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Der er flere regenter med navnet Ferdinand 2. Se:
Ferdinand 2. af Aragonien (1452-1516) – konge af Aragonien 1452-1516
Ferdinand 2. (Tysk-romerske rige) (1578-1637) – tysk-romersk kejser 1619-1637
Ferdinand 2. af Begge Sicilier (1810-1859) – konge af Begge Sicilier 1830-1859
Ferdinand 2. af Portugal (1816-1885) – kongegemal af Portugal 1837-1853
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,282
|
I had such a great time cross-country skiing over the holidays in Michigan, I was anxious to find a place close to home in NE Wisconsin that I could keep up the activity. Thank goodness for the internet search engines, I found a gem, Gordon Bubolz Nature Preserve, just ten minutes up the road. I found that they charge to ski the trails or you can add your name to their membership for a mere $25 and have unlimited access to their 775 acres. The area features a huge peat bog, prairie, and woodland landscape. The ski trails are centered mostly in the peat bog with trails of varying distances. The trails interconnect so you can make your outing as long as you want without retracing your steps. I have skied the Deer Run Trail several times which is about 2.5 mi. But if you don't want to ski the trails, they offer hiking and snowshoeing pathways as well. The Nature Center has rental equipment for the times you may have guests, but no equipment and I found the prices very reasonable. Most important of all, the Nature Center has nice warm restroom facilities and often they have something hot available to heat up your insides. While you are warming up, you can wander around the lodge-like interior of the center where they have maps of the nature preserve, history, and lots and lots of stuffed animals and fish mounted on the walls. They offer monthly activities to the public (I am looking forward to the Maple Syrup activities in March). While skiing I noted many boardwalk and docklike structures for visitors to get "up close" for warmer weather bird watching and general wildlife observation. Local schools use this resource to enhance their curriculum. My neighbor Ray said his grand daughter's class went out there last spring. He asked her what she did there. She gave the standard kid answer "nothing" . Upon more inquiry, Ray found out her class participated in the Maple Syrup tapping, collecting, and tasting activites. They also offer monthly moonlight activities (skiing and snowshoeing). If we get more snow it might be a fun adventure!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,208
|
Q: How can I convert CT Nifti Files to Hounsfield Units? What is the correct way to convert CT scan nifti files to Hounsfield units? My code is as follows:
path = 'input/volume/volume-0.nii'
img_obj = nib.load(path)
img_data = img_obj.get_fdata()
slope = img_obj.dataobj.slope
intercept = img_obj.dataobj.inter
img_data[img_data >= 1200] = 0 #trying to remove bone area
images = slope * img_data.astype(np.float32)
hu_images = images + intercept
But when I try to normalize this hu converted image to [0,1], it yields a black image.
A: nib.get_fdata() should already scale your data if the corresponding header fields are set correctly. For an explanation, see this nipy article.
Note that if you're trying to set high HU densities to 0 (with img_data[img_data >= 1200] = 0), you're giving it the same density as water. What you probably want is to set it to air density (HU -1,000).
In order to scale the image to [0, 1] you could use:
from skimage.exposure import rescale_intensity
rescale_intensity(img_data, in_range='image', out_range=(0., 1.))
