segments\/1427131297622.30\/warc\/CC-MAIN-20150323172137-00088-ip-10-168-14-71.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
Studies of magneto-elastic coupling in intermetallic compounds have attracted considerable interest \cite{dung,wang,li}. Manganese telluride (MnTe) is a crossroad material between NiAs-type metallic transition-metal compounds and NaCl-type insulating manganese chalcogenides \cite{allen,bane,gos}. MnTe is a particularly interesting because it is a p-type semiconductor with a very high density of impurity charge carriers \cite{gos}. In spite of ordering antiferromagnetically with a Neel temperature, $T_N$ = 323K, it exhibits magnon drag effect \cite{was}. MnTe shows anomalies in transport and magnetic properties, like negative coefficient of resistance below 100 K and a sharp rise in susceptibility at around 83 K similar to a ferromagnetic transition which are closely related to its structural parameters \cite{efrem1}.
On the other hand, MnSe is an insulator with NaCl type structure at room temperature. Magnetic properties of MnSe show two antiferromagnetic transitions at about 265K and 130K respectively. Neutron diffraction studies have shown that MnSe undergoes a structural transition at about 270 K wherein a part of itself (~30\%) converts to NiAs phase which is antiferromagnetically ordered when it appears. While the high temperature cubic phase orders antiferromagnetically at 130K \cite{efrem2}. This partial transformation to NiAs phase can be attributed to presence of stacking faults in MnSe.
It is therefore of interest to explore the magnetic properties of solid solutions of MnTe and MnSe. Such attempts have been made in past which provide only a partial understanding of magnetic properties \cite{chehab,demi}. In light of new insights available on magnetic properties of MnTe and MnSe through neutron diffraction studies \cite{efrem1,efrem2} it would be interesting to revisit the solid solutions of the type MnTe$_x$Se$_{1-x}$, especially in the Se rich region and understand their magnetic properties.
In this paper we present detailed neutron diffraction studies on two compositions of MnTe - MnSe solid solutions. We have chosen MnTe$_{0.3}$Se$_{0.7}$ because 30\% of MnSe transforms to NiAs phase at 270K. The second composition chosen is MnTe$_{0.5}$Se$_{0.5}$ as it is on the NiAs to NaCl structural phase boundary in the phase diagram presented in Ref. \cite{chehab,demi}.
\section{Experimental}
The samples were prepared by mixing stoichiometric amounts of finely powdered Mn, Se, and Te and then pelletized and sealed in evacuated quartz ampoule below 10$^{-6}$ Torr. Subsequently these ampoules were slowly heated to 650$^\circ$C, annealed for 20 h and furnace-cooled. The samples were characterized by X-ray diffraction and were found to be single phase crystallizing in NaCl-type structure. Minor impurity peaks corresponding to NiAs type phase were noticed in MnTe$_{0.5}$Se$_{0.5}$. Magnetic susceptibility measurements were performed using an ac susceptometer in the temperature range 80 - 300 K in a field of 100 Oe. Neutron diffraction experiments were carried out in the temperature range 10 - 300 K and a wavelength of 1.24\AA~ using powder diffractometer at Dhruva, Trombay.
\section{Results}
Magnetic susceptibility of MnTe$_{0.3}$Se$_{0.7}$ and MnTe$_{0.5}$Se$_{0.5}$, measured as a function of temperature were very similar to those presented in \cite{demi} and showed broad transition with maximum at 130K and 120K respectively representing a transition from paramagnetic to antiferromagnetic state. Unlike in the case of MnSe \cite{efrem2}, no high temperature magnetic transition arising from the hexagonal NiAs phase was seen indicating a possibility of stable crystal structure in these two Te doped compounds. This is indeed interesting because the reported phase diagram shows a structural transformation from NaCl to NiAs type crystal structure up on Te doping. The transformation occurs close to 50\% doping \cite{chehab}. About 30\% of MnSe converts to antiferromagnetically ordered NiAs phase at around 270K \cite{efrem2} and therefore these doped systems should either have had NiAs type structure at room temperature or should have undergone a phase transformation from NaCl to NiAs phase at lower temperatures.
In order to understand this rather surprising magnetic behavior, temperature dependent neutron diffraction measurements on the the two polycrystalline MnTe$_x$Se$_{1-x}$ (x = 0.3 and 0.5) have been carried out. Rietveld refinements of the diffraction data at various temperatures were carried out using the FULLPROF suite \cite{car}. The neutron diffraction patterns recorded in the angular range of $3^\circ \le 2\theta \le 130^\circ$ at room temperature (RT) for the two compounds are presented in Figs. \ref{NDTe3Se7} and \ref{NDTe5Se5} respectively.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{Te3Se7NDRT.eps}
\caption{\label{NDTe3Se7} Rietveld refined ND pattern of MnTe$_{0.3}$Se$_{0.7}$ at RT. The open circles show the observed counts and the continuous line passing through these counts is the calculated profile. The difference between the observed and calculated patterns is shown as a continuous line at the bottom of the two profiles. The calculated positions of the reflections are shown as vertical bars. The $R$-factors obatained were $R_P$ = 5.67, $R_{wp}$ = 8.36 and $R_{exp}$ = 5.13}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{Te5Se5NDRT.eps}
\caption{\label{NDTe5Se5} Rietveld refined ND pattern of MnTe$_{0.5}$Se$_{0.5}$ at RT. The open circles show the observed counts and the continuous line passing through these counts is the calculated profile. The difference between the observed and calculated patterns is shown as a continuous line at the bottom of the two profiles. The calculated positions of the reflections are shown as vertical bars. The $R$-factors obtained were $R_P$ = 7.14, $R_{wp}$ = 10.31 and $R_{exp}$ = 5.79}
\end{figure}
MnTe$_{0.3}$Se$_{0.7}$ (Fig. \ref{NDTe3Se7}) presents a single phase NaCl type structure with lattice constant $a$ = 5.581 \AA. No additional peaks corresponding to hexagonal NiAs phase are visible. This is in agreement with the room temperature structure obtained from X-ray diffraction (not shown). In Fig. \ref{NDTe5Se5}, the diffraction pattern shows presence of a minor impurity phase in addition to major NaCl type cubic phase. The impurity phase can be refined with NiAs type hexagonal phase. The chemical composition of the hexagonal phase is similar to the nominal MnTe$_{0.5}$Se$_{0.5}$ and its relative composition is about 10\%. The presence of NiAs phase in the RT pattern is not surprising as this composition is close to the NaCl to NiAs transformation boundary its \% composition is much less. In MnSe, a partial conversion to NiAs phase occurs at 270K with about 30\% of the sample converting to hexagonal structure. Te doping should have facilitated formation of NiAs type phase. It may be mentioned here that amount of hexagonal phase does not change with lowering in temperature. Another point to be noted is that appearance of hexagonal phase in MnSe was also associated with its antiferromagnetic ordering. No peaks corresponding to antiferromagnetic ordering of the hexagonal phase are visible in the ND pattern recorded at RT.
In MnTe$_{0.3}$Se$_{0.7}$, with lowering of temperature to about 120K, a weak peak at exactly half the angle (2$\theta$ $\sim$ 11$^\circ$) of the most intense Bragg reflection emerges (Fig \ref{Te3Se7ndcompare}). This corresponds to antiferromagnetic ordering of the NaCl type parent phase with $T_N$ lying at about 130K. The appearance of magnetic (111) reflection suggests G-type antiferromagnetic order wherein Mn spins are aligned antiferromagnetically along the [111] direction. Similar spin alignment was also seen in MnSe. Between 100K and 50K the magnetic reflections grow in intensity indicating build up of Mn magnetic moment with decrease in temperature. However, with further lowering of temperature to 40K, the magnetic Bragg peak nearly disappears only to reemerge much more strongly at 30K. Additional reflections are also visible at this temperature and the diffraction pattern can only be refined by taking into account a small fraction of NiAs type hexagonal phase. This hexagonal phase is magnetically ordered at the temperature it appears with an antiparallel alignment of Mn spins along $c$ axis. This is again an interesting result and as will be discussed later, its origin lies in magnetostructural coupling.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Te3Se7NDcompare.eps}
\caption{\label{Te3Se7ndcompare} A comparison of neutron diffractions patterns of MnTe$_{0.3}$Se$_{0.7}$ recorded at various temperatures. The data is presented in limited 2$\theta$ range for clarity. Vertical bars shown at the bottom of the figure represent the positions of Bragg reflections calculated for cubic and hexagonal phases (nuclear and magnetic).}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Te5Se5NDCompare.eps}
\caption{\label{Te5Se5ndcompare} A comparison of neutron diffractions patterns of MnTe$_{0.5}$Se$_{0.5}$ recorded at various temperatures. The data is presented in limited 2$\theta$ range for clarity. Vertical bars presented at the bottom correspond to Bragg positions calculated for nuclear and magnetic contributions of cubic and magnetic phases.}
\end{figure}
The minor hexagonal phase present in MnTe$_{0.5}$Se$_{0.5}$ can be seen in Fig. \ref{Te5Se5ndcompare}, to order antiferromagnetically at about 270K. The hexagonal phase of MnSe also orders around the same temperature \cite{efrem2}. The major cubic phase orders antiferromagnetically at 100K. This is clearly evident from additional magnetic reflections that can be seen in diffraction patterns recorded below 265K and 100K respectively. There is no difference between the spin alignments of the two constituent phases in this compound and those found in parent MnSe suggesting thereby the mechanism of antiferromagnetic alignment to be same in these NaCl type Mn-chalcogenide semiconductors.
The low temperature patterns were Rietveld refined using parameters obtained at RT as inputs. The parameters refined were cell parameters and magnetic moment parameters. The variation of cell parameters for each of the phases present in both the compounds is depicted in Fig. \ref{lattice}. In the case of MnTe$_{0.5}$Se$_{0.5}$ the lattice parameters of both cubic and hexagonal phases show nearly monotonic decrease with temperature. Anomalies seen around 270K and 110K can be identified with magnetic ordering of cubic and hexagonal phases. Like in case of MnSe, the hexagonal phase is identified with stacking faults along the $c$-axis \cite{and,jac}. These faults arise due to the incomplete structural transition of MnSe wherein only $\sim$ 30\% of the compound is converted to hexagonal phase below 270K. In the present case, about 10\% of NiAs type phase is present throughout the temperature range studied here. This hexagonal phase orders antiferromagnetically below 270K. The anomalies present in lattice parameters at the magnetic ordering temperatures hint towards a correlation between magnetic and structural degrees of freedom.
The neutron diffraction patterns of MnTe$_{0.3}$Se$_{0.7}$ provide an interesting trend. The compound exhibits purely cubic structure at RT and does not undergo any structural transition down to 40K. This is irrespective of the fact that 30\% of Se is replaced with Te and in MnSe almost about the same fraction of the sample converts to hexagonal phase below 270K. Furthermore, it must be mentioned that MnTe crystallizes in NiAs type hexagonal structure \cite{a}. However, the variation of lattice parameters exhibit very distinct anomalies. The cubic cell parameter shows a step like increase at about 270K which corresponds to the structural transition temperature in MnSe. A small kink like structure is visible at about 130K which corresponds to antiferromagnetic ordering temperature of the cubic phase. Further down, another strong anomaly is visible at about 40K which is just above the appearance of hexagonal phase. It is at this temperature, that magnetic ordering of the cubic phase was nearly destroyed (see Fig. \ref{Te3Se7ndcompare}). The variation of Mn magnetic moment presented below will make this point even more clear. Nevertheless, the intimate relationship between lattice parameters, and magnetic ordering indicates presence of magnetostructural coupling in both these compounds.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Latticeparameter.eps}
\caption{\label{lattice} Evolution of lattice parameters of cubic and hexagonal phases as a function of temperature in MnTe$_{x}$Se$_{1-x}$, $x$ = 0.3 and 0.5.}
\end{figure}
Variation of magnetic moment per Mn ion in both, MnTe$_{0.3}$Se$_{0.7}$ and MnTe$_{0.5}$Se$_{0.5}$, obtained from Rietveld refinement of low temperature neutron diffraction data is presented in Fig. \ref{magmom}. In MnTe$_{0.5}$Se$_{0.5}$, the magnetic moment of both the phases increases with lowering of temperature. In case of hexagonal phase Mn ion exhibits near saturation below 100K and acquires a moment of about 1.5 $\mu_B$ at 12K. In the case of cubic phase however, the magnetic moment seems to continuously rise reaching about 1.0 $\mu_B$ at the lowest investigated temperature. In the other compound, the magnetic moment of the cubic phase remains quite low ($\sim$ 0.5 $\mu_B$) till about 50K below which it shows a sharp fall only to rise even more sharply at lower temperatures. Here the Mn ion acquires a maximum moment of 1.4 $\mu_B$ at the lowest temperature. Below 40K, there is also co-existing antiferromagnetically ordered hexagonal phase and the magnetic moment of Mn ions belonging to this phase show a sharp increase reaching a maximum value of about 2.5 $\mu_B$. A comparison of magnetic moment in MnSe and the present two compositions suggests an interesting correlation. In MnSe, the magnetic moment per Mn ion at 12K in both the phases was reported to be 3.3 $\mu_B$. With addition of Te in place of Se, the magnetic moment per Mn ion steadily decreases to 1.5 $\mu_B$ and 1.0 $\mu_B$ in hexagonal and cubic phases respectively in MnTe$_{0.5}$Se$_{0.5}$.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{magmom.eps}
\caption{\label{magmom} Variation of magnetic moment per Mn atom as a function of temperatures in MnTe$_{0.3}$Se$_{0.7}$ and MnTe$_{0.5}$Se$_{0.5}$.}
\end{figure}
\section{Discussion}
MnTe$_{x}$Se$_{1-x}$ exhibits structural transition from NaCl cubic to NiAs hexagonal structure with increasing Te doping. Along with this structural transition the covalency between Mn $3d$ and chalcogen $p$ band is also affected giving rise unusual behavior of transport properties like magnon drag effect in MnTe \cite{a}. Although both MnSe and MnTe are antiferromagnets with high enough ordering temperatures, interplay between ferromagnetic and antiferromagnetic interactions has been reported in MnTe \cite{efrem1} and transition metal doped MnTe \cite{li}. The NiAs structure is a varient of hexagonal close packed structure and has a $c/a$ ratio very close to the ideal value of $\sqrt{8/3}$.
A hexagonal structure can be visualized in a NaCl unit cell with the lattice parameters satisfying a relation, $a_{hexa} = a_{cubic}/\sqrt2$ and $c_{hexa} = \sqrt3 a_{cubic}/\sqrt2$. This gives a $c/a$ ratio of $\sqrt 3$ which is slightly higher than the ideal $c/a$ ratio normally satisfied by a NiAs structure. If we compare the lattice parameters of the two phases present in MnSe \cite{efrem2}, it can clearly be seen that $c$ parameter of the hexagonal phase is distinctly less than the one calculated from the cell parameter of NaCl cell. This mismatch in lattice parameters is perhaps the reason for the presence of stacking faults in NiAs type phase. In MnTe$_{0.5}$Se$_{0.5}$, hexagonal unit cell parameters are closer to the ones calculated from cubic cell constant over the entire temperature range and hence there is not much broadening of hexagonal peaks due to stacking faults. Interesting aspect however is that in spite of 50\% Te content, the amount of hexagonal phase is only about 10\%.
Surprising aspect however, is the near absence of hexagonal phase in MnTe$_{0.3}$Se$_{0.7}$. However, anomalies in lattice parameters are visible at 270K - magnetic ordering temperature of hexagonal phase in MnSe as well as MnTe$_{0.5}$Se$_{0.5}$. It is indeed surprising that hexagonal phase is not visible in this compound to temperatures down to 40K and could be related to the lattice parameter of the cubic phase. It is possible that c/a ratio of the hexagonal phase in this compound is greater than the ideal value of 1.63 and closer to $\sqrt 3$. In such a case the transformation of part of the compound to hexagonal phase will not be visible in presence of NaCl type cubic phase. With lowering of temperatures, the evolution of lattice parameters of the two phases changes and an antiferromagnetic NiAs type phase appears below 40K. The presence of competing cubic and hexagonal phases could be also the reason for a very small increase in magnetic moment of the cubic phase until the appearance of hexagonal phase. This also indicates the presence of magnetostructural coupling in this compound.
Antiferromagnetic ordering is prevalent in both the samples. The NiAs phase has type I ordering with Mn moments ferromagnetically aligned in the basal plane and these planes stacked antiferromagnetically along $c$ axis. The ordering temperature of the hexagonal phase in MnTe$_{0.5}$Se$_{0.5}$ is 270K which is nearly the same as in MnSe. In MnTe$_{0.3}$Se$_{0.7}$ the hexagonal phase is antiferromagnetic at the temperature it appears (T $\approx$ 30K). The parent cubic phase also orders antiferromagnetically and the antiferromagnetic ordering temperature decreases with increasing Te content. This weakening of magnetic interactions in the cubic phase is linked to the structure. The expansion of cubic lattice parameter with addition of Te, facilitates the transformation from cubic to hexagonal structure at higher Te contents.
\section{Conclusions}
Crystal and magnetic structure of Se rich MnTe$_x$Se$_{1-x}$ have been studied using neutron diffraction. The sample with $x$ = 0.5 is close to R.T. structural phase transition boundary between NaCl type cubic to NiAs type hexagonal phase. NiAs phase is absent in MnTe$_{0.3}$Se$_{0.7}$ at higher temperatures and only appears at temperatures below 40K. This is because of the unique relationship between the lattice parameters of the cubic NaCl and hexagonal NiAs phases. Both the phases order antiferromagnetically. The magnetic ordering temperature of NiAs phase is 270K and is always magnetically ordered at T $<$ 270K it appears. The N$\acute{\rm e}$el temperature of the cubic phase slightly decreases with increasing Te content. The magnetic and structural transitions in these mixed chalcogenides are driven by a strong coupling between magnetic and structural degrees of freedom.
\section*{Acknowledgements}
Department of Science and Technology (DST), Government of India is acknowledged for financial support under the project No. SR/S2/CMP-57.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,802 |
"""Support for UK Met Office weather service."""
import logging
from homeassistant.components.weather import WeatherEntity
from homeassistant.const import LENGTH_KILOMETERS, TEMP_CELSIUS
from homeassistant.core import callback
from homeassistant.helpers.typing import ConfigType, HomeAssistantType
from .const import (
ATTRIBUTION,
CONDITION_CLASSES,
DEFAULT_NAME,
DOMAIN,
METOFFICE_COORDINATOR,
METOFFICE_DATA,
METOFFICE_NAME,
VISIBILITY_CLASSES,
VISIBILITY_DISTANCE_CLASSES,
)
_LOGGER = logging.getLogger(__name__)
async def async_setup_entry(
hass: HomeAssistantType, entry: ConfigType, async_add_entities
) -> None:
"""Set up the Met Office weather sensor platform."""
hass_data = hass.data[DOMAIN][entry.entry_id]
async_add_entities(
[
MetOfficeWeather(
entry.data,
hass_data,
)
],
False,
)
class MetOfficeWeather(WeatherEntity):
"""Implementation of a Met Office weather condition."""
def __init__(self, entry_data, hass_data):
"""Initialise the platform with a data instance."""
self._data = hass_data[METOFFICE_DATA]
self._coordinator = hass_data[METOFFICE_COORDINATOR]
self._name = f"{DEFAULT_NAME} {hass_data[METOFFICE_NAME]}"
self._unique_id = f"{self._data.latitude}_{self._data.longitude}"
self.metoffice_now = None
@property
def name(self):
"""Return the name of the sensor."""
return self._name
@property
def unique_id(self):
"""Return the unique of the sensor."""
return self._unique_id
@property
def condition(self):
"""Return the current condition."""
return (
[
k
for k, v in CONDITION_CLASSES.items()
if self.metoffice_now.weather.value in v
][0]
if self.metoffice_now
else None
)
@property
def temperature(self):
"""Return the platform temperature."""
return (
self.metoffice_now.temperature.value
if self.metoffice_now and self.metoffice_now.temperature
else None
)
@property
def temperature_unit(self):
"""Return the unit of measurement."""
return TEMP_CELSIUS
@property
def visibility(self):
"""Return the platform visibility."""
_visibility = None
if hasattr(self.metoffice_now, "visibility"):
_visibility = f"{VISIBILITY_CLASSES.get(self.metoffice_now.visibility.value)} - {VISIBILITY_DISTANCE_CLASSES.get(self.metoffice_now.visibility.value)}"
return _visibility
@property
def visibility_unit(self):
"""Return the unit of measurement."""
return LENGTH_KILOMETERS
@property
def pressure(self):
"""Return the mean sea-level pressure."""
return (
self.metoffice_now.pressure.value
if self.metoffice_now and self.metoffice_now.pressure
else None
)
@property
def humidity(self):
"""Return the relative humidity."""
return (
self.metoffice_now.humidity.value
if self.metoffice_now and self.metoffice_now.humidity
else None
)
@property
def wind_speed(self):
"""Return the wind speed."""
return (
self.metoffice_now.wind_speed.value
if self.metoffice_now and self.metoffice_now.wind_speed
else None
)
@property
def wind_bearing(self):
"""Return the wind bearing."""
return (
self.metoffice_now.wind_direction.value
if self.metoffice_now and self.metoffice_now.wind_direction
else None
)
@property
def attribution(self):
"""Return the attribution."""
return ATTRIBUTION
async def async_added_to_hass(self) -> None:
"""Set up a listener and load data."""
self.async_on_remove(
self._coordinator.async_add_listener(self._update_callback)
)
self._update_callback()
@callback
def _update_callback(self) -> None:
"""Load data from integration."""
self.metoffice_now = self._data.now
self.async_write_ha_state()
@property
def should_poll(self) -> bool:
"""Entities do not individually poll."""
return False
@property
def available(self):
"""Return if state is available."""
return self.metoffice_now is not None
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,357 |
Q: Error while trying to start PostgreSQL installed via Homebrew: "Operation not permitted" I have recently installed PostgreSQL on my Mac via Homebrew. (I may have already had it installed; it was not running.)
brew install postgres
ln -sfv /usr/local/opt/postgresql/*.plist ~/Library/LaunchAgents
Now I am trying to start it with launchctl:
launchctl load ~/Library/LaunchAgents/homebrew.mxcl.postgresql.plist
...but I am getting an error:
/usr/local/Cellar/postgresql/9.4.0/homebrew.mxcl.postgresql.plist: Operation not permitted
What does this error mean? What am I doing wrong? How can I fix the problem?
A: It could be that you're using launchctl inside of Tmux or Screen.
Tmux and Screen are terminal multiplexers that spawn multiple "screens" that you can easily switch between in a single terminal.
For some reason unknown to me, running launchctl inside of Tmux never works, and emits the error Operation not permitted. Run it inside of a normal shell and it will probably work just fine.
A: Here are the steps you may need to take:
Remove a previous installation of PostgreSQL:
brew remove postgres
rm ~/Library/LaunchAgents/homebrew.mxcl.postgresql.plist
Install the new version:
brew install postgres
ln -sfv /usr/local/opt/postgresql/*.plist ~/Library/LaunchAgents
The data from your previous installation will need to be upgraded to be compatible with PostgreSQL 9.4+: http://www.postgresql.org/docs/9.4/static/upgrading.html
It seems like you need two installations of PostgreSQL in order to upgrade your database, and I didn't care to bother with that, so I just recreated the database with the new version:
mv /usr/local/var/postgres /usr/local/var/old-postgres
initdb -D /usr/local/var/postgres
Now launch PostgreSQL (outside of tmux if you are using that):
launchctl load ~/Library/LaunchAgents/homebrew.mxcl.postgresql.plist
Check the logs for any issues:
tail /usr/local/var/postgres/server.log
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,662 |
Sub-categories of "economy"
economic development:
Economic Partnership Agreements (EPA)
economic development …
cash transfer
low and middle income countries (LMIC)
income …
infrastructure …
microeconomic
political economy:
political economy …
trade …
Sub-categories of "private sector"
transnational corporation
Filtering for small and medium enterprise (SME) (remove)
Acceptability of E-Filing of Taxes by Micro-Entrepreneurs in Northwestern Nigeria
2019/05 – Institute of Development Studies (IDS), Brighton; research paper; Author(s): Abdulsalam Mas'ud
Details for "Acceptability of E-Filing of Taxes by Micro-Entrepreneurs in Northwestern Nigeria"
Nacionalna mreža srednjih preduzeća u Republici Srbiji (NMSP) Vol.3 (PDF)
2018/10 – Institute of Economic Sciences (IEN); Journal: The National Network of Medium Enterprises in the Republic of Serbia (NMSP); Editor(s): Slavica Stevanović et al.
Details for "Nacionalna mreža srednjih preduzeća u Republici Srbiji (NMSP) Vol.3 (PDF)"
Informal Sector Innovation in Ghana: Data Set and Descriptive Analysis
2018/08 – United Nations University, Maastricht Economic and Social Research Institute on Innovation and Technology (UNU-MERIT); Working Paper 2018-030; Author(s): Elvis Avenyo
Details for "Informal Sector Innovation in Ghana: Data Set and Descriptive Analysis"
Financial Power and Development Potential of Environmentally Responsible Medium Sized Enterprises in the Serbian Industrial Sector
2018/05 – Institute of Economic Sciences (IEN); Sustainable Growth and Development in Small Open Economies, pp.124-142; Author(s): Sonja Đuričin, Isidora Beraha
Details for "Financial Power and Development Potential of Environmentally Responsible Medium Sized Enterprises in the Serbian Industrial Sector"
Financial Inclusion and Women Entrepreneurship: Evidence From Mexico
2017/09 – OECD Development Centre (OECD/DC); OECD Economics Department Working Papers No. 1411; Author(s): Fozan Fareed, Mabel Gabriel, Patrick Lenain, Julien Reynaud
Details for "Financial Inclusion and Women Entrepreneurship: Evidence From Mexico"
Internationalization Strategies of African Firms: A Survey of 210 Food Processing Firms From Tanzania, Kenya and Zambia
2017/12 – Copenhagen Business School, Department of Intercultural Communication and Management (IKL); Special CBDS Working Paper Series; Author(s): Amelie Schmidt, Michael Wendelboe Hansen
Details for "Internationalization Strategies of African Firms: A Survey of 210 Food Processing Firms From Tanzania, Kenya and Zambia"
Differentials in Market Constraints and Value Addition Among Micro, Small, and Medium Enterprises in Viet Nam
2017/04 – United Nations University World Institute for Development Economics Research (UNU-WIDER); WIDER Working Paper 82/2017; Author(s): Christine Ngoc Ngo, Miao Chi
Details for "Differentials in Market Constraints and Value Addition Among Micro, Small, and Medium Enterprises in Viet Nam"
The Impact of Stress and Health on Quality of Working Life of Women in SMEs in Republic of Serbia (PDF)
2017/07 – Institute of Economic Sciences (IEN); Oliver Momčilović et al.; Author(s): zorica.bozic
Details for "The Impact of Stress and Health on Quality of Working Life of Women in SMEs in Republic of Serbia (PDF)"
Redefining Inclusive Growth in Asia: How APEC Can Achieve an Economy That Leaves No One Behind
2017/11 – Oxfam GB; Briefing paper; Author(s): Maria Dolores Bernabe
Details for "Redefining Inclusive Growth in Asia: How APEC Can Achieve an Economy That Leaves No One Behind"
Financial Education for MSMEs and Potential Entrepreneurs
2017/09 – OECD Development Centre (OECD/DC); OECD Working Papers on Finance, Insurance and Private Pensions No. 43; Author(s): Adele Atkinson
Details for "Financial Education for MSMEs and Potential Entrepreneurs" | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,977 |
Q: What is the best way to change a user-password remotely in Unix? What is the best way to change a user-password remotely in Unix?
This must be performed by the user, in a Web-app or Windows-App, without using SSH or any direct connection between the user and the server (direct command line not allowed).
Thanks
Webmin seemed to be a good application to do that, but I found it extremely hard to configure it right. My Unix users are unable to login to Webmin or Usermin.
Do you know any other alternatives to Webmin and Usermin?
Thanks
A: Use Webmin (more specifically the UserMin module).
Webmin provides a mini webserver, so you just need to install and configure it slightly. You'll get a lot more than just password-changing, and you can remove functionality you don't want the user to have.
A: @Rich Bradshaw
Just make sure you don't introduce security issues. The solution should use https encryption (the password should be never sent in clear text). It should be protected against shell injection attacks (strip any newlines from input, escape it properly etc). More details depend on choosen implementation.
A: I've done this in the past to change passwords on several servers at once by using a script written in Expect. It's perfect for the job but you will need the servers to be listening via SSH.
Once written, the script will execute on your local workstation and will connect to the remote host, do the interaction you've scripted, and then you should be gold. All the while, using the encryption you're already trusting if you're running SSH. Just don't save the passwords in your script: you should be able to prompt yourself for them (even taking them by command line argument is generally considered poor practice.)
Expect is a great language too: lots of fun!
A: You could write a server side script that ran passwd, you could do that in any language that allows shell commands to be run.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,803 |
'use strict';
describe('ReactDOMEventListener', () => {
let React;
let ReactDOM;
beforeEach(() => {
jest.resetModules();
React = require('react');
ReactDOM = require('react-dom');
});
it('should dispatch events from outside React tree', () => {
const mock = jest.fn();
const container = document.createElement('div');
const node = ReactDOM.render(<div onMouseEnter={mock} />, container);
const otherNode = document.createElement('h1');
document.body.appendChild(container);
document.body.appendChild(otherNode);
otherNode.dispatchEvent(
new MouseEvent('mouseout', {
bubbles: true,
cancelable: true,
relatedTarget: node,
}),
);
expect(mock).toBeCalled();
});
describe('Propagation', () => {
it('should propagate events one level down', () => {
const mouseOut = jest.fn();
const onMouseOut = event => mouseOut(event.currentTarget);
const childContainer = document.createElement('div');
const parentContainer = document.createElement('div');
const childNode = ReactDOM.render(
<div onMouseOut={onMouseOut}>Child</div>,
childContainer,
);
const parentNode = ReactDOM.render(
<div onMouseOut={onMouseOut}>div</div>,
parentContainer,
);
parentNode.appendChild(childContainer);
document.body.appendChild(parentContainer);
const nativeEvent = document.createEvent('Event');
nativeEvent.initEvent('mouseout', true, true);
childNode.dispatchEvent(nativeEvent);
expect(mouseOut).toBeCalled();
expect(mouseOut.mock.calls.length).toBe(2);
expect(mouseOut.mock.calls[0][0]).toEqual(childNode);
expect(mouseOut.mock.calls[1][0]).toEqual(parentNode);
document.body.removeChild(parentContainer);
});
it('should propagate events two levels down', () => {
const mouseOut = jest.fn();
const onMouseOut = event => mouseOut(event.currentTarget);
const childContainer = document.createElement('div');
const parentContainer = document.createElement('div');
const grandParentContainer = document.createElement('div');
const childNode = ReactDOM.render(
<div onMouseOut={onMouseOut}>Child</div>,
childContainer,
);
const parentNode = ReactDOM.render(
<div onMouseOut={onMouseOut}>Parent</div>,
parentContainer,
);
const grandParentNode = ReactDOM.render(
<div onMouseOut={onMouseOut}>Parent</div>,
grandParentContainer,
);
parentNode.appendChild(childContainer);
grandParentNode.appendChild(parentContainer);
document.body.appendChild(grandParentContainer);
const nativeEvent = document.createEvent('Event');
nativeEvent.initEvent('mouseout', true, true);
childNode.dispatchEvent(nativeEvent);
expect(mouseOut).toBeCalled();
expect(mouseOut.mock.calls.length).toBe(3);
expect(mouseOut.mock.calls[0][0]).toEqual(childNode);
expect(mouseOut.mock.calls[1][0]).toEqual(parentNode);
expect(mouseOut.mock.calls[2][0]).toEqual(grandParentNode);
document.body.removeChild(grandParentContainer);
});
// Regression test for https://github.com/facebook/react/issues/1105
it('should not get confused by disappearing elements', () => {
const container = document.createElement('div');
document.body.appendChild(container);
class MyComponent extends React.Component {
state = {clicked: false};
handleClick = () => {
this.setState({clicked: true});
};
componentDidMount() {
expect(ReactDOM.findDOMNode(this)).toBe(container.firstChild);
}
componentDidUpdate() {
expect(ReactDOM.findDOMNode(this)).toBe(container.firstChild);
}
render() {
if (this.state.clicked) {
return <span>clicked!</span>;
} else {
return <button onClick={this.handleClick}>not yet clicked</button>;
}
}
}
ReactDOM.render(<MyComponent />, container);
container.firstChild.dispatchEvent(
new MouseEvent('click', {
bubbles: true,
}),
);
expect(container.firstChild.textContent).toBe('clicked!');
document.body.removeChild(container);
});
it('should batch between handlers from different roots', () => {
const mock = jest.fn();
const childContainer = document.createElement('div');
const handleChildMouseOut = () => {
ReactDOM.render(<div>1</div>, childContainer);
mock(childNode.textContent);
};
const parentContainer = document.createElement('div');
const handleParentMouseOut = () => {
ReactDOM.render(<div>2</div>, childContainer);
mock(childNode.textContent);
};
const childNode = ReactDOM.render(
<div onMouseOut={handleChildMouseOut}>Child</div>,
childContainer,
);
const parentNode = ReactDOM.render(
<div onMouseOut={handleParentMouseOut}>Parent</div>,
parentContainer,
);
parentNode.appendChild(childContainer);
document.body.appendChild(parentContainer);
const nativeEvent = document.createEvent('Event');
nativeEvent.initEvent('mouseout', true, true);
childNode.dispatchEvent(nativeEvent);
// Child and parent should both call from event handlers.
expect(mock.mock.calls.length).toBe(2);
// The first call schedules a render of '1' into the 'Child'.
// However, we're batching so it isn't flushed yet.
expect(mock.mock.calls[0][0]).toBe('Child');
// The first call schedules a render of '2' into the 'Child'.
// We're still batching so it isn't flushed yet either.
expect(mock.mock.calls[1][0]).toBe('Child');
// By the time we leave the handler, the second update is flushed.
expect(childNode.textContent).toBe('2');
document.body.removeChild(parentContainer);
});
});
it('should not fire duplicate events for a React DOM tree', () => {
const mouseOut = jest.fn();
const onMouseOut = event => mouseOut(event.target);
class Wrapper extends React.Component {
getInner = () => {
return this.refs.inner;
};
render() {
const inner = <div ref="inner">Inner</div>;
return (
<div>
<div onMouseOut={onMouseOut} id="outer">
{inner}
</div>
</div>
);
}
}
const container = document.createElement('div');
const instance = ReactDOM.render(<Wrapper />, container);
document.body.appendChild(container);
const nativeEvent = document.createEvent('Event');
nativeEvent.initEvent('mouseout', true, true);
instance.getInner().dispatchEvent(nativeEvent);
expect(mouseOut).toBeCalled();
expect(mouseOut.mock.calls.length).toBe(1);
expect(mouseOut.mock.calls[0][0]).toEqual(instance.getInner());
document.body.removeChild(container);
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,768 |
Poängtabell
Spanien mot Japan
Honduras mot Marocko
Spanien mot Honduras
Japan mot Marocko
Japan mot Honduras
Spanien mot Marocko
Grupp D | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,783 |
/**
* @fileOverview
* @author seanxphuang
* @version 1
* Created: 13-9-23 上午11:00
*/
LBF.define('ui.widget.FileUploader.FileUploader', function(require){
var $ = require('lib.jQuery'),
Node = require('ui.Nodes.Node');
require('{theme}/lbfUI/css/FileUploader.css');
if ( window.FormData ) {
require.async(['ui.widget.FileUploader.ajaxUpload.ajaxUpload']);
}
/**
* 文件上传组件,会根据浏览器不同选用ajax、iframe、flash等三种上传方式.
* @class FileUploader
* @namespace ui.widget
* @module ui
* @submodule ui-widget
* @extends ui.Nodes.Node
* @constructor
* @param {Object} opts Options for initialization
* @param {String} opts.url url地址,必需,指定后台cgi
* @param {String|Dom|$} [opts.selector] 对象或选择符,指定触发按钮的初始化对象,一个实例可以有多个触发按钮
* @param {String|Dom|$} [opts.container] 对象或选择符,指定生成触发按钮的容器,一个实例可以有多个触发按钮
* @param {String} [opts.flashUrl] 字符串,指定swfupload文件路径,默认跟url参数指定路径同目录
* @param {Boolean} [opts.singleUpload] 布尔值,单文件还是多文件上传
* @param {String} [opts.dropbox] 选择符,文件拖拽上传时的感应容器
* @param {Number} [opts.triggerWidth] 数值,指定触发按钮的宽度
* @param {Number} [opts.triggerHeight] 数值,指定触发按钮的高度
* @param {String} [opts.file_size_limit] 字符串,限定文件大小,'0' | '20 b/kb/mb/gb',默认为0表示不限制大小,默认单位KB
* @param {String} [opts.file_types] 字符串,限定文件类型,默认为 "*.jpg;*.gif;*.jpeg;*.png;*.bmp"
* @param {String} [opts.buttonText] 字符串,指定触发按钮上显示的文字
* @param {Object} [opts.data] 对象,附加自定义参数
* @example
* new FileUploader({
* url: 'demo.php',
* flashUrl: 'swfupload.swf',
* dropbox: '.dragdrop',
* selector: '.lbf-ImageCrop',
* data: {
* user: 'sean',
* age: 18
* },
* container: '.lbf-ImageCrop',
* singleUpload: true,
* file_size_limit: '200kb'
* })
*/
var FileUploader = Node.inherit({
initialize: function(opts){
var node = this;
node.mergeOptions(opts);
var settings = node.attributes();
if ( window.FormData ) {
require.async(['ui.widget.FileUploader.ajaxUpload.ajaxUpload'], function(ajaxUpload){
var instance = new ajaxUpload(settings);
node['instance'] = instance;
node._attachEvent();
node.trigger('loaded');
});
} else if( !settings.singleUpload ) {
require.async(['ui.widget.FileUploader.swfUpload.init'], function(swfUpload){
var instance = new swfUpload(settings);
node['instance'] = instance;
node._attachEvent();
node.trigger('loaded');
});
} else {
require.async(['ui.widget.FileUploader.iframeUpload.iframeUpload'], function(iframeUpload){
var instance = new iframeUpload(settings);
node['instance'] = instance;
node._attachEvent();
node.trigger('loaded');
});
}
},
/**
* 取消上传
* @method cancel
*/
cancel: function(){
var node = this;
if( !node['instance']){
node.bind('loaded', function(){
node['instance'].cancel();
});
} else {
node['instance'].cancel();
node.cancel = function(){
var node = this;
node['instance'].cancel();
};
}
},
/**
* 禁用上传
* @method disable
* @param {Number} [index] 指定第几个按钮被禁用, 不传入会禁用所有按钮
*/
disable: function(index){
var node = this;
if( !node['instance']){
node.bind('loaded', function(){
node['instance'].disable(index);
});
} else {
node['instance'].disable(index);
node.disable = function(index){
var node = this;
node['instance'].disable(index);
};
}
},
/**
* 启用上传
* @method enable
* @param {Number} [index] 指定第几个按钮被启用,不传入会启用所有按钮
*/
enable: function(index){
var node = this;
if( !node['instance']){
node.bind('loaded', function(){
node['instance'].enable(index);
});
} else {
node['instance'].enable(index);
node.enable = function(index){
var node = this;
node['instance'].enable(index);
};
}
},
_attachEvent: function(){
var node = this;
this['instance'].bind('uploadStart', function(){
var args = Array.prototype.slice.call(arguments, 1);
/**
* Fire when upload start
* @event uploadStart
* @param {Event} event JQuery event
* @param {Node} node Node object
* @param {Object} stats uploaded files stats
*/
node.trigger('uploadStart', args);
});
this['instance'].bind('uploadCancel', function(){
var args = Array.prototype.slice.call(arguments, 1);
/**
* Fire when upload canceled
* @event cancel
* @param {Event} event JQuery event
* @param {Node} node Node object
*/
node.trigger('uploadCancel', args);
});
this['instance'].bind('uploadError', function(){
var args = Array.prototype.slice.call(arguments, 1);
/**
* Fire when upload error
* @event uploadError
* @param {Event} event JQuery event
* @param {Object} opts server response data
* @param {Object} stats uploaded files stats
*/
node.trigger('uploadError', args);
});
this['instance'].bind('uploadSuccess', function(){
var args = Array.prototype.slice.call(arguments, 1);
/**
* Fire when upload successed
* @event uploadSuccess
* @param {Event} event JQuery event
* @param {Object} opts server response data
* @param {Object} opts.r status code
* @param {Object} opts.msg message
* @param {Object} opts.data {url: '', show_url:''}
* @param {Object} stats uploaded files stats
*/
node.trigger('uploadSuccess', args);
});
this['instance'].bind('uploadComplete', function(){
var args = Array.prototype.slice.call(arguments, 1);
/**
* Fire when upload complete
* @event uploadComplete
* @param {Event} event JQuery event
* @param {Object} opts server response data
* @param {Object} stats uploaded files stats
*/
node.trigger('uploadComplete', args);
});
}
});
FileUploader.include({
/**
* Default settings
* @property settings
* @type Object
* @static
* @protected
*/
settings: {
//单文件还是多文件上传;
singleUpload: true,
buttonText: '选择文件',
//container: '.lbf-FileUploader',
name: 'lbf-file-upload',
//用户附加自定义参数;
data: {},
//对即将发送的数据进行预处理;
processData: function(data){
//data是上面用户附加的参数;
},
file_size_limit : 0,
file_types : "*.jpg;*.gif;*.jpeg;*.png;*.bmp"
}
});
return FileUploader;
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,277 |
\section{Introduction}
In \cite{Sims57} Sims obtained an extension of the Weyl limit-point, limit-circle classification for the differential equation
\begin{equation}
M[y] =-y^{''}+qy=\lambda y,\;\;\; \lambda \in {\bf C}, \label{eq:1.1}
\end{equation}
on an interval $[a,b)$, where $q$ is complex-valued, and the end-points $a,b$ are respectively regular and singular. Under the assumption that ${\rm Im} q(x) \leq 0$
for all $x \in [a,b)$, Sims proved that for $ \lambda \in {\bf C_+}$, there exists at least one solution of (\ref{eq:1.1})
which lies in the weighted space $L^2(a,b; {\rm Im}[ \lambda-q] dx)$; such a solution lies in $L^2(a,b)$. There are now three distinct possibilities for $ \lambda \in {\bf C_+} $:
(I) there is, up to constant multiples, precisely one solution of (\ref{eq:1.1}) in $L^2(a,b; {\rm Im}[ \lambda-q] dx)$ and $L^2(a,b)$, (II)
one solution in $L^2(a,b; {\rm Im}[ \lambda-q] dx)$ but all in $L^2(a,b)$, and (III) all in $L^2(a,b; {\rm Im}[ \lambda-q] dx)$. This classification is independent
of $\lambda \in {\bf C_+}$ and, indeed, if all solutions of (\ref{eq:1.1})
are in $L^2(a,b; {\rm Im}( \lambda-q) dx)$ or in $L^2(a,b)$, for some $\lambda$,
it remains so for all $\lambda \in {\bf C }$.
At the core of Sims' analysis is an analogue for (\ref{eq:1.1}) of the Titchmarsh-Weyl $m$-function whose properties determine the self-adjoint realisations of
$-\frac{ d^2}{dx^2}
+q$ in $L^2(0,\infty)$
when $q$ is real and appropriate boundary conditions are prescribed at $a$ and $b$. Sims made a thorough study of the ``appropriate" boundary conditions and the spectral properties of the resulting operators in the case of complex $q$. The extension of the theory for an interval $(a,b)$ where both end points are singular follows in a standard way.
\par
We have two objectives in this paper. Firstly, we construct an analogue
of the Sims theory to the equation
\begin{equation}
-(py^{'})^{'} + q y = \lambda w y \label{eq:1.2}
\end{equation}
where $p$ and $q$ are both complex-valued, and $w$ is a positive weight function.
This is not simply a straightforward generalisation of \cite{Sims57}, for the
general problem exposes problems and properties of (\ref{eq:1.2}) which are
hidden in the special case
considered by Sims; some of these features may also be seen in \cite{BK76}
where a system of the form (\ref{eq:1.2}) with $p=\omega=1$ is considered (see Remark 2.5 below).
Secondly, once we have our analogue of the
Titchmarsh-Weyl-Sims $m-$function, we are (like Sims) in a position to
define
natural quasi m-accretive operators generated by $-\frac{1}{w}\{\frac{ d }{dx }(p\frac{ d }{dx })
+q\}$ in $L^2(a,b;wdx)$ and to investigate their spectral properties; these, of course, depend on the analogue of the 3 cases of Sims.
Our concern, in particular, is to relate these spectral properties to those
of the $m-$function, in a way reminiscent of that achieved for the case of real $p,q$ by Chaudhuri and Everitt \cite{chaudhurieveritt}.
We establish the correspondence between the eigenvalues and poles of the $m-$function, but, unlike in the self-adjoint case considered in \cite{chaudhurieveritt}, there is in general a part of the spectrum which is inaccessible from the subset of ${\bf C}$ in which the $m-$function is initially
defined and its properties determined.
However, even within this region we are able to define an $m-$function (Definition 4.10).
\par
We are grateful to the referees for comments which have helped to improve the presentation in the paper.
\section{The limit-point, limit-circle theory}
Let
\begin{equation}
M[y]= \frac{1}{w} [ - (py^{'})^{'} + q y ] \;\;\;{\rm on}\;\;[a,b)
\label{eq:2.1}
\end{equation}
where
\newcounter{rem1}
\begin{list}%
{( \roman{rem1} )}{\usecounter{rem1}
\setlength{\rightmargin}{\leftmargin}}
\item
$w>0$, $p \neq 0$ a.e. on $[a,b)$ and $w, 1/p \in L^1_{loc}[a,b)$;
\item
$p,q$ are complex-valued, $q \in L^1_{loc}[a.b)$ and
\begin{equation}
Q=\overline{co} \{ \frac{q(x)}{w(x)} + r p(x):
x \in [a,b), \; 0 < r < \infty \} \neq {\bf C}, \label{eq:2.2}
\end{equation}
\end{list}
where $\overline{co}$ denotes the closed convex hull.
\par
The assumptions on $w,p,q$ ensure that $a$ is a regular end-point of the equation $M[y]= \lambda w y$. We have in mind that $b$ is a singular end-point,
i.e. at least one of $b = \infty$ or
\begin{displaymath}
\int_a^b(w + \frac{1}{ \mid p \mid } + \mid q \mid ) dx = \infty
\end{displaymath}
holds; however the case of regular $b$ is included in the analysis.
The conditions i) and ii) will be assumed hereafter without further mention.
\par
The complement in ${\bf C}$ of the closed convex set $Q$ has one or two connected components. For $\lambda_0 \in {\bf C} \backslash Q$, denote by $K=K(\lambda_0)$ its (unique) nearest point in $Q$ and denote by $L=L(\lambda_0)$ the tangent to $Q$ at $K$ if it exists (which it does for almost all points on the boundary of $Q$), and otherwise any line touching $Q$ at $K$.
Then if the complex plane is subjected to a translation $z \mapsto z -K$ and a rotation through an
appropriate angle $\eta=\eta(\lambda_0) \in (-\pi, \pi]$, the image of $L$ coincides with the imaginary axis and the images of $\lambda_0$ and $Q$ lie in the new negative and non-negative half-planes respectively:
in other words, for all $x \in [a,b)$ and $r \in (0,\infty)$
\begin{equation}
Re[ \{ r p(x) + \frac{ q(x)}{w(x)}-K \} e^{i \eta} ] \geq 0 \label{eq:2.3}
\end{equation}
and
\begin{displaymath}
Re [ (\lambda_0-K) e^{i \eta} ] <0.
\end{displaymath}
For such {\it admissible} $K, \eta$ (corresponding to some $\lambda_0 \in {\bf C } \backslash Q $), define the half-plane
\begin{equation}
\Lambda_{\eta,K} := \{ \lambda \in {\bf C}: Re[(\lambda-K)e^{i \eta}] <0 \}.
\label{eq:2.4}
\end{equation}
Note that for all $\lambda \in \Lambda_{\eta,K}$
\begin{equation}
Re[(\lambda-K)e^{i \eta}]=-\delta <0
\label{eq:2.5}
\end{equation}
where $\delta=\delta_{\eta,K}(\lambda)$ is the distance from $\lambda$ to the boundary $\partial \Lambda_{\eta,K}$.
Also ${\bf C}\backslash Q$ is the union of the half-planes $ \Lambda_{\eta,K}$ over the set $S$ of admissible values of $\eta$ and $K$.
\par
We shall initially establish the analogue of the Sims-Titchmarsh-Weyl theory on the half-planes $ \Lambda_{\eta,K}$, but subject to the condition
\begin{equation}
Re[ e^{i \eta} \cos \alpha \;\overline{ \sin \alpha}] \leq 0
\label{eq:2.6}
\end{equation}
for some fixed $\alpha \in {\bf C}$:
the parameter $\alpha$ appears in the boundary condition at $a$ satisfied by functions in the domain of the underlying operator (see Section 4).
Denote by $S(\alpha)$ the set $\{ (\eta,K)\in S:\; (\ref{eq:2.6})\;{\rm is \;satisfied} \}$.
We assume throughout that
\begin{equation}
Q(\alpha) := {\bf C} \backslash \cup_{S(\alpha)} \Lambda_{\eta,K} = \cap_{S(\alpha)}
( {\bf C} \backslash \Lambda_{\eta,K} ) \neq \emptyset.
\label{eq:2.7a}
\end{equation}
The set $Q(\alpha)$ is clearly closed and convex, and
$Q(\alpha) \supseteq Q$ in general: for the important special cases $\alpha=0, \frac{\pi}{2}$, corresponding to the Dirichlet and Neumann problems, $Q(\alpha)=Q$.
In \cite{Sims57} Sims assumes that $p=w=1$ and the values of $q$ lie in ${\bf C_-}$;
thus $\eta =\pi/2, \; K=sup_{[a,b)}[ {\rm Im }q(x)]$, are admissible values, and $(\eta, K) \in S(\alpha)$
if
\begin{displaymath}
{\rm - Im }[ \cos \alpha \;\overline{ \sin \alpha}]= \sinh [ 2 {\rm Im} \alpha] \leq 0,
\end{displaymath}
the assumption made by Sims. If $\alpha$ is real, then (\ref{eq:2.6}) requires $\mid \eta \mid \leq \pi/2$ if $\alpha \in [\pi/2,\pi ]$, and
$\mid \eta \mid \geq \pi/2$ if $\alpha \in [0,\pi/2]$.
\par
We shall prove below that the spectrum of the differential operators defined in a natural way by the problems considered lie in the set $Q(\alpha)$.
This and related results can be interpreted as implying a restriction on the range of values of boundary condition parameter $\alpha$ permitted:
if $\alpha$ satisfies (\ref{eq:2.6}) for all $\eta$ which are such that $(\eta,K) \in S$ for some $K \in {\bf C}$, then $Q(\alpha)=Q$.
However, if $\alpha \in {\bf C}$ is given, it is the set $Q(\alpha)$ and not $Q$, which plays the central role in general.
\par
Let $\theta,\phi$ be the solutions of (\ref{eq:1.2}) which satisfy
\begin{eqnarray}
\phi(a,\lambda)=\sin \alpha, & \theta(a,\lambda)= \cos \alpha \nonumber \\
p\phi^{'}(a,\lambda)=-\cos \alpha, & p\theta^{'}(a,\lambda)= \sin \alpha \label{eq:2.6a}
\end{eqnarray}
where $\alpha \in {\bf C}$.
On integration by parts we have, for $a \leq Y < X < b$ and $u,v \in D(M)$ defined by
\begin{equation}
D(M)= \{y : y,py^{'} \in AC_{loc}[a,b)\}, \label{eq:2.7}
\end{equation}
that
\begin{equation}
\int_Y^X u M[v] w dx = - puv^{'} \mid_{Y}^X + \int_Y^X(pu{'}v^{'} + q uv)dx, \label{eq:2.8}
\end{equation}
\begin{equation}
\int_Y^X (u M[v]-vM[u])w dx = -[u,v](X)+[u,v](Y), \label{eq:2.9}
\end{equation}
where
\begin{equation}
[u,v](x)= p(x)(u(x)v^{'}(x)-v(x)u^{'}(x)), \label{eq:2.10}
\end{equation}
and
\begin{eqnarray}
&&\int_Y^X (u\overline{ M[v]}-\overline{v}M[u])w dx \nonumber \\
&&= (pu^{'}\overline{v}-\overline{p}u \overline{v}^{'})(X)-(pu^{'}\overline{v}-\overline{p}u \overline{v}^{'})(Y) +\int_Y^X [ ( \overline{p}-p)u^{'}\overline{v}^{'}+( \overline{q}-q)u \overline{v}] dx. \label{eq:2.11}
\end{eqnarray}
\par
Let $ \psi=\theta + l \phi$ satisfy
\begin{displaymath}
\psi(X) \cos \beta +(p \psi^{'})(X) \sin \beta = 0, \;\;\; \beta \in {\bf C}.
\end{displaymath}
Then
\begin{displaymath}
l \equiv l_X(\lambda, \cot \beta)=
-\frac{ \theta(X,\lambda) \cot \beta + p(X) \theta^{'}(X,\lambda)}{ \phi(X,\lambda) \cot \beta + p(X) \phi^{'}(X,\lambda)}.
\end{displaymath}
Let
\begin{equation}
l_X(\lambda, z):=
-\frac{ \theta(X,\lambda) z + p(X) \theta^{'}(X,\lambda)}{ \phi(X,\lambda) z + p(X) \phi^{'}(X,\lambda)},\;\;\;\;\; z \in {\bf C}. \label{eq:2.12}
\end{equation}
This has inverse
\begin{equation}
z= z_X(\lambda, l)=
-\frac{ p(X) \phi^{'}(X,\lambda) l + p(X) \theta^{'}(X,\lambda)}{ \phi(X,\lambda) l + \theta (X,\lambda)}. \label{eq:2.13}
\end{equation}
For $\eta$ satisfying (\ref{eq:2.6}), the M\"obius transformation (\ref{eq:2.12}) ( note that $p( \theta \phi^{'}-\phi \theta^{'})(X)=[ \theta,\phi](X)=-1)$ is such that, for $\lambda \in \Lambda_{\eta,K}, \;
z \mapsto l_X(\lambda,z)$ maps the half-plane ${\rm Re}[z e^{ i \eta}] \geq 0$
onto a closed disc $D_X(\lambda)$ in ${\bf C}$. To see this, set $\tilde{z}=z e^{i \eta}$ and
\begin{equation}
\tilde{l}_X(\lambda, \tilde{z})=
-\frac{ \theta(X,\lambda) \tilde{z} + p(X) \theta^{'}(X,\lambda)e^{i \eta}}
{ \phi(X,\lambda) \tilde{z} + p(X) \phi^{'}(X,\lambda)e^{i \eta}}
=l_X(\lambda,z). \label{eq:2.14}
\end{equation}
This has critical point
$\tilde{z}=-e^{i \eta} p(X) \phi^{'}(X,\lambda)/\phi(X,\lambda)$,
and we require this to satisfy ${\rm Re}[ \tilde{z}]<0$. We have
\begin{displaymath}
{\rm Re}[\tilde{z}]=
-{\rm Re}[ e^{i \eta}p(X) \phi^{'}(X,\lambda) \overline{\phi}(X,\lambda)/\mid \phi(X,\lambda)\mid^2 ]
\end{displaymath}
and, from (\ref{eq:2.8})
\begin{displaymath}
\int_a^X \overline{\phi}M[\phi]w dx = -p(X) \phi^{'}(X,\lambda) \overline{\phi}(X,\lambda)-
\cos \alpha \overline{ \sin \alpha }
+ \int_a^X(p \mid \phi^{'} \mid^2 + q \mid \phi \mid^2 ) dx.
\end{displaymath}
This yields
\begin{eqnarray}
&&\mid \phi (X,\lambda )\mid^2 Re [
e^{i \eta }p(X) \phi^{'}(X,\lambda) \overline { \phi} (X,\lambda) / \mid \phi(X,\lambda )\mid^2 ] = -{\rm Re}[ e^{i \eta} \cos \alpha \overline{ \sin \alpha}]
\nonumber \\
&+& {\rm Re}[ \int_a^X e^{i \eta}\{ \frac{p}{w} \mid \phi^{'}\mid^2 + ( \frac{q}{w}
-\lambda ) \mid \phi \mid^2 \}w ]dx
\nonumber \\
&>&0 \label{eq:2.15}
\end{eqnarray}
by (\ref{eq:2.3}).
Thus, when (\ref{eq:2.6}) is satisfied, $ z \mapsto l_X(\lambda,z)$ maps
${\rm Re}[z e^{i \eta}] \geq 0$ onto $D_X(\lambda)$, a closed disc with centre
\begin{equation}
\sigma_X(\lambda) = \tilde{l}_X(\lambda, e^{-i \eta} \overline { p(X) \phi^{'}
(X,\lambda)}/\overline{\phi(X,\lambda)}). \label{eq:2.17}
\end{equation}
Furthermore $\tilde{z}=0$ is mapped onto a point on the circle $C_X(\lambda)$ bounding $D_X(\lambda)$, namely the point
\begin{equation}
\tilde{l}_X(\lambda,0)=- \theta^{'}(X,\lambda)/\phi^{'}(X,\lambda),
\label{eq:2.18}
\end{equation}
and a calculation gives for the radius $\rho_X(\lambda)$ of $C_X(\lambda)$
\begin{eqnarray}
\rho_X(\lambda) &=& (2 \mid {\rm Re }[ e^{i \eta} p(X) \phi^{'}(X,\lambda) \overline{ \phi}(X,\lambda)]\mid)^{-1} \nonumber \\
&=& \frac{1}{2} \{ -{\rm Re} [ e^{i \eta} \cos \alpha \overline { \sin \alpha}] + \int_a^X {\rm Re }[ e^{i\eta} ( p\mid \phi^{'}\mid^2 +(q-\lambda w ) \mid \phi \mid^2]dx \} ^{-1} \label{eq:2.19}
\end{eqnarray}
by (\ref{eq:2.15}).
\par
The next step is to establish that the circles $C_X(\lambda)$ are nested as $X \rightarrow b$. Set $\psi_l=\theta+l\phi$ so that (\ref{eq:2.13}) gives
\begin{displaymath}
z=z_X(\lambda,l)=-p(X)\psi_l^{'}(X,\lambda)/\psi_l(X,\lambda).
\end{displaymath}
We have already seen that $l=l(\lambda) \in D_X(\lambda)$ if and only if
${\rm Re}[ e^{i \eta }z_X(\lambda,l)] \geq 0$, that is,
\newline
$ {\rm Re}[ e^{i \eta} p(X) \psi^{'}_l
(X,\lambda) \overline{ \psi}_l(X,\lambda)] \leq 0$.
As in (\ref{eq:2.15}), this can be written as
\begin{displaymath}
0 \geq Re [ e^{i \eta} \{ p(a) \psi^{'}_l(a,\lambda ) \overline{\psi}_l(a,\lambda) + \int_a^X( p \mid \psi^{'}_l\mid^2 + ( q - \lambda w ) \mid \psi_l\mid^2 ) dx \} ].
\end{displaymath}
On substituting (\ref{eq:2.6a}), this gives that $l \in D_X(\lambda)$ if
and only if
\begin{eqnarray}
&& \int_a^X Re [ e^{i\eta}\{ p \mid \psi^{'}_l \mid^2 + ( q - \lambda w ) \mid \psi_l \mid ^2 \}] dx \nonumber \\
&\leq& - Re [ e^{i\eta}( \sin \alpha - l \cos \alpha)( \overline { \cos \alpha} + \overline{ l \sin } \alpha )] \nonumber \\
&=:& {\cal {A}}( \alpha, \eta; l(\lambda)) \label{eq:2.20}
\end{eqnarray}
say. Note that $l \in C_X(\lambda)$ if and only if equality holds in (\ref{eq:2.20}). In view of (\ref{eq:2.3}) and (\ref{eq:2.5}), the integrand on the left-hand side of (\ref{eq:2.20}) is positive and so $D_Y(\lambda) \subset D_X(\lambda)$ if $X<Y$.
Hence the discs $D_X(\lambda), \; a < X< b$ are nested, and as $X \rightarrow b$ they converge to a disc $D_b(\lambda)$ or a point $m(\lambda)$: these are the {\it limit-circle} and {\it limit-point} cases respectively. The disc $D_b(\lambda)$ and point $m(\lambda)$ depend on $\eta$ and $K$ in general, but we shall only indicate this dependence
explicitly when necessary for clarity.
\par
Let
\begin{equation}
\psi(x,\lambda):= \theta(x,\lambda)+ m(\lambda) \phi(x,\lambda), \;\; \lambda
\in \Lambda_{\eta,K} \label{eq:2.21}
\end{equation}
where $m(\lambda)$ is either a point in $D_b(\lambda)$ in the limit-circle case, or the limit-point otherwise. The nesting property and (\ref{eq:2.20})
imply that
\begin{equation}
\int_a^b{\rm Re}[ e^{i \eta} \{ p\mid \psi^{'} \mid^2 + ( q - \lambda w ) \mid \psi \mid ^2\}] dx \leq {\cal {A}}(\alpha, \eta; m(\lambda)).
\label{eq:2.22}
\end{equation}
Moreover in the limit-point case, it follows from (\ref{eq:2.19}) that
\begin{equation}
\int_a^b{\rm Re}[ e^{i \eta} \{ p\mid \phi^{'} \mid^2 + ( q - \lambda w ) \mid \phi \mid^2 \}] dx = \infty,
\label{eq:2.23}
\end{equation}
whereas in the limit-circle case the left-hand side of (\ref{eq:2.23}) is finite. Also note that, by (\ref{eq:2.5}),
a solution $y$ of (\ref{eq:1.2}) for $\lambda \in \Lambda_{\eta,K}$ satisfies
\begin{equation}
\int_a^b{\rm Re}[ e^{i \eta} \{ p \mid y^{'} \mid^2 +(q-\lambda w ) \mid y \mid^2 \}]dx < \infty
\label{eq:2.24}
\end{equation}
if and only if
\begin{equation}
\int_a^b{\rm Re}[ e^{i \eta} \{ p \mid y^{'} \mid^2 +(q-K w ) \mid y \mid^2 \}]dx + \int_a^b \mid y \mid^2 w dx < \infty;
\label{eq:2.25}
\end{equation}
in particular this yields
\begin{equation}
y \in L^2(a,b;wdx). \label{eq:2.26}
\end{equation}
In the limit-point case there is a unique solution of (\ref{eq:1.2}) for $\lambda \in \Lambda_{\eta,K}$ satisfying (\ref{eq:2.25}), but it may be that all solutions satisfy (\ref{eq:2.26}). We therefore have the following analogue of Sims' result. The uniqueness referred to in the theorem is only up to
constant multiples.
\begin{theorem}
For $\lambda \in \Lambda_{\eta,K}$, $(\eta,K) \in S(\alpha)$ the Weyl circles converge either to a limit-point $m(\lambda)$ or a limit-circle $C_b(\lambda)$. The following distinct cases are possible, the first two being sub-cases of the limit-point case:
\begin{itemize}
\item
Case I : there exists a unique solution of (\ref{eq:1.2}) satisfying (\ref{eq:2.25}), and this is the only solution satisfying (\ref{eq:2.26});
\item
Case II : there exists a unique solution of (\ref{eq:1.2}) satisfying (\ref{eq:2.25}),but all solutions of(\ref{eq:1.2}) satisfy (\ref{eq:2.26});
\item
Case III: all solutions of (\ref{eq:1.2}) satisfy (\ref{eq:2.25}) and hence (\ref{eq:2.26}).
\end{itemize}
\end{theorem}
\begin{rem}
It follows by a standard argument involving the variation of parameters formula
(c.f.\cite[Section 3 Thm. 2]{Sims57}) that the classification of (\ref{eq:1.2}) in Theorem 2.1 is independent of $\lambda$ in the following sense:
\begin{list}%
{( \roman{rem1} )}{\usecounter{rem1}
\setlength{\rightmargin}{\leftmargin}}
\item
if all solutions of (\ref{eq:1.2}) satisfy (\ref{eq:2.25}) for some $\lambda^{'} \in \Lambda_{\eta,K}$ (i.e. Case III) then all solutions of (\ref{eq:1.2}) satisfy (\ref{eq:2.25}) for all $\lambda \in {\bf C}$;
\item
if all solutions of (\ref{eq:1.2}) satisfy (\ref{eq:2.26}) for some $ \lambda^{'} \in {\bf C}$ then all solutions of (\ref{eq:1.2}) satisfy ( \ref{eq:2.26}) for all $ \lambda \in {\bf C}$.
\end{list}
\end{rem}
\begin{rem}
Suppose that $p$ is real and non-negative and that for some $\eta \in [ -\frac{\pi}{2},\frac{\pi}{2}]$ and $K \in {\bf C}$,
\begin{equation}
\theta_{K,\eta}(x) = Re [ e^{i \eta}( q(x)-Kw(x))] \geq 0\;\; a.e. \; x \in(a,b).
\label{eq:2.27}
\end{equation}
Then the condition (\ref{eq:2.25}) in the Sims characterisation of (\ref{eq:1.2}) in Theorem 2.1 for $ \lambda \in \Lambda_{\eta,K}$, $(\eta,K) \in S(\alpha)$, becomes
\begin{equation}
\cos \eta \int_a^b p\mid y^{'} \mid^2dx + \int_a^b \theta_{K\eta}(x) \mid y(x) \mid^2 dx + \int_\alpha ^b \mid y(x)\mid^2w(x) dx < \infty.
\label{eq:2.28}
\end{equation}
\end{rem}
In this case Remark 2.2 (i) can be extended to the following:
\begin{list}%
{( \roman{rem1} )}{\usecounter{rem1}
\setlength{\rightmargin}{\leftmargin}}
\item
if for some $\lambda^{'} \in {\bf C}$ all the solutions of (\ref{eq:1.2}) satisfy (\ref{eq:2.28}); then for all $\lambda \in {\bf C}$ all solutions of (\ref{eq:1.2}) satisfy (\ref{eq:2.28});
\item
if for some $\lambda^{'} \in {\bf C}$ all the solutions of (\ref{eq:1.2}) satisfy one of
\begin{equation}
\cos \eta \int_a^b p \mid y^{'} \mid^2 dx < \infty \label{eq:2.29}
\end{equation}
\begin{equation}
\int_a^b \theta_{K\eta }\mid y\mid^2dx < \infty \label{eq:2.30}
\end{equation}
then the same applies for all $\lambda \in {\bf C}$.
\end{list}
\par
The case considered by Sims in \cite{Sims57} is when $\eta = \frac{\pi}{2},
K=0$ in (\ref{eq:2.27}). This overlooks the interesting features present in (\ref{eq:2.28}) when $\eta \in ( -\frac{\pi}{2},\frac{\pi}{2})$, namely, that the classification in Theorem 2.1 involves a weighted Sobolev space as well as $L^2(a,b;wdx)$.
\begin{rem}
We have not been able to exclude the possibility in Cases II and III that there exists a solution $y$ of (\ref{eq:1.2}) for $ \lambda \in \Lambda_{\eta_1,K_1} \cap \Lambda_{\eta_2,K_2}$ such that
\begin{equation}
\int_a^bRe [ e^{i\eta_1} (p \mid y^{'} \mid^2 + ( q -K_1w)\mid y \mid^2 )] dx
+ \int_a^b \mid y \mid^2 w dx < \\\infty \label{eq:2.31}
\end{equation}
\begin{equation}
\int_a^bRe [ e^{i\eta_2 }(p \mid y^{'} \mid^2 + ( q -K_2w)\mid y \mid^2 ) ]dx
+ \int_a^b \mid y \mid^2 w dx = \\\infty \label{eq:2.32}
\end{equation}
for different values of $\eta_1,\eta_2$ and $K_1,K_2$. In Case I this is not possible by Remark 2.2. Thus, in Cases II and III, the classification appears to depend on $K, \eta$, even under the circumstances of Remark 2.3.
\end{rem}
\begin{rem}
In \cite{BK76} a generalisation of Weyl's limit-circles theory, which includes that of
Sims, is obtained in the case of a system of the form (\ref{eq:1.2}) with $p =
\omega = 1, \lambda = 0$ and $Im[e^{-i \eta} \; q(x)] \leq -k < 0$. The
existence of solutions which satisfy (\ref{eq:2.25}) is established, and it is shown
that the analogue of Case I holds when $\eta \neq \pm \frac{\pi}{2}$.
\end{rem}
\section{Properties of $m$}
Throughout the paper hearafter we shall assume that $(\eta,K) \in S(\alpha) $.
We denote by $ m_{\eta,K}(\cdot)$ the function $m(\cdot)$ defined in Section 2 on
$ \Lambda_{\eta,K}$ whenever there is a risk of confusion. The argument in \cite[ Section 2.2]{ECT58} and
\cite[Theorem 3]{Sims57}
remains valid in our problem to give
\begin{lemma}
In Cases I and II, $ m_{\eta,K}$ is analytic throughout $ \Lambda_{\eta,K}$. In Case I the function defined by
\begin{equation}
m(\lambda)= m_{\eta,K}(\lambda),\;\;\lambda \in \Lambda_{\eta,K} \label{eq:3.1}
\end{equation}
is well-defined on each, of the possible two connected components of ${\bf C \backslash}Q (\alpha) = \cup_{S(\alpha)} \Lambda_{\eta,K}$, (see (\ref{eq:2.7a})); the restriction to a connected component is
analytic on that set.
\par
In Case III, given $m_0 \in C_b(\lambda_0), \lambda_0 \in \Lambda_{\eta,K}$, there exists a function $ m_{\eta,K}$ which is analytic in $ \Lambda_{\eta,K}$ and $ m_{\eta,K}(\lambda_0)=m_0$,
moreover, a function $ m_{\eta,K}$ can be found such that $ m_{\eta,K}(\lambda) \in C_b(\lambda)$ for all $\lambda \in \Lambda_{\eta,K}$.
\end{lemma}
{\bf Proof} The only part not covered by the argument in \cite[Theorem 3]{Sims57}
is that pertaining to (\ref{eq:3.1}) on ${\bf C } \backslash Q(\alpha)$ in Case I. We need only show that $ m_{\eta_1,K_1}(\lambda)= m_{\eta_2,K_2}(\lambda)$ if $ \lambda \in \Lambda_{\eta_1,K_1} \cap \Lambda_{\eta_2,K_2}$. Since in Case I, the function in
(\ref{eq:2.21}) (now denoted by $\psi_{\eta,K}(\cdot,\lambda)$ for $ \lambda \in \Lambda_{\eta,K}$) is the unique solution of (\ref{eq:1.2}) in $L^2(a,b;wdx)$ it follows that
\begin{displaymath}
\psi_{\eta_1,K_1}(x,\lambda)=K(\lambda) \psi_{\eta_2,K_2}(x,\lambda)
\end{displaymath}
for some $K(\lambda)$.
On substituting the initial conditions (\ref{eq:2.6a})
we obtain $m_{\eta_1,K_1}(\lambda)=m_{\eta_2,K_2}(\lambda)$.
\par
In Case I, if ${\bf C} \backslash Q(\alpha)$ has two connected components $C_1$, $C_2$ say and $m^{(1)}, m^{(2)}$ are the $m-$functions defined on $C_1,C_2$ respectively by Lemma 3.1,
we define $m$ on ${\bf C} \backslash Q(\alpha)$ by
\begin{displaymath}
m(\lambda)= \left \{
\begin{array}{cc}
m^{(1)} & \lambda \in C_1, \\
m^{(2)} & \lambda \in C_2.
\end{array}
\right .
\end{displaymath}
\begin{rem}
Let $\alpha \in \{ 0,\pi \}$ in (\ref{eq:2.20}). Then $l \in D_X(\lambda)$ implies that ${\rm Re}[ e^{i \eta} l] \geq 0$. Thus $z \mapsto l_X(\lambda,z)$
maps the half-plane ${\rm Re}[ e^{i \eta}z] \geq0$ into itself and, in particular,
$m(\cdot)$ possesses an analogue of the Nevanlinna property enjoyed by the Titchmarsh-Weyl function in the formally symmetric case. If $\alpha= \frac{\pi}{2}$, then $l \in D_X(\lambda)$ implies that ${\rm Re}[ e^{i\eta} \overline{l}] \leq 0$.
\end{rem}
\par
The argument in \cite[ Lemma 2.3]{ECT62} requires only a slight modification to give the important lemma
\begin{lemma}
Let $\lambda, \lambda^{'} \in \Lambda_{\eta,K}$ and $\psi(\cdot,\lambda)=\theta(\cdot,\lambda)+m(\lambda) \phi(\cdot,\lambda)$,
where $m(\lambda)$ is either the limit point or an arbitrary point in $D_b(\lambda)$ in the limit-circle case. Then
\begin{equation}
\lim_{X\rightarrow b} [ \psi(\cdot,\lambda), \psi(\cdot,\lambda^{'})](X) \equiv
\lim_{X\rightarrow b} \{p(X)[ \psi(X,\lambda)\psi^{'}(X,\lambda^{'})- \psi^{'}(X,\lambda)\psi(X,\lambda^{'})]\} =0. \label{eq:3.2}
\end{equation}
In Case I, (\ref{eq:3.2}) continues to hold for all $\lambda,\lambda^{'} \in {\bf C \backslash} Q(\alpha).$
\end{lemma}
{\bf Proof }
The starting point is the observation that if Re $[z e^{i \eta}]\geq 0,$ and hence $l_X(\lambda,z)$ in (\ref{eq:2.12}) lies on the disc $D_X(\lambda)$, then with $\psi_X=\theta+l_X\phi$
\begin{displaymath}
z\psi_X(X,\lambda) + p \psi^{'}_X(X,\lambda)=0
\end{displaymath}
and similarly for $\lambda^{'}$. Then
\begin{displaymath}
[ \psi_X(\cdot, \lambda), \psi_X(\cdot, \lambda^{'})](X)=0
\end{displaymath}
and the argument proceeds as in \cite{ECT62}.
\par
Lemma 3.3 and (\ref{eq:2.8}) yield
\begin{cor}
For all $\lambda, \lambda^{'} \in \Lambda_{\eta,K}$
\begin{equation}
(\lambda^{'}-\lambda) \int_a^b \psi(x,\lambda)\psi(x,\lambda^{'})w(x)dx = m(\lambda)-m(\lambda^{'}); \label{eq:3.3}
\end{equation}
this holds for all $\lambda,\lambda^{'} \in {\bf C \backslash }Q$ in Case I.
It follows that in Case II and III, for a fixed $\lambda^{'} \in \Lambda_{\eta,K}$,
\begin{equation}
m(\lambda)= \frac{ m(\lambda^{'})-(\lambda-\lambda^{'})\int_a^b \theta(x,\lambda)\psi(x,\lambda^{'})w(x) dx}
{1 + (\lambda-\lambda^{'})\int_a^b \phi(x,\lambda)\psi(x,\lambda^{'})w(x)dx}
\label{eq:3.4}
\end{equation}
defines $m(\lambda)$ as a meromorphic function in ${\bf C}$; it has a pole at $\lambda $ if and only if
\begin{equation}
1 +(\lambda-\lambda^{'})\int_a^b \phi(x,\lambda) \psi(x,\lambda^{'})w(x)=0.
\label{eq:3.5}
\end{equation}
\end{cor}
{\bf Proof}
The identity (\ref{eq:3.3}) follows easily from (\ref{eq:2.9}) and Lemma 3.3. In Cases II and III, $\theta(\cdot,\lambda),\phi(\cdot,\lambda) \in L^2(a,b,wdx)$, and (\ref{eq:3.4}) is derived from (\ref{eq:3.3}) on writing $\psi(\cdot,\lambda)=
\theta(\cdot,\lambda)+m(\lambda)\phi(\cdot,\lambda)$.
\begin{theorem}
Suppose that (\ref{eq:1.2}) is in Case I. Define
\begin{eqnarray}
Q_c &:= &\overline{co} \{ \frac{ q(x)}{w(x)}+r p(x):
x \in [c,b), \; r \in (0,\infty)\}, \label{eq:3.6} \\
Q_b &:=& \cap_{ c \in (a,b)} Q_c , \;\;\; Q_b(\alpha) = \cap_{c \in (a,b) }
Q_c(\alpha), \label{eq:3.7}
\end{eqnarray}
where $Q_c(\alpha)$ is the set $Q(\alpha)$ defined in (\ref{eq:2.7a}) when the
underlying interval is $[c,b)$ rather than $[a,b)$. Then $m(\lambda)$ is
defined throughout ${\bf C \backslash} Q (\alpha)$ and has a meromorphic
extension to ${\bf C \backslash }Q_b(\alpha)$, with poles only in
$ Q(\alpha) \backslash Q_b (\alpha)$.
\end{theorem}
{\bf Proof}
Let $m_c(\cdot)$ denote the limit point in the problem on $[c,b)$ with $c$ now replacing $a$ in the initial conditions (\ref{eq:2.6a}); it is defined and analytic throughout each of the possible two connected components of ${\bf C \backslash} Q_c(\alpha)$, by Lemma 3.1. Also $\psi_c := \theta _c+ m_c \phi_c $ can be
uniquely extended to $[a,b)$ with $\psi_c(x,\cdot)$ and $p\psi_c^{'}(x,\cdot)$ analytic in ${\bf C \backslash} Q_c(\alpha) $ for fixed $x$. Since we are in Case I there exists $K(\lambda)$ such that
\begin{displaymath}
\psi(x,\lambda)= K(\lambda) \psi_c (x,\lambda).
\end{displaymath}
On substituting (\ref{eq:2.6a}), we obtain
\begin{equation}
m(\lambda) = \frac{ \sin \alpha \psi_c(a,\lambda)-\cos \alpha p \psi_c^{'}(a,\lambda)}
{ \cos \alpha \psi_c(a,\lambda)+\sin \alpha p \psi_c^{'}(a,\lambda)}.
\label{eq:3.8}
\end{equation}
This defines $m(\lambda)$ as a meromorphic function in ${\bf C \backslash} Q_c(\alpha)$
with isolated poles at the zeros of the denominator in (\ref{eq:3.8}).
In the case $b=\infty$, $Q_b$ appears in \cite[section 35]{G65}.
\section{Operator realisations of $M$}
For $\lambda \in \Lambda_{\eta,K}, \; (\eta,K) \in S(\alpha)$ define
\begin{equation}
G(x,y;\lambda)= \left \{
\begin{array}{cc}
-\phi(x,\lambda)\psi(y,\lambda), & a < x<y<b, \\
-\psi(x,\lambda)\phi(y,\lambda), & a<y<x<b,
\end{array} \right .\label{eq:4.1}
\end{equation}
where $\phi,\psi$ are the solutions of (\ref{eq:1.2}) in (\ref{eq:2.6a}) and
(\ref{eq:2.21}).
Recall that $m$, and hence $\psi$, depends on $(\eta,K)$ in general, but for simplicity of notation we suppress this dependency. In Case I however, Lemma 3.1 shows that $m$ is properly defined throughout ${\bf C } \backslash Q(\alpha)$.
In Cases II and III, we know from Theorem 3.5 that $m(\cdot)$ can be continued as
a meromorphic function throughout ${\bf C}$ (but apparently still depends on $\eta$ and $K$).
For $\lambda \in \Lambda_{\eta,K}$ and $f \in L^2(a,b;wdx)$ define
\begin{equation}
R_\lambda f(x):= \int_a^bG(x,y;\lambda)f(y)w(y)dx. \label{eq:4.2}
\end{equation}
It is readily verified that $p (R_\lambda f)^{'} \in AC_{loc}[a,b)$ and from
\begin{displaymath}
[\phi,\psi](x)=[\phi,\psi](a)=1\;\;\;(x \in (a,b))
\end{displaymath}
(see (\ref{eq:2.9}) and (\ref{eq:2.10})) that for a.e. $ x \in (a,b)$
\begin{equation}
(M-\lambda)R_\lambda f(x)=f(x). \label{eq:4.3}
\end{equation}
Also, for any $\lambda^{'} \in {\bf C}$
\begin{equation}
[ R_\lambda f, \phi( \cdot,\lambda^{'})] (a)=-[ \phi(\cdot,\lambda) ,\phi(\cdot,\lambda^{'})](a) \int_a^b \psi f w dx =0. \label{eq:4.4}
\end{equation}
Moreover, if $f$ is supported away from $b$, then, by Lemma 3.3, for any $\lambda, \lambda^{'} \in \Lambda_{\eta,K}$,
\begin{eqnarray}
[ R_\lambda f, \psi(\cdot,\lambda^{'})](b) &:=&
\lim_{X \rightarrow b}[ R_\lambda f, \psi(\cdot, \lambda^{'})](X) \nonumber \\
&=& -\lim_{X\rightarrow b} \{ [ \psi(\cdot,\lambda), \psi(\cdot,\lambda^{'})](X)
\int_a^X\phi f w dx \} \nonumber \\
&=&0. \label{eq:4.5}
\end{eqnarray}
In Cases II and III (\ref{eq:4.5}) holds for all $f \in L^2(a,b,;wdx)$
since then the integral on the right-hand side remains bounded as $X \rightarrow b$ and $\lim_{X\rightarrow b} [ \psi(\cdot, \lambda), \psi(\cdot, \lambda^{'})](X)$ is zero by (\ref{eq:3.2}).
In Case I (\ref{eq:4.5}) continues to be true for all $\lambda, \lambda^{'}
\in {\bf C}\backslash Q(\alpha)$.
\par
Before preceding to define the realisations of $M$ which are natural to the problem,
we need the following theorem which provides our basic tool. In the theorem $\parallel \cdot \parallel $ denotes the $L^2(a,b;wdx)$ norm.
\par
\begin{theorem}
Let $f \in L^2(a,b;wdx)$ and $\lambda \in \Lambda_{\eta,K}$, $(\eta, K) \in S(\alpha)$. Then, in every case, with $\Phi \equiv R_\lambda f,$ and $\delta = dist (\lambda, \partial \Lambda_{\eta,K}),$
\begin{equation}
\int_a^b {\rm Re} [ e^{i \eta} ( p \mid \Phi ^{'} \mid^2 + ( q - K w ) \mid \Phi \mid^2 )]dx + ( {\rm Re} [ (K-\lambda )e^{i \eta}]-\epsilon) \int_a^b \mid \Phi \mid^2 w dx \leq \frac{1}{4 \epsilon} \int_a^b \mid f \mid ^2 w dx \label{eq:4.6}
\end{equation}
for any $\epsilon >0$. In particular, $R_\lambda$ is bounded and
\begin{equation}
\parallel R_\lambda f \parallel \leq \frac{1}{\delta} \parallel f \parallel.
\label{eq:4.7}
\end{equation}
\end{theorem}
{\bf Proof}
Let $f_X=\chi_{(a,X)}f$ and $\Phi_X=R_\lambda f_X$. Then, by (\ref{eq:2.8}) and (\ref{eq:4.3})
\begin{eqnarray*}
&& \int_a^X(p \mid \Phi^{'}_X \mid^2 + (q-\lambda w )\mid \Phi_X\mid^2 ) dx= p \overline{\Phi_X}\Phi^{'}_X \mid_a^X + \int_a^X \overline{\Phi}_Xfw dx \\
&&= p(X)\overline{\psi(X)} \psi^{'}(X) \mid \int_a^X \phi fwdx \mid^2 -p(a) \overline{\phi}(a) \phi^{'}(a) \mid \int_a^X \psi f w dx \mid^2 +
\int_a^X \overline{\Phi_X}fwdx \\
&&= \{ \int_a^X ( p \mid \psi^{'} \mid^2 +(q-\lambda w ) \mid \psi\mid^2 ) dx +( \overline { \cos \alpha + m \sin \alpha})(\sin \alpha - m \cos \alpha )\}
\mid \int_a^X \phi f w dx \mid^2\\
&& + \overline{\sin \alpha }\cos \alpha \mid \int_a^X \psi fw dx \mid^2
+ \int_a^X\overline{\Phi}_Xfwdx \end{eqnarray*}
from (\ref{eq:2.8}) again, and (\ref{eq:2.6a}).
Hence, by (\ref{eq:2.20}) and (\ref{eq:2.22}),
\begin{eqnarray*}
&\int_a^X&{\rm Re}[ e^{i\eta}( p \mid \Phi^{'}_X\mid^2 +(q-\lambda w ) \mid \Phi_X \mid^2)]dx \\
&=& \int_a^X \{ {\rm Re} [ e^{i \eta}( p \mid \psi^{'} \mid^2 + ( q-\lambda w ) \mid \psi \mid ^2 )] dx - {\cal A}(\alpha, \eta;m(\lambda))\}\mid \int_a^X \phi fwdx \mid^2 \\
&+& {\rm Re} [ e^{i\eta} \overline{\sin \alpha} \cos \alpha ] \mid \int_a^X \psi fw dx \mid^2 + {\rm Re}[ e^{i\eta} \int_a^X \overline{\Phi}_Xfw]dx \\
&\leq& \int_a^X \mid \Phi_X \mid \mid f \mid w dx \leq \epsilon \int_a^X \mid \Phi_X \mid^2 w dx + \frac{1}{4\epsilon} \int_a^X \mid f_X \mid^2w dx,
\end{eqnarray*}
whence
\begin{eqnarray*}
\int_a^b{\rm Re} [ e^{i \eta} ( p \mid \Phi^{'}_X \mid^2 + (q-Kw) \mid \Phi_X \mid^2 ) dx ]
+ ( {\rm Re} [ e^{i \eta}(K-\lambda)]-\epsilon ) \int_a^b \mid \Phi_X \mid^2 w dx \\
\leq \frac{1}{4\epsilon}\int_a^b \mid f_X \mid^2 w dx.
\end{eqnarray*}
As $X \rightarrow b, \; \Phi_X(x) \rightarrow \Phi(x)$ and (\ref{eq:4.6}) follows by Fatou's lemma. We also obtain from (\ref{eq:4.6}), (\ref{eq:2.3}),(\ref{eq:2.5} and (\ref{eq:2.8}) that
\begin{displaymath}
(\delta-\epsilon)\int_a^b \mid \Phi \mid^2 w dx \leq \frac{1}{4\epsilon} \int_a^b \mid f \mid^2w dx.
\end{displaymath}
The choice $\epsilon = \frac{\delta}{2}$ yields (\ref{eq:4.7}).
\par
Theorem 4.1 enables us to establish (\ref{eq:4.5}) for all $f \in L^2(a,b;wdx)$
in Case I (and hence in all Cases).
\begin{lemma}
For $\lambda,\lambda^{'} \in \Lambda_{\eta,K}, \;\; (\eta, K) \in S(\alpha),$ and $f \in L^2(a,b;wdx)$
\begin{displaymath}
[R_\lambda f, \psi(\cdot,\lambda^{'}) ](b)=0.
\end{displaymath}
\end{lemma}
{\bf Proof}
Let $f_c=\chi_{[a,c]}f$, so that as $ c \rightarrow b$ we have
\begin{equation}
f_c \rightarrow f, \;\; R_\lambda f_c \rightarrow R_\lambda f \;{\rm in }\; L^2(a,b;wdx),
\label{eq:4.8a}
\end{equation}
\begin{equation}
[R_\lambda f_c, \psi(\cdot,\lambda^{'})](a) \rightarrow [R_\lambda f, \psi(\cdot,\lambda^{'})](a), \label{eq:4.9}
\end{equation}
since
\begin{displaymath}
(R_\lambda f_c)(a)=-\phi(a,\lambda)\int_a^b \psi(y,\lambda)f_c(y)w dy
\rightarrow (R_\lambda f)(a),
\end{displaymath}
\begin{displaymath}
[p(R_\lambda f_c)^{'}](a) = -p\phi^{'}(a,\lambda)\int_a^b\psi(y,\lambda)f_c(y)w(y)dy \rightarrow[p(R_\lambda f)^{'}](a),
\end{displaymath}
and, by (\ref{eq:4.5}),
\begin{equation}
[R_\lambda f_c, \psi(\cdot,\lambda^{'})](b)=0.
\label{eq:4.10}
\end{equation}
Hence, by (\ref{eq:2.9}),
\begin{eqnarray*}
[R_\lambda f, \psi(\cdot,\lambda^{'})](X) &=& [ R_\lambda (f-f_c), \psi(\cdot,\lambda^{'})](a)+[R_\lambda f_c, \psi(\cdot,\lambda^{'})](X) \\
&+& \int_a^X \{ (\lambda-\lambda^{'}) \psi(x,\lambda^{'})R_\lambda [ f-f_c](x)
+\psi(x,\lambda^{'})[f-f_c](x)\}w(x)dx \\
\rightarrow [ R_\lambda (f-f_c), \psi(\cdot,\lambda^{'})](a) & +&
\int_a^b \{ (\lambda-\lambda^{'})\psi(x,\lambda^{'}) R_\lambda [f-f_c](x)+\psi(x,\lambda^{'})[f-f_c](x) \} w(x) dx
\end{eqnarray*}
as $X \rightarrow b$, by (\ref{eq:4.10}),
\begin{displaymath}
\rightarrow 0
\end{displaymath}
by (\ref{eq:4.8a}) and (\ref{eq:4.9})
\begin{rem}
In Cases II and III, $R_\lambda$ is obviously Hilbert-Schmidt for any $ \lambda \in \Lambda_{\eta,K}, \; (\eta, K) \in S(\alpha)$.
\end{rem}
In view of Theorem 4.1 and preceding remarks, it is natural to define the following operators. Let $\lambda^{'} \in \Lambda_{\eta,K}, \; (\eta,K) \in S(\alpha)$, be fixed and set
\begin{eqnarray}
D(\tilde{M})&:=&\{ u:u,
pu^{'} \in AC_{loc}[a,b), u,Mu \in L^2(a,b;wdx),[u,\phi(\cdot,\lambda^{'})](a)=0\;\;{\rm and}\; [ u,\psi(\cdot,\lambda^{'})](b)=0 \}, \nonumber \\
\tilde{M}u&:=&Mu, \;\;\; u \in D(\tilde{M}). \label{eq:4.8}
\end{eqnarray}
The dependence, or otherwise, of $D(\tilde{M})$ on $\lambda^{'}$ is made clear in
\begin{theorem}
In Case I
\begin{equation}
D(\tilde{M})= D_1 := \{ u:u, pu^{'} \in AC_{loc}[a,b),\; u,\; Mu \in
L^2(a,b;wdx), \; (\cos \alpha) u (a) + (\sin \alpha) p(a) u^{'}(a)=0\}.
\label{eq:4.12}
\end{equation}
In Case II and III, $D_1$ is the direct sum
\begin{equation}
D_1=D(\tilde{M}) \stackrel{.}{+} [ \phi(\cdot,\lambda^{'})] \label{eq:4.13}
\end{equation}
where $[\cdot ]$ indicates the linear span.
\end{theorem}
{\bf Proof}
Clearly $D( \tilde{M}) \subset D_1$: note that the boundary condition at
$a$ in (\ref{eq:4.12}) can be written as $[u,\phi(\cdot,\lambda^{'})](a)=0$.
Let $u \in D_1$, and for $\lambda^{'} \in \Lambda_{\eta,K}$ set $v=R_{\lambda^{'}}[(M-\lambda^{'})u]$. Then $(M-\lambda^{'})v = (M-\lambda^{'})u$
and $[ v-u, \phi(\cdot,\lambda^{'})](a)=0$.
It follows that $v-u=K_1\phi(\cdot,\lambda^{'})$ for some constant $K_1$.
In Case I, this implies that $K=0$ since $v \in D( \tilde{M})$ and
$\phi(\cdot,\lambda^{'}) \not \in
L^2 (a,b;wdx)$. The decomposition (\ref{eq:4.13}) also follows since the right-hand side of (\ref{eq:4.13}) is obviously in $D_1$ in Cases II and III.
\par
In the next theorem $J$ stands for the conjugation operator $u \mapsto \overline{u}$. An operator $T$ is $J-$symmetric if $JTJ \subset T^{\ast}$ and
$J-$self-adjoint if $JTJ=T^{\ast}$ (see \cite[section III.5)]{EE87}.
Also $T$ is m-accretive if Re $\lambda < 0$ implies that $\lambda \in \rho(T)$, the resolvent set of $T$,
and $\parallel ( T-\lambda I)^{-1} \parallel \leq \mid {\rm Re } \lambda \mid ^{-1}$.
If for some $K \in {\bf C}$ and $\eta \in (-\pi,\pi)$, $ e^{i \eta }(T-K)$ is m-accretive, we shall say that $T$ is quasi-m-accretive;
note this is slightly different to the standard notion which does not involve the rotation $e^{i\eta}$ ( cf. \cite[section III.]{EE87}).
\par
Let $\sigma(\tilde{M})$ denote the spectrum of $\tilde{M}$.
We define the essential spectrum, $\sigma_e ( \tilde{M})$, of $\tilde{M}$
to be the complement in ${\bf C}$ of the set
\begin{displaymath}
\Delta ( \tilde{M})=\{ \lambda : ( \tilde{M}-\lambda I )
{\rm \;is \;a\; Fredholm \; operator\; and \; ind }( \tilde{M}-\lambda I)=0\}.
\end{displaymath}
Recall that a Fredholm operator $A$ is one with closed range, finite nullity nul $A$ and finite deficiency def $A$, and ind $A$= nul $A-$def $A$.
Thus any $\lambda \in \sigma ( \tilde{M}) \backslash \sigma_e(\tilde{M})$
is an eigenvalue of finite (geometric) multiplicity.
\begin{theorem}
The operators defined in (\ref{eq:4.8}) for any $\lambda^{'} \in \Lambda_{\eta,K}, \; (\eta,K) \in S(\alpha)$ (or
(\ref{eq:4.12}) in Case I) are
$J-$self-adjoint and quasi-m-accretive, and $\sigma(\tilde{M}) \subseteq
{\bf C}\backslash \Lambda_{\eta,K}$. For any $\lambda \in \Lambda_{\eta,K}, ( \tilde{M}-\lambda)^{-1}=R_\lambda$.
\par
In Case I, $\sigma (\tilde{M}) \subseteq Q(\alpha)$ and $\sigma_e (\tilde{M})\subseteq Q_b(\alpha)$, where $Q_b(\alpha)$ is defined in (\ref{eq:3.7}): in $Q(\alpha)\backslash Q_b(\alpha)$, $\sigma(\tilde{M})$ consists only of eigenvalues of finite geometric multiplicity.
\par
In Cases II and III, $R_\lambda$ is compact for any $\lambda \in \rho(\tilde{M})$ and $\sigma(\tilde{M})$ consists only of isolated eigenvalues (in ${\bf C } \backslash \Lambda_{\eta,K}$) having finite algebraic multiplicity.
\end{theorem}
{\bf Proof}
From $JMJ=M^+$, the Lagrange adjoint of $M$, it follows that $M$ is $J$-symmetric. Since $(\tilde{M}-\lambda)^{-1}=R_\lambda$ and $ \Lambda_{\eta,K} \subseteq \rho(\tilde{M})$ are
established in Theorem 4.1 and the preceding remarks, it follows that $\tilde{M}$ is quasi-m-accretive, and hence also $J$-self-adjoint by Theorem III 6.7 in \cite{EE87}.
\par
In Case I, Theorem 4.1 holds for any $\lambda \in {\bf C} \backslash Q(\alpha)$ and hence $\sigma(\tilde{M}) \subseteq Q(\alpha).$
Also, by the ``decomposition principle" (see \cite[Theorem IX 9.3 and Remark IX 9.8]{EE87}) $\sigma_e(\tilde{M}) \subseteq Q_b(\alpha)$.
\par
The compactness of $R_\lambda$ for $\lambda \in \Lambda_{\eta,K}$ in Cases II and III is noted in Remark 4.3, and the rest of the theorem follows.
\begin{rem}
The argument in \cite[ Theorem 35.29]{G65} can be used to prove that in Case I of
Theorem 4.5, either $\sigma( \tilde{M}) \backslash Q_b(\alpha)$ consists of isolated points of finite algebraic multiplicity and with no limit-point outside $Q_b(\alpha)$ or else each point of at least one of the (possible two) connected components of $Q(\alpha)\backslash Q_b(\alpha)$ is an eigenvalue.
We now prove that the latter is not possible.
\end{rem}
\begin{theorem}
Let (\ref{eq:1.2}) be in Case I. Then $\sigma(\tilde{M}) \subseteq Q(\alpha),\; \sigma_e(\tilde{M}) \subseteq Q_b(\alpha)$
and in $Q(\alpha)\backslash Q_b(\alpha),\; \sigma(\tilde{M})$
consists only of isolated eigenvalues of finite algebraic multiplicity,
these points being the poles of the meromorphic extension of $m$ defined in Theorem 3.5.
\end{theorem}
{\bf Proof } Let $\lambda \in Q(\alpha) \backslash Q_b(\alpha)$ be such that the meromorphic
extension of $m$ in Theorem 3.5 is regular at $\lambda$, and for $ c \in (a,b)$,
let $\psi(\cdot,\lambda)=K(\lambda) \psi_c(\cdot,\lambda)$ in the notation of the proof of Theorem 3.5.
Then $\psi(\cdot,\lambda)=\theta(\cdot,\lambda) + m(\lambda) \phi(\cdot,\lambda)
\in L^2(a,b;wdx)$ and the operator $R_\lambda^c$ defined by
\begin{displaymath}
R^c_\lambda f(x):= -\psi_c(x,\lambda)\int_c^x \phi(y,\lambda)f(y)w(y)dy -\phi(x,\lambda) \int_x^b \psi_c(y,\lambda)f(y)w(y)dy
\end{displaymath}
is bounded on $L^2 (c,b;wdx)$ for $c$ sufficiently close to $b$ (so that $\lambda \not \in Q_c(\alpha))$, by Theorem 4.1 applied to $[c,b)$.
Moreover (\ref{eq:4.3}) and (\ref{eq:4.4}) are satisfied by $R_\lambda$, now defined for this $\lambda \in Q(\alpha)\backslash Q_b(\alpha)$, and hence if we can prove that $R_\lambda$ is bounded on $L^2(a,b;wdx)$, it will follow that $\lambda \in \rho(\tilde{M})$, whence the theorem in view of Remark 4.6. But, for any $f \in L^2 (a,b;wdx)$, it is readily verified that
\begin{displaymath}
\parallel R_\lambda f \parallel \leq {\rm const} \{
\parallel \phi \parallel_{(a,c)}\parallel \psi \parallel +
\parallel R^c_\lambda \parallel \} \parallel f\parallel.
\end{displaymath}
Hence $\lambda \in \rho( \tilde{M})$. In Lemma 4.12 below we shall prove that $m$ is analytic on $\rho( \tilde{M})$, hence any pole of $m$ in $Q(\alpha)\backslash Q_b(\alpha)$ lies in $\sigma(\tilde{M})$.
The theorem is therefore proved.
\begin{rem}
Suppose that Case I holds.
In the notation of \cite[section IX.1]{EE87} our essential spectrum $\sigma_e$
is $\sigma_{e4}$.
However, since the operator $\tilde{M}$ is $J-$self-adjoint, by Theorem 4.5, all the essential spectra $\sigma_{ek}(\tilde{M}),\;k=1,2,3,4$ defined in \cite[Section IX.1]{EE87} coincide, by \cite[Section IX.1.6]{EE87}.
Furthermore, for any $\alpha$,
$\tilde{M}$ is a $2$-dimensional extension of the closed minimal operator generated by $M$ on
\begin{displaymath}
D_0= \{ u : u, pu^{'} \in {\rm AC}_{loc}[a,b),u,Mu,\in L^2(a,b,wdx), u(a)=p(a)u^{'}(a) =0 \}
\end{displaymath}
(cf. \cite[Theorem III 10.13 and Lemma IX 9.2]{EE87}).
It therefore follows from \cite[ IX.1, 4.2]{EE87} that the essential spectrum $\sigma_e(\tilde{M})$ is independent of $\alpha$.
Thus in Theorem 4.7 $\sigma_e(\tilde{M})\subseteq Q_b$, since $Q_b(0)=Q_b$.
\end{rem}
\par
We now proceed to analyse the connections between the spectrum of $\tilde{M}$ and the singularities of extensions of the $m(\cdot)$ function as is done for the Sturm-Liouville problem in \cite{chaudhurieveritt}. An important observation for this analysis is
the following lemma. In it $(\cdot,\cdot)$ denotes the $L^2(a,b;wdx)$ inner-product.
\begin{lemma}
For all $\lambda, \lambda^{'} \in \Lambda_{\eta,K}, \; (\eta, K) \in S(\alpha)$,
\begin{equation}
m(\lambda)=m(\lambda^{'})-(\lambda-\lambda^{'})\int_a^b\psi^2(x,\lambda^{'})w(x)dx -(\lambda-\lambda^{'})^2 ( R_\lambda \psi(\cdot,\lambda^{'}),
\overline{\psi}(\cdot,\lambda^{'})),
\label{eq:4.11}
\end{equation}
\begin{equation}
m(\lambda)=[\psi(\cdot,\lambda),\theta(\cdot,\lambda^{'})](a), \label{eq:4.12a}
\end{equation}
and
\begin{equation}
\psi(\cdot,\lambda)=\psi(\cdot,\lambda^{'})+(\lambda-\lambda^{'})R_\lambda \psi(\cdot,\lambda^{'}). \label{eq:4.13a}
\end{equation}
\end{lemma}
{\bf Proof }
The identity (\ref{eq:4.11}) is an immediate consequence of (\ref{eq:3.3})
and (\ref{eq:4.13a}), and (\ref{eq:4.12a}) follows from (\ref{eq:2.6a}) and (\ref{eq:2.21}). To prove (\ref{eq:4.13a}), set $u=\psi(\cdot, \lambda )-\psi(\cdot,\lambda^{'})$.
Then $ u \in D(\tilde{M})$ by Lemma 3.3 and since
\begin{equation}[\psi(\cdot,\lambda), \phi(\cdot,\lambda^{'})](a)-[\psi(\cdot,\lambda^{'}), \phi(\cdot,\lambda^{'})](a) =0.
\label{eq:4.17aa}
\end{equation}
Also $(\tilde{M}-\lambda)u=(\lambda-\lambda^{'})\psi(\cdot,\lambda^{'})$.
This yields $u=(\lambda-\lambda^{'})R_\lambda\psi(\cdot,\lambda^{'})$ and (\ref{eq:4.13a}) is established. The lemma is therefore proved.
\par
Motivated by (\ref{eq:4.12a}) and (\ref{eq:4.13a}) in Lemma 4.9, we have
\begin{defn}
For $\lambda^{'} \in \Lambda_{\eta,K}, \; (\eta, K) \in S(\alpha),$ and $R_\lambda= ( \tilde{M}-\lambda)^{-1}$, we define $m$ on $\rho(\tilde{M})$ by
\begin{equation}
m(\lambda)= [ \Psi(\cdot,\lambda ),\theta(\cdot,\lambda^{'})](a), \label{eq:4.14a}
\end{equation}
where
\begin{equation}
\Psi(\cdot,\lambda) = \psi(\cdot,\lambda^{'})+(\lambda-\lambda^{'})R_\lambda\psi(\cdot,\lambda^{'}). \label{eq:4.15a}
\end{equation}
\end{defn}
\begin{rem}
In Cases II and III, the points $m(\lambda^{'})$ on the limit-circle for
$ \lambda^{'} \in \Lambda_{\eta,K}$ seem to depend on $\eta,K$ (see Remark \ref{eq:2.4})
and hence so does the extension to $\rho ( \tilde{M})$ in Definition 4.6. This is not so in Case I, in view of Lemma 3.1.
\end{rem}
\begin{lemma}
Let $\lambda ^{'} \in \Lambda_{\eta,K}, \; (\eta, K) \in S(\alpha),$
and define $m$ by (\ref{eq:4.14a})
on $\rho(\tilde{M})$, where
$R_\lambda =( \tilde{M}-\lambda)^{-1}$.
Then in (\ref{eq:4.15a})
\begin{equation}
\Psi(\cdot,\lambda)= \theta (\cdot,\lambda)+m(\lambda) \phi(\cdot,\lambda).
\label{eq:4.22a}
\end{equation}
Also (\ref{eq:3.3}) and (\ref{eq:4.11}) hold for all $\lambda \in \rho(\tilde{M})$. Hence $m$ is analytic on $\rho(\tilde{M})$, and in Cases II and III, (\ref{eq:4.14a}) and (\ref{eq:3.4}) define the same meromorphic extension of $m$, while in Case I, (\ref{eq:4.14a}) defines the same meromorphic extension to ${\bf C} \backslash Q_b(\alpha)$ as that described in Theorem 3.5.
\end{lemma}
\par
{\bf Proof}
Since
\begin{displaymath}
(M-\lambda)\Psi(\cdot,\lambda )=[ (\lambda^{'}-\lambda)+(\lambda-\lambda^{'})]
\psi(\cdot,\lambda^{'})=0
\end{displaymath}
we have that
\begin{displaymath}
\Psi(\cdot,\lambda)=A \theta(\cdot,\lambda)+B\phi(\cdot,\lambda)
\end{displaymath}
for some constants $A,B$. On using (\ref{eq:2.6a}) and (\ref{eq:4.8})
it is readily verified that
\begin{eqnarray*}
A &=& -A[\theta(\cdot,\lambda ),\phi(\cdot,\lambda^{'})](a) \\
&=& -[\Psi(\cdot,\lambda ),\phi(\cdot,\lambda^{'})](a) \\
&=& -[\psi(\cdot,\lambda^{'}),\phi(\cdot,\lambda^{'})](a)-(\lambda-\lambda^{'}) [ R_\lambda\psi(\cdot,\lambda^{'}), \phi(\cdot,\lambda^{'})](a) \\
&=& 1,
\end{eqnarray*}
and
\begin{eqnarray*}
B &=&B [\phi(\cdot,\lambda),\theta(\cdot,\lambda^{'})](a) \\
&=& [ \Psi(\cdot,\lambda),\theta(\cdot,\lambda^{'})](a) \\
&=& m(\lambda)
\end{eqnarray*}
whence (\ref{eq:4.22a}).
Also, from (\ref{eq:4.15a})
\begin{eqnarray*}
&&(\lambda-\lambda^{'})^2 ( R_\lambda \psi(\cdot,\lambda^{'}), \overline{\psi}(\cdot,\lambda^{'}))+ (\lambda-\lambda^{'})\int_a^b \psi^2(x,\lambda^{'})w(x)dx \\
&=& (\lambda-\lambda^{'})\int_a^b \Psi(x,\lambda)\psi(x,\lambda^{'})w(x)dx \\
&=&-\int_a^b \{ \Psi (x,\lambda)M \psi(x,\lambda^{'})-\psi(x,\lambda^{'})M \Psi(x,\lambda) \} w dx \\
&=& [ \Psi(\cdot,\lambda), \psi(\cdot,\lambda^{'})](b)-[ \Psi(\cdot,\lambda), \psi(\cdot,\lambda^{'})](a)
\end{eqnarray*}
by (\ref{eq:2.9})
\begin{displaymath}
= - [\Psi(\cdot,\lambda),\psi(\cdot,\lambda^{'})](a)
\end{displaymath}
by (\ref{eq:4.15a}) and since $\lambda \in \rho( \tilde{M})$,
\begin{eqnarray*}
&=& -m(\lambda)-m(\lambda^{'}) [ \Psi (\cdot,\lambda), \phi(\cdot,\lambda^{'})](a) \\
&=& m(\lambda^{'})-m(\lambda)
\end{eqnarray*}
on account of (\ref{eq:4.14a}) and again using $\lambda \in \rho(\tilde{M})$. The lemma is therefore proved.
\par
We now define, for $\lambda \in \rho( \tilde{M})$ and $f \in L^2(a,b;wdx)$,
\begin{equation}
\tilde{G}(x,y;\lambda) = \left \{\begin{array}{ll}
-\phi(x,\lambda) \Psi(y,\lambda) & a < x<y<b, \\
-\Psi(x,\lambda) \phi(y,\lambda) & a < y<x <b,
\end{array}
\right . \label{eq:4.17}
\end{equation}
\begin{equation}
\tilde{R}_\lambda f(x) := \int_a^b \tilde{G}(x,y;\lambda)f(x)w(x)dy,
\label{eq:4.18}
\end{equation}
where $\Psi$ is defined in (\ref{eq:4.22a}) and $m$ in Definition 4.10. Thus, for $\lambda \in {\bf C} \backslash Q(\alpha)$, $(\lambda \in \Lambda_{\eta,K},
\; (\eta,K) \in S(\alpha),$ in Cases II and III), we have that $R_\lambda = \tilde{R}_\lambda$. We can say more, for
(\ref{eq:4.3}), (\ref{eq:4.4}) and (\ref{eq:4.5}) hold for $\tilde{R}_\lambda$,
whenever $m(\lambda)$ is defined, and thus $\tilde{R}_\lambda=R_\lambda$ for every $\lambda$ which is such that $\tilde{R}_\lambda$ is bounded.
This is true for every $\lambda$ at which $m$ is regular in Cases II and III.
From (\ref{eq:4.14a}) and Lemma 4.12 we know that in Cases II and III $\lambda$ is a pole of $m(\lambda)$ if and only if
$\lambda$ is an eigenvalue of $\tilde{M}$;
this is also true in Case I for $\lambda \not \in Q_b(\alpha)$.
\begin{theorem}
In Cases II and III $\lambda_0$ is a pole of $m$ of order $s$ if and only if $\lambda_0$ is an eigenvalue of $\tilde{M}$ of algebraic multiplicity $s$.
\end{theorem}
{\bf Proof}
For any $f\in L^2(a,b;wdx)$, $R_\lambda f(x)$ has a pole of order $s$ at $\lambda_0$ with residue
\begin{displaymath}
\left \{
\frac{1}{(s-1)!} \frac{\partial^{s-1}}{\partial \lambda^{s-1}}
[ (\lambda-\lambda_0)^sm(\lambda) \int_a^b \phi(x,\lambda) \phi(y,\lambda) f(y) w(y)dy]
\right \}_{\lambda=\lambda_0}.
\end{displaymath}
This is of the form
\begin{equation}\sum_{j=0}^{s-1} \frac{\partial^{j}}{\partial \lambda^{j}}
\phi(x,\lambda_0) c_j(\lambda_0,f) \label{eq:4.25}
\end{equation}
where the coefficients $c_j(\lambda_0,f)$ are linear combinations of
\begin{equation}
\int_a^b \frac{\partial^{j}}{\partial \lambda^{j}}
\phi(y,\lambda_0) f(y)w(y)dy,\;\;\;j=0,1,...,s-1.
\label{eq:4.26}
\end{equation}
From $(M-\lambda)\phi(\cdot,\lambda)=0$, it follows that for $j=0,1,...s-1$,
\begin{equation}
(M-\lambda_0)\phi_j=j\phi_{j-1}, \label{eq:4.25d}
\end{equation}
\begin{equation}
(M-\lambda_0)^{j+1}\phi_j=0, \label{eq:4.26d}
\end{equation}
where
\begin{equation}
\phi_j=\frac{\partial^j}{\partial\lambda^j}\phi(\cdot,\lambda_0),\; j=0,s-1.
\label{eq:4.27d}
\end{equation}
It follows inductively from (\ref{eq:4.25d}), on using the variation of parameters, that
\begin{equation}
\phi_j \in L^2(a,b;wdx),\;\;j=0,1,...s-1.
\label{eq:4.28d}
\end{equation}
Let $\Gamma_{\lambda_0}$ be a positively oriented small circle enclosing $\lambda_0$ but excluding the other eigenvalues of $\tilde{M}$. We have
\begin{equation}
\frac{1}{2\pi i} \int_{\Gamma_{\lambda_0}} R_\lambda d\lambda=P_{\Gamma_{\lambda_0}}
\label{eq:4.27}
\end{equation}
where $P_{\Gamma_{\lambda_0}}$ is a bounded operator of finite rank given by
(\ref{eq:4.25}): its range is spanned by $\phi_j,\; j=0,1,...,s-1$.
The identity (\ref{eq:4.26d}) readily implies that the functions in (\ref{eq:4.27d})
are linearly independent. Thus $P_{\lambda_0}$ is of rank $s$, and $s$ is the algebraic multiplicity of $\lambda_0$.
The functions in (\ref{eq:4.27d}) span the algebraic eigenspace of $\tilde{M}$
at $\lambda_0$ and are the generalised eigenfunctions corresponding to $\lambda_0$: they satisfy
\begin{equation}
(\tilde{M}-\lambda_0)^{j+1}\phi_j\neq 0,\;\;\; (\tilde{M}-\lambda_0)^j\phi_j =0\;\;j=0,1,...,s-1; \label{eq:4.30}
\end{equation}
see \cite[Section III.4]{K76} and \cite{jbm61}.
In Case I, we expect Theorem 4.12 to remain true for $\lambda_0 \in Q (\alpha)\backslash Q_b(\alpha)$, but we have been unable to prove (\ref{eq:4.28d}) in this case.
\section{Examples}
\subsection{The sets $Q$ and $Q(\alpha)$}
Suppose that $[a,b)=[1,\infty)$ and the coefficients are of the form
\begin{equation}
p(x)= \mid p(x) \mid e^{i \phi},\;\;\; q(x)=q_1 x^{b_1} + i q_ 2 x ^{b_2},
\;\;\;w(x) = x^{\omega} \label{eq:5.1n}
\end{equation}
where $\phi,\; q_1 ,\;q_2 ,\;b_1, \;b_2, w$ are real constants.
Then $q(x)/w(x)$, $x \in [1,\infty)$, lie on the curve
\begin{equation}
C:= \{ z \in {\bf C}:\;\; z = q_1x^{b_1-\omega} + i q_2 x^{b_2 - \omega},\;\; x\in [1,\infty)\}
\end{equation}
The determination of the sets $Q$ and $Q(\alpha)$ is a straightforward exercise. As an illustration, we consider the case $\phi \in [ -\pi/2,\pi/2],\; q_1 <0,\; q_2 \leq 0,\; b_2 >b_1 > \omega$ in the Figures 1,2,3.
The arrows indicate addition by $r p(x),\; 0 < r <\infty$, to the point
$q(x)/w(x)$ on $C$, and the other shading in each figure is the fill-in
required to produce the closed convex set $Q$. We set $z_0=q_1+i q_2$,
$\tan \theta_0$ is the gradient of the tangent to $C$ at $z_0$, and $z_1$ the point
on $C$ where the gradient is $\tan \phi$ when $\phi \geq \theta_0$ and $z_1 = z_0$ if $\phi <
\theta_0$.
\par
The admissible values of $\eta$ (for an appropriate $K$) and the sets $Q(\alpha)$
for real values of the boundary value
parameters $\alpha \in (-\pi,\pi]$ are as follows :
(recall that $Q(\alpha)$ is defined in (\ref{eq:2.7a}), where the
admissible values of $\eta$ must now satisfy $sin 2 \alpha \;\; cos \eta \;\;
\leq 0$)
\noindent Figure 1 : $ 0 < \eta \leq \pi/2 - \phi < \pi/2$;
\begin{center}
$Q(\alpha) = \left\{\begin{array}{ll}
Q & \rm{if}\;\; \alpha \in [-\pi/2,0] \cup [\pi/2,\pi], \\
{\bf C} & \rm{if} \;\; \alpha \in (-\pi,-\pi/2) \cup (0,\pi/2). \\
\end{array}
\right. $
\end{center}
\noindent Figure 2 : $ 0 < \eta \leq \pi/2 - \phi < \pi$;
\begin{center}
$Q(\alpha) = \left\{\begin{array}{ll}
Q \;\;{\rm if}\;\; \alpha \in \{-\pi/2,0,\pi/2,\pi\}, \\
Q \cup \{z : \phi < arg(z - z_0) \leq 0 \} & \rm{if} \;\;\alpha \in (-\pi/2,0) \cup (\pi/2,\pi), \\
\{z : -\pi \leq arg(z - z_0) \leq \phi \} & \rm{if} \;\; \alpha \in (-\pi,-\pi/2)
\cup (0,\pi/2). \\
\end{array}
\right. $
\end{center}
\noindent Figure 3 : $ \eta = \pi $;
\begin{center}
$Q(\alpha) = \left\{\begin{array}{ll}
Q & \rm{if} \;\; \alpha \in [-\pi,-\pi/2] \cup [0,\pi/2], \\
{\bf C} & \rm{if} \;\; \alpha \in (-\pi/2,0) \cup (\pi/2,\pi). \\
\end{array}
\right. $
\end{center}
\begin{figure}[htbp]
\centerline{
\epsfysize=7cm
\epsffile{pic1.eps}
}
\caption{$0 < \phi < \frac{\pi}{2}$}
\end{figure}
\newpage
\begin{figure}[ht]
\centerline{
\epsfysize=7cm
\epsffile{pic3.eps}
}
\caption{$-\frac{\pi}{2} < \phi \le 0 $}
\end{figure}
\vspace{1in}
\begin{figure}[htbp]
\centerline{
\epsfysize=7cm
\epsffile{pic2.eps}
}
\caption{$\phi = \frac{\pi}{2}$}
\end{figure}
\newpage
\subsection{The classification of $(1.2)$}
In this section we
analyse the Sims classification of (\ref{eq:1.2}) when the coefficients are
\begin{equation}
p(x)=p_1x^{ a_1}+ip_2x^{ a_2},\;q(x)=q_1x^{b_1}+iq_2x^{b_2},
\;w(x)=x^\omega,
\label{eq:5.1}
\end{equation}
where $p_j,q_j, a_j,b_j\;(j=1,2)$ and $\omega$ are real, and $x \in [1,\infty)$. We write $A={\rm max }\; (a_1,a_2)$ and
$B={\rm max }\; (b_1,b_2,\omega)$. Our results follow from an analysis of the asymptotic behaviour of
linearly independent solutions of (\ref{eq:1.2}) at infinity as given by
the Liouville-Green formulae \cite{Eastham1}. A general description covering all
cases is far too complicated and hardly helpful. Instead, we provide a
prescription for determining the classification. In each specific case the details are
straightforward, though tedious.
\subsubsection{The case $A-B <2$}
In this case, linearly independent solutions $y_\pm$ exist which are such that,
as $x \rightarrow \infty$
\begin{equation}
y_\pm(x) \sim [ p(x)s(x)] ^{-1/4} \exp \left ( \pm \int^x _1 {\rm Re}
[( s/p)^{1/2}] dt \right )
\label{eq:5.2}
\end{equation}
\begin{equation}
p(x)y_\pm^{'}(x) \sim [ p(x)s(x)] ^{1/4} \exp \left ( \pm \int^x_1
{\rm Re} [( s/p)^{1/2}] dt \right )
\label{eq:5.3}
\end{equation}
where
$s(x)=q(x)-\lambda w(x)$ (see \cite[page 58]{Eastham1}). We use the notation $f(x)
\sim g(x)$ to mean that $f(x)/g(x) \rightarrow 1$ as $ x \rightarrow \infty$,
and $f(x) \stackrel{\smile}{\frown} g(x)$ if $\mid f(x)/g(x) \mid$ is bounded above and below by positive
constants.
Note that, for $z=re^{i \theta} \;\in {\bf C},\; 0\leq \theta < 2 \pi$, $r >0$,
we define the $n^{th}$ root of $z$ to be the complex number
$r^{1/n}e^{i\theta/n}$.
\par
Suppose that for some $ \Lambda_{\eta,K}$, $(\eta, K) \in S(\alpha)$, and
$\lambda \in \Lambda_{\eta,K}$, as $x \rightarrow \infty$,
\begin{equation}
{\rm Re} \left [ \left (\frac{ s(x)}{p(x)} \right )^{1/2}\right ] =
D x^\tau \left ( 1 + O(\frac{1}{x^\epsilon}) \right ) \;\;\; D
\neq 0,\; \epsilon >0,
\;\; D, \tau \in {\bf R}\label{eq:5.4}
\end{equation}
and
\begin{equation}
|p(x)s(x)| \stackrel{\smile}{\frown} x^\gamma\;\; \gamma \in {\bf R}.
\label{eq:5.5}
\end{equation}
\par
In each of the following cases, at least one of the solutions
$y_+$ and $y_-$ is not in $L^2(1,\infty;w dx)$, and hence (\ref{eq:1.2}) is
in Case I :
\begin{enumerate}
\item
$\tau >-1$;
\item
$\tau =-1$ and $2 \mid D \mid + \omega -\gamma/2 + 1 \geq 0$;
\item
$\tau <-1$ and $\omega -\gamma/2 + 1 \geq 0$;.
\end{enumerate}
In all other cases when $A-B <2$, and (\ref{eq:5.4}), and (\ref{eq:5.5})
hold, we are either in Case II or Case III: on setting
\begin{equation}
W_\pm(x):={\rm Re} \left [
e^{i\eta} \left ( p(x) \mid y^{'}_\pm (x) \mid^2 + s(x) \mid y_\pm(x)^2
\mid\right ) \right ] \label{eq:5.7}
\end{equation}
we have that Case III prevails if $W_+$ and $W_-$ are both integrable
(which can be verified using (\ref{eq:5.2}) and (\ref{eq:5.3}) ) and Case II
otherwise.
\subsubsection{The case $A-B=2$}
In this case the equation (\ref{eq:1.2}) is asymptotically of Euler type.
Here the results of \cite[page 75]{Eastham1} give, with $c=1/4( \sqrt{17}-1)$
\begin{displaymath}
\mid y_+ \;\mid \stackrel{\smile}{\frown}\; x^{ 2(A-1)c},\;\;\;
\mid py_+^{'} \mid \; \stackrel{\smile}{\frown} \;x^{ 2(A-1)(\frac{1}{2}+c)}
\end{displaymath}
and
\begin{displaymath}
\mid y_- \mid \; \stackrel{\smile}{\frown} x^{- 2(A-1)(\frac{1}{2}+c)},\;\;\;
\mid py_-^{'}\; \mid \stackrel{\smile}{\frown} x^{ -2(A-1)c}.
\end{displaymath}
At least one of the solutions $y_+$, $y_-$ is not in $L^2(1,\infty; w dx)$,
and hence (\ref{eq:1.2}) is in Case I, in each of the following cases:
\begin{enumerate}
\item
$A >1$ and $\omega + 4 ( A -1)c +1 \geq 0$;
\item
$A =1$ and $\omega \geq -1$;
\item
$A <1$ and $\omega - 4 ( A -1)(\frac{1}{2}+c) +1 \geq 0$.
\end{enumerate}
In all other cases when $A-B =2$, we are in Case III when
$W_+$ and $W_-$ defined in (\ref{eq:5.7}) are both integrable, and
Case II otherwise.
\subsubsection{The case $A-B >2$}
Here the relevant analysis is that in \cite[page 78]{Eastham1}.
It follows that
\begin{displaymath}
\mid y_+ \mid \; \stackrel{\smile}{\frown} 1,\;\;\; \mid y_+^{'}\mid \; \stackrel{\smile}{\frown} \; x^{ (B-A)/2},
\end{displaymath}
\begin{displaymath}
\mid y_- \mid \; \stackrel{\smile}{\frown} \; x^{-(A+B)/2},\;\;\; \mid y_-^{'}\mid \; \stackrel{\smile}{\frown} \;x^{ -A }.
\end{displaymath}
\par
At least one of the solutions $y_+$,$y_-$ is not in $L^2(1,\infty; w dx)$,
and hence (\ref{eq:1.2}) is in Case I, if $\omega - \rm{min} \{0,A+B\} \geq -1$.
If $\omega - \rm{min} \{0,A+B\} <-1$, (\ref{eq:1.2}) is in Case III if $W_\pm$ are both integrable and Case II otherwise.
\par
The case $p=w=1$ is covered in detail in \cite[Theorem III, 10.28]{EE87};
this includes the original example of Sims \cite[p. 257]{Sims57} establishing the existence of Case II.
\subsection{The spectra}
Finally, we investigate the spectra of the operators $\tilde{M}$ generated
in $L^2(0,\infty)$ by expressions $M$ of the form
\begin{equation}
M[y]=-y^{''}+c x^\beta y, \;\;\;0\leq x<\infty, \label{eq:1}
\end{equation}
where $\beta >0$ and $c \in {\bf C}$ with arg$ \;c \in [0,\pi]$;
the case arg $ c \in (\pi,2 \pi)$ is similar.
\par
If arg $c \neq \pi$, we have
\begin{equation}
Q=\{z:0\leq {\rm arg} z \leq {\rm arg}\; c\},\;\;\; Q_\infty = \emptyset.
\label{eq:2}
\end{equation}
Suppose that
\begin{equation}
{\rm Im } [ \overline { \sin\alpha} \cos \alpha ] \geq 0.
\label{eq:3}
\end{equation}
Then, (\ref{eq:2.6}) is satisfied for $\eta =-\frac{\pi}{2}$ and, for any
$K>0$, $(-\pi/2,K) \in S(\alpha)$.
Consequently
\begin{equation}
Q(\alpha) \subseteq {\bf C}\backslash \Lambda_{-\pi/2,K}= \overline{\bf C_+}
\label{eq:4}
\end{equation}
and, similarly,
\begin{equation}
Q_\infty(\alpha)=\emptyset \label{eq:5}
\end{equation}
(see (\ref{eq:3.7})).
Also, it follows from Section 5.2.1 (item 1) that Case I holds.
Hence, by Theorem 4.7 and Remark 4.8, for arg $c\neq \pi$, the operator realisation $\tilde{M}$ of $M$ defined in (\ref{eq:4.12})
has empty essential spectrum $\sigma_e(\tilde{M})$.
Such a result is given in \cite[Theorem 30]{G65} for the analogous problem on $(-\infty, \infty)$.
\par
If arg $c=\pi$, we have
\begin{equation}
Q=Q_\infty={\bf R} \label{eq:6}
\end{equation}
and, if (\ref{eq:3}) is satisfied,
$
Q(\alpha) \subseteq ({\bf C} \backslash \Lambda_{-\pi/2,K}) \cap
({\bf C }
\backslash \Lambda_{\pi/2,K})={\bf R}$, and hence
\begin{equation}
Q(\alpha) = Q_{\infty}(\alpha) = {\bf R}. \label{eq:8}
\end{equation}
For $\lambda = i$ and $\eta = \pm \pi/2$, we now have $\mid W_\pm \mid =
\mid y_\pm \mid ^2$ and in Section 5.2.1
\begin{displaymath}
y_\pm(x) \stackrel{\smile}{\frown}
\left \{
\begin{array}{cc}
x^{-\beta/4} & {\rm if\;} \beta >2, \\
x^{-\frac{1}{2} \mp \frac{1}{2 \mid c \mid ^{1/2}}} & {\rm if \;} \beta = 2, \\
x^{-\frac{\beta}{4}} exp [ \mp \frac{x^{1-\beta/2}}{\mid c \mid ^{1/2}
(2-\beta)} ] & {\rm if \;} \beta < 2.
\end{array}
\right .
\end{displaymath}
It follows that Case I holds if $\beta \leq 2$ and Case III if $\beta > 2$;
note that Case III is now the Weyl limit-circle case since $M$ is formally
symmetric.
Hence, if arg $c=\pi$, by Theorem 4.5,
\begin{equation}
\sigma_e(\tilde{M})
\left \{
\begin{array}{cc}
= \emptyset & {\rm if\;} \beta >2, \\
\subseteq {\bf R} &{\rm if \;} \beta \leq 2.
\end{array}
\label{eq:9}
\right .
\end{equation}
If $\alpha$ is real, (\ref{eq:3}) is satisfied. In this case, when $\beta
\leq 2$, $M$ is in the Weyl limit-point
case at $\infty$ (so that $\tilde{M}$ is self-adjoint)
and $\sigma_e(\tilde{M})={\bf R}$ (see \cite[Theorem V.5.10]{ECT62}).
\par
In Case I the identity (\ref{eq:5.17a}) below (which holds for (\ref{eq:1.2})
in general) is often useful and reinforces Remark 4.8.
Denote the functions $\theta,\; \phi$ in (\ref{eq:2.6a})
by $\theta_\alpha, \; \phi_\alpha$ respectively,
and the corresponding $m-$function by $m_\alpha$.
Since $\alpha=0,\pi/2$ satisfy (\ref{eq:2.6})
for any $\eta$, we have $Q(0)=Q(\pi/2)=Q$.
Also, for $\lambda \not \in Q(\alpha)$, there exist $K \neq 0$ such that
\begin{displaymath}
\theta_\alpha (x,\lambda)+m_\alpha (\lambda) \phi_\alpha(x,\lambda)
=K[ \theta_{\pi/2}(x,\lambda)+m_{\pi/2}(\lambda) \phi_{\pi/2}(x,\lambda)].
\end{displaymath}
On substituting (\ref{eq:2.6a}) we have
\begin{equation}
m_\alpha(\lambda) =
\frac{ m_{\pi/2}(\lambda) \sin \alpha-\cos \alpha}
{m_{\pi/2}(\lambda) \cos \alpha + \sin \alpha}.
\label{eq:5.17a}
\end{equation}
Hence, if $m_{\pi/2}$ is meromorphic in ${\bf C}$,
the same is true of $m_\alpha$, for any $\alpha$.
\par
An important special case of (\ref{eq:1}) is the expression for the harmonic oscillator
\begin{displaymath}
M[y]= -y^{''} +c x^2 y \;\;\;\; 0 \leq x < \infty.
\end{displaymath}
On setting
$x=\frac{z }{\sqrt{2} c^{1/4}}$, the equation $(M-\lambda)[y]=0$ becomes
\begin{equation}
-y^{''} +\frac{1}{4}z^2 y = \mu y \label{eq:10}
\end{equation}
where $'$ now denotes differentiation with respect to $z$ along the ray with argument $\frac{1}{4} {\rm arg }\; c$, and $\mu= \frac{\lambda}{2 \sqrt{c}}$.
From \cite[page 341]{WW15}, for $0 \leq {\rm arg } c < \pi$,
the unique solution of (\ref{eq:10}) in $L^2(0,\infty)$ is the parabolic
cylinder function
$D_{\mu-1/2}(z)$.
It follows from (\ref{eq:2.6a}) and the fact that our function $\psi$ in
(\ref{eq:2.21}) must be a
constant multiple of $D_{\mu-1/2}(z)$ that
\begin{displaymath}
m_{\pi/2}(\lambda) = \frac{ D_{\mu-1/2}(0)}{D^{'}_{\mu-1/2}(0)}
\end{displaymath}
and this gives
\begin{equation}
m_{\pi/2}(\lambda) = -\frac{1}{2 c^{1/4}} \frac{\Gamma(1/4 - \frac{\lambda}{ 4 \sqrt{c}}
)}{ \Gamma(3/4 - \frac{\lambda}{ 4 \sqrt{c}})}.
\end{equation}
This is meromorphic with poles at
\begin{displaymath}
\lambda_n = ( 4 n+1)\sqrt{c},\;\;\; n=0,1,2,...
\end{displaymath}
When arg $c=\pi$, $Q=Q_\infty ={\bf R}$, and for $\alpha=\pi/2$,
there are $m$-functions defined in ${\bf C_+}$ and ${\bf C_- }$ :
\begin{eqnarray*}
m^{(1)}_{\pi/2} &=& -\frac{e^{- i\pi/4}}{2 \mid c\mid ^{1/4}} \frac{\Gamma(1/4 - \frac{\lambda}{ 4 \sqrt{c}}
)}{ \Gamma(3/4 - \frac{\lambda}{ 4 \sqrt{c}})} \;\;\; (\lambda \in {\bf C_+}) \\
m^{(2)}_{\pi/2}
&=& -\frac{e^{i\pi/4}}{2 \mid c\mid ^{1/4}} \frac{\Gamma(1/4 +\frac{\lambda}{ 4 \sqrt{c}}
)}{ \Gamma(3/4 + \frac{\lambda}{ 4 \sqrt{c}})} \;\;\; (\lambda \in {\bf C_-});
\end{eqnarray*}
${\bf C_+},{\bf C_-}$ are the connected components $C_1,C_2$ referred to in Lemma 3.1 and the following comment.
These functions are not analytic continuations of each other and the self adjoint operator $\tilde{M}$, with $\alpha = \pi/2$, has $\sigma_e(\tilde{M})={\bf R}$: this is therefore true for all values of $\alpha$ by Remark 4.8.
Criteria on $q$ for $\sigma_e(\tilde{M}) \supseteq [0,\infty)$ in the case $p=w=1$ are given in \cite{K79}; see also \cite[Chapter VII]{G65}.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,282 |
{"url":"https:\/\/curriculum.illustrativemathematics.org\/MS\/teachers\/2\/8\/9\/index.html","text":"# Lesson 9\n\nMulti-step Experiments\n\n## 9.1: True or False? (5 minutes)\n\n### Warm-up\n\nThe purpose of this warm-up is to gather strategies and understandings students have for averaging numbers. Understanding these strategies will\u00a0help students develop fluency and will be useful later in this unit when students will need to be able to compute averages of values.\n\nWhile 3 problems are given, it may not be possible to share every strategy for all the problems. Consider gathering only 2\u00a0or 3\u00a0different strategies per problem, saving most of the time for the final question.\n\n### Launch\n\nReveal one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all previous problems displayed throughout the talk. Follow with a whole-class discussion.\n\n### Student Facing\n\nIs each equation true or false? Explain your reasoning.\n\n$$8=(8+8+8+8)\\div3$$\n\n$$(10+10+10+10+10)\\div5=10$$\n\n$$(6+4+6+4+6+4)\\div6=5$$\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Activity Synthesis\n\nAsk students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:\n\n\u2022 \u201cWho can restate ___\u2019s reasoning in a different way?\u201d\n\n\u2022 \u201cDid anyone have the same strategy but would explain it differently?\u201d\n\n\u2022 \u201cDid anyone solve the problem in a different way?\u201d\n\n\u2022 \u201cDoes anyone want to add on to _____\u2019s strategy?\u201d\n\n\u2022 \u201cDo you agree or disagree? Why?\u201d\n\n## 9.2: Spinning a Color and Number (10 minutes)\n\n### Activity\n\nIn this activity, students are reminded how to calculate probability based on the number of outcomes in the sample space, then apply that to multi-step experiments. The events are described in everyday language, so students need to reason abstractly (MP2) to identify\u00a0the outcomes described. This lesson begins with students returning to a problem they have previously seen when writing out the sample space. This will save students some time if they can recall or refer back to the initial problem.\u00a0In the following activities, students will work with situations for which they have not written out the sample space to practice finding probabilities using all the necessary steps.\n\n### Launch\n\nArrange students in groups of 2.\n\nDisplay the two spinners for all to see. Ask students, \u201cWhat do you notice? What do you wonder?\u201d\n\nGive students 1 minute to think about the image. Record their responses for all to see.\n\nStudents may notice:\n\n\u2022 The number of sections in each spinner.\n\u2022 The labels for the two spinners.\n\u2022 Within each spinner, the sections are equally sized.\n\nStudents may wonder:\n\n\u2022 Do you choose which one to spin or do you spin both?\n\u2022 If you spin both, how many different outcomes will there be?\n\u2022 Is this part of a game? If so, what is a \u201cgood\u201d spin?\n\nTell students: \u201cFor sample spaces where each outcome is equally likely, recall that the probability of an event can be computed by counting the number of outcomes in the event and dividing that number by the total number of outcomes in the sample space.\u201d For example, in the previous lesson, students found that there were 12 possible outcomes when flipping a coin and rolling a number cube. If we wanted the probability of getting heads and rolling an even number, we count that there are 3 ways to do this (H2, H4, and H6) out of the 12 outcomes in the sample space. So, the probability of getting heads and an even number should be $$\\frac{3}{12}$$ or $$\\frac{1}{4}$$ or 0.25.\n\nRemind students that they have already drawn out the sample space for this chance experiment in a previous activity,\u00a0and they may use that to help answer the questions.\n\nGive students of 5 minutes quiet work time followed by partner and whole-class discussion.\n\nRepresentation: Access for Perception. Provide access to concrete manipulatives. Provide spinners for students to view or manipulate. These hands-on models will help students identify characteristics or features, and support finding outcomes for calculating probabilities.\nSupports accessibility for: Visual-spatial processing; Conceptual processing\n\n### Student Facing\n\nThe other day, you wrote the sample space for spinning each of these spinners once.\n\nWhat is the probability of getting:\n\n1. Green and 3?\n2. Blue and any odd number?\n3. Any color other than red and any number other than 2?\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Activity Synthesis\n\nThe purpose of the discussion is for students to explain their interpretations of the questions and share methods for solving.\n\nSome questions for discussion:\n\n\u2022 \u201cHow did you calculate the number of outcomes in the sample space?\u201d (Counting the items in the tree, table, or list, or using the multiplication idea from an earlier lesson.)\n\u2022 \u201cAlthough we had the sample space for this situation in a previous problem, how could you find the sample space if you did not know it already?\u201d (Draw a tree, table, or list.)\n\u2022 \u201cFor each problem, how many outcomes were in the event that was described? How did you count them?\u201d\nRepresenting, Speaking, Listening: MLR2 Collect and Display. As pairs discuss strategies for calculating the probability of each outcome, circulate and write down the words and phrases students use to explain their reasoning. Listen for students who reference their representation of the sample space (e.g., list, table, or tree) to determine the probability of each outcome. As students review the language collected in the visual display, encourage students to revise and improve how ideas are communicated. For example, a phrase such as: \u201cThe probability is $$\\frac{1}{20}$$, because there are 20 outcomes\u201d can be improved with the phrase \u201cThe probability is $$\\frac{1}{20}$$, because there is only 1 outcome in the event, and there are 20 equally likely outcomes in the sample space.\u201d This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.\nDesign Principle(s): Support sense-making; Maximize meta-awareness\n\n## 9.3: Cubes and Coins (20 minutes)\n\n### Activity\n\nIn this activity, students continue to compute probabilities for multi-step experiments using the number of outcomes in the sample space. The first problem involves a situation for which students have already seen the sample space. Following this problem, the class will discuss the merits of the different representations for writing out the sample space.\u00a0The next two problems involve situations in which students may need to write out the sample space on their own.\u00a0Students are also reminded that some events have a probability of 0, which represents an event that is impossible. In the discussion following the activity, students are asked to think about the probabilities of two events that make up the entire sample space and have no outcomes common to both events (MP2).\n\n### Launch\n\nKeep students in groups of 2.\n\nAssign each group a representation for writing out the sample space: a tree, a table, or a list. Tell students that they should write out the sample space for the first problem using the representation they were assigned. (This was done for them in a previous lesson and they are allowed to use those as a guide if they wish.)\n\nTell students that they should work on the first problem only and then pause for a discussion before proceeding to the next problems.\n\nGive students 2 minutes of partner work time for the first problem followed by a pause for a whole-class discussion centered around the different representations for sample space.\n\nAfter all groups have completed the first question, select at least one group for each representation and have them explain how they arrived at their answer. As the groups explain, display the appropriate representations for all to see. Ask each of the groups how they counted the number of outcomes in the sample space as well as the number of outcomes in the event using their representation.\n\nList:\n\nTable:\n\n1 2 3 4 5 6\nH1 H2 H3 H4 H5 H6\nT1 T2 T3 T4 T5 T6\n\nTree:\n\nAfter students have had a chance to explain how they used the representations, ask students to give some pros and cons for using each of the representations. For example, the list method may be easy to write out and interpret, but could be very long and is not the easiest method for keeping track of which outcomes have been written and which still need to be included.\n\nAllow the groups to continue with the remaining problems, telling them they may use any method they choose to work with the sample space for these problems. Give students 10 minutes of partner work time followed by a whole-class discussion about the activity as a whole.\n\n### Student Facing\n\nThe other day you looked at a list, a table, and a tree that showed the sample space for rolling a number cube and flipping\u00a0a coin.\n\n1. Your teacher will assign you one of these three structures to use to answer these questions. Be prepared to explain your reasoning.\n\n1. What is the probability of getting tails and a 6?\n2. What is the probability of getting heads and an odd number?\n\n2. Suppose you roll two number cubes. What is the probability of getting:\n\n1. Both cubes showing the same number?\n2. Exactly one cube\u00a0showing an even number?\n3. At least one cube\u00a0showing an even number?\n4. Two values that have a sum of 8?\n5. Two values that have a sum of 13?\n3. Jada flips three quarters. What is the probability that all three will land showing the same side?\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Anticipated Misconceptions\n\nSome students may not recognize that rolling a 2 then a 3 is different from rolling a 3 then a 2. Ask students to imagine the number cubes are different colors to help see that there are actually 2 different ways to get these results.\n\nSimilarly, some students may think that HHT counts the same as HTH and THH. Ask the student to think about the coins being flipped one at a time rather than all tossed at once. Drawing an entire tree and seeing all the branches may further help.\n\n### Activity Synthesis\n\nThe purpose of the discussion is for students to explain their methods for solving the problems and to discuss how writing out the sample space aided in their solutions.\n\nPoll the class on how they computed the number of outcomes in the sample space and the number of outcomes in the event for the second set of questions given these options: List, Table, Tree, Computed Outcomes Without Writing Them All Out, Another Method.\n\nConsider these questions for discussion:\n\n\u2022 \u201cWhich representation did you use for each of the problems?\u201d\n\u2022 \u201cDo you think you will always try to use the same representation, or can you think of situations when one representation might be better than another?\u201d\n\u2022 \u201cDid you have a method for finding the number of outcomes in the sample space or event that was more efficient than just counting them?\u201d (The number of outcomes in the sample space for the number cubes could be found using $$6 \\boldsymbol{\\cdot} 6 = 36$$. To find the number of outcomes with at least 1 even number, I knew there would be 6 for each time an even was rolled first and only 3 for each time an odd number was rolled first, so I found the number of outcomes by $$3 \\boldsymbol{\\cdot} 6 + 3 \\boldsymbol{\\cdot} 3 = 27$$.)\n\u2022 \u201cOne of the events had a probability of zero. What does this mean?\u201d\n(It is impossible.)\n\u2022 \u201cWhat would be the probability of an event that was certain?\u201d (1)\n\u2022 \u201cJada was concerned with having all the coins show the same side. What would be the probability of having at least 1 coin not match the others?\u201d ($$\\frac{6}{8}$$, since there are 6 outcomes where at least 1 coin does not match: HHT, TTH, HTH, THT, HTT, THH.)\n\u2022 \u201cHow do the answers to Jada\u2019s question and the one we just answered relate to one another?\u201d (Since every outcome in the sample space has either \u201cat least one heads\u201d or \u201call tails,\u201d and there is no outcome that applies to both events, together the sum of their probabilities must be 100% or 1.)\nSpeaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and\/or revoice what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to produce language as they interpret the reasoning of others.\nDesign Principle(s): Support sense-making\n\n## 9.4: Pick a Card (15 minutes)\n\n### Optional activity\n\nThe activity provides further practice in finding probabilities of events.\n\nIn this activity, students see an experiment that has two steps where the result of the first step influences the possibilities for the second step. Often this process is referred to as doing something \u201cwithout replacement.\u201d At this stage, students should approach these experiments in a very similar way to all of the other probability questions they have encountered, but they must be very careful about the number of outcomes in the sample space (MP6).\n\n### Launch\n\nKeep students in groups of 2. Give students 5\u20137 minutes of quiet work time followed by partner and whole-class discussion.\n\nIdentify students who are not noticing that it is impossible to draw the same color twice based on the instructions. Refocus these students by asking them to imagine drawing a red card on the first pick and thinking about what\u2019s possible to get for the second card.\n\nRepresentation: Access for Perception. Provide access to concrete manipulatives. Provide five different colored cards for students to view or manipulate. These hands-on models will help students identify characteristics or features, and support finding outcomes for calculating probabilities.\nSupports accessibility for: Visual-spatial processing; Conceptual processing\nConversing: MLR5 Co-Craft Questions. Display the initial task statement that begins, \u201cImagine there are five cards...\", before revealing the questions that follow. Ask pairs to write down possible mathematical questions that can be answered about the situation. As pairs share their questions with the class, listen for and highlight questions that ask how the \u201cwithout putting the card back\u201d part of the scenario would change the outcome of the sample space. This will help students consider the differences between this problem and previous problems and how that impacts the sample space.\nDesign Principle(s): Optimize output; Maximize meta-awareness\n\n### Student Facing\n\nImagine there are 5 cards. They are colored red, yellow, green, white, and black. You mix up the cards and select one of them without looking. Then, without putting that card back, you mix up the remaining cards and select another one.\n\n1. Write the sample space\u00a0and tell how many possible outcomes there are.\n\n2. What structure did you use to write all of the outcomes (list, table, tree, something else)? Explain why you chose that structure.\n\n3. What is the probability that:\n\n1. You get a white card and a red card (in either order)?\n\n2. You get a black card (either time)?\n\n3. You do\u00a0not get a black card (either time)?\n\n4. You get a blue card?\n\n5. You get 2 cards of\u00a0the same color?\n\n6. You get 2 cards of different colors?\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Student Facing\n\n#### Are you ready for more?\n\nIn a game using five cards numbered 1, 2, 3, 4, and 5, you take two cards and add the values together. If the sum is 8, you win. Would you rather pick a card and put it back before picking the second card, or keep the card in your hand while you pick the second card? Explain your reasoning.\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Anticipated Misconceptions\n\nStudents may misread the problem and think that they replace the card before picking the next one. Ask these students to read the problem more carefully and ask the student, \u201cWhat is possible to get when you draw the second card while you already have a red card in your hand?\u201d\n\n### Activity Synthesis\n\nThe purpose of the discussion is for students to compare the same context with replacement and without replacement.\n\nConsider asking these questions for discussion:\n\n\u2022 \u201cWhat would change about your calculations if the experiment required replacing the first card before picking a second card?\u201d (There would be 25 outcomes in the sample space. The probability of getting the same color twice would be $$\\frac{5}{25}$$. The probability of getting different colors would be $$\\frac{20}{25}$$. The probability of getting red and white would be $$\\frac{2}{25}$$. The probability of getting a black card would be $$\\frac{9}{25}$$ and not getting a black card would be $$\\frac{16}{25}$$. It would still be impossible to get a blue card, so its probability would be 0.)\n\u2022 \u201cWhat do you notice about the sum of the probability of getting a black card and the probability of not getting a black card?\u201d (They have a sum of 1.)\n\u2022 \u201cExplain why these outcomes might have probabilities with this relationship.\u201d (Since you either get a black card or not, together their probabilities should be 1 or 100%.)\n\n## Lesson Synthesis\n\n### Lesson Synthesis\n\nThese discussion questions will help students reflect on their learning:\n\n\u2022 \u201cWhen the outcomes in the sample space are equally likely, how is the size of the sample space used to calculate the probability of an event?\u201d\n\u2022 \u201cNow that you\u2019ve have plenty of practice, do you have a favorite method for writing out the sample space?\u201d\n\u2022 \u201cAre there times that one strategy for writing out the sample space makes more sense than others?\u201d\n\n## 9.5: Cool-down - A Number Cube and 10 Cards (5 minutes)\n\n### Cool-Down\n\nFor access, consult one of our IM Certified Partners.\n\n## Student Lesson Summary\n\n### Student Facing\n\nSuppose we have two bags. One contains 1\u00a0star block\u00a0and 4\u00a0moon blocks. The other contains 3 star blocks and 1\u00a0moon block.\n\nIf we select one block at random from each, what is the probability that we will get two star blocks or two moon blocks?\n\nTo answer this question, we can draw a tree diagram to see all of the possible outcomes.\n\nThere are $$5 \\boldcdot 4 = 20$$ possible outcomes. Of these,\u00a03 of them are both stars, and 4 are both moons. So the probability of getting 2 star blocks or 2 moon blocks is $$\\frac{7}{20}$$.\n\nIn general, if all outcomes in an experiment are equally likely, then the probability of an event is the fraction of outcomes in the sample space for which the event occurs.","date":"2021-06-18 06:07:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5605846643447876, \"perplexity\": 720.8853173429607}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487635724.52\/warc\/CC-MAIN-20210618043356-20210618073356-00172.warc.gz\"}"} | null | null |
National Defence Fund (NDF) is an Indian Government Institution, set up in the year 1962 to receive voluntary donations for the promotion and welfare of the members of the Indian Armed Forces (including paramilitary forces) and their dependents. Members of the executive committee include the Prime Minister of India—as chairperson, Home Minister, Defence Minister and Finance Minister—Treasurer. Donations to the National Defence Fund are 100% tax exempt. Donations can be made through an online government portal also.
Income and expenditure
The income and expenditure of the NDF for the period 2013 to 2019 is as follows (in crore rupees):
Notable contributions
Osman Ali Khan, Asaf Jah VII, the last Nizam of Hyderabad of the former Hyderabad State, made a donation of 425 kg of gold to the National Defence Fund in 1965.( Hindu 11 NOV 2018)
References
1962 establishments in India
Government agencies of India
Ministry of Defence (India) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 315 |
28S ribosomal protein S22, mitochondrial is a protein that in humans is encoded by the MRPS22 gene.
Mammalian mitochondrial ribosomal proteins are encoded by nuclear genes and help in protein synthesis within the mitochondrion. Mitochondrial ribosomes (mitoribosomes) consist of a small 28S subunit and a large 39S subunit. They have an estimated 75% protein to rRNA composition compared to prokaryotic ribosomes, where this ratio is reversed. Another difference between mammalian mitoribosomes and prokaryotic ribosomes is that the latter contain a 5S rRNA. Among different species, the proteins comprising the mitoribosome differ greatly in sequence, and sometimes in biochemical properties, which prevents easy recognition by sequence homology. This gene encodes a 28S subunit protein that does not seem to have a counterpart in prokaryotic and fungal-mitochondrial ribosomes. This gene lies telomeric of and is transcribed in the opposite direction from the forkhead box L2 gene. A pseudogene corresponding to this gene is found on chromosome Xq.
References
Further reading
Ribosomal proteins | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,808 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.