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,197
|
using System;
using stCore;
using stNet;
using stNet.Syslog;
using stDokuWiki;
using stDokuWiki.WikiEngine;
using stDokuWiki.Data;
using System.Collections.Generic;
namespace stCoCServerConfig.CoCServerConfigData
{
public class Configuration : IDisposable
{
public Configuration()
{
}
~Configuration()
{
this.Dispose();
}
public bool Disposed = false;
private IrcClient _irc = null;
public IrcClient Irc
{
get { return this._irc; }
set { this._irc = value; }
}
private stCoCAPI.CoCAPI _cocapi = null;
public stCoCAPI.CoCAPI Api
{
get { return this._cocapi; }
set { this._cocapi = value; }
}
private stCore.IOFile _logfile = null;
public stCore.IOFile LogDump
{
get { return this._logfile; }
set { this._logfile = value; }
}
private dynamic _irccmd = null;
public dynamic IrcCmd
{
get { return this._irccmd; }
set { this._irccmd = value; }
}
private stNet.stWebServer _web = null;
public stNet.stWebServer HttpSrv
{
get { return this._web; }
set { this._web = value; }
}
private stNet.stWebServerUtil.HtmlTemplate _tmpl = null;
public stNet.stWebServerUtil.HtmlTemplate HtmlTemplate
{
get { return this._tmpl; }
set { this._tmpl = value; }
}
private stDokuWiki.WikiEngine.WikiFile _wiki = null;
public stDokuWiki.WikiEngine.WikiFile WikiEngine
{
get { return this._wiki; }
set { this._wiki = value; }
}
private stGeo.GeoFilter _gf = null;
public stGeo.GeoFilter Geo
{
get { return this._gf; }
set { this._gf = value; }
}
private stSysLogNG _syslog = null;
public stSysLogNG SysLog
{
get { return this._syslog; }
set { this._syslog = value; }
}
private IMessage _iLog = null;
public IMessage ILog
{
get { return this._iLog; }
set { this._iLog = value; }
}
public CoCServerConfigData.Option Opt;
public DateTime StatTime;
public void Dispose()
{
this.Disposed = true;
if (this.Irc != null)
{
this.Irc.Disconnect(true);
this.Irc.Dispose();
this.Irc = null;
}
if (this.HttpSrv != null)
{
this.HttpSrv.Stop();
this.HttpSrv.Dispose();
this.HttpSrv = null;
}
if (this.Api != null)
{
this.Api.Stop();
this.Api.Dispose();
this.Api = null;
}
if (this.Geo != null)
{
this.Geo.Dispose();
this.Geo = null;
}
if (this.LogDump != null)
{
this.LogDump.Close();
this.LogDump = null;
}
if (this.SysLog != null)
{
this.SysLog.Close();
this.SysLog = null;
}
}
}
public class Option
{
public Option()
{
}
public object this[string pName]
{
get {
var Obj = this.GetType().GetProperty(pName);
if (Obj != null)
{
return Obj.GetValue(this, null);
}
return null;
}
set {
var Obj = this.GetType().GetProperty(pName);
if (Obj != null)
{
Obj.SetValue(this, value, null);
}
}
}
public OptionItem IRCPort { get; set; }
public OptionItem IRCServer { get; set; }
public OptionItem IRCPassword { get; set; }
public OptionItem IRCAdminPassword { get; set; }
public OptionItem IRCChannel { get; set; }
public OptionItem IRCNik { get; set; }
public OptionItem IRCLanguage { get; set; }
public OptionItem IRCFloodTimeOut { get; set; }
public OptionItem IRCSOutDirName { get; set; }
public OptionItem IRCSOutFileName { get; set; }
public OptionItem IRCLogTimeFormat { get; set; }
public OptionItem IRCServerMessage { get; set; }
public OptionItem IRCNoticeMessage { get; set; }
public OptionItem IRCKickRespawn { get; set; }
public OptionItem IRCSetNewChannel { get; set; }
public OptionItem IRCPluginSayEnable { get; set; }
public OptionItem IRCPluginClanEnable { get; set; }
public OptionItem IRCPluginHelpEnable { get; set; }
public OptionItem IRCPluginModeEnable { get; set; }
public OptionItem IRCPluginTimeEnable { get; set; }
public OptionItem IRCPluginTopicEnable { get; set; }
public OptionItem IRCPluginUpTimeEnable { get; set; }
public OptionItem IRCPluginVersionEnable { get; set; }
public OptionItem IRCPluginUrlShortEnable { get; set; }
public OptionItem IRCPluginNotifySetupEnable { get; set; }
public OptionItem IRCPluginLangSetupEnable { get; set; }
public OptionItem IRCPluginContextUrlTitleEnable { get; set; }
public OptionItem IRCPluginLoopClanNotifyEnable { get; set; }
public OptionItem IRCPluginLoopClanNotifyPeriod { get; set; }
public OptionItem SYSAppName { get; set; }
public OptionItem SYSROOTPath { get; set; }
public OptionItem SYSCONFPath { get; set; }
public OptionItem SYSGEOPath { get; set; }
public OptionItem SYSIRCLOGPath { get; set; }
public OptionItem SYSTMPLPath { get; set; }
public OptionItem SYSLANGConsole { get; set; }
public OptionItem CLANTag { get; set; }
public OptionItem CLANAPIKey { get; set; }
public OptionItem CLANInformerStaticEnable { get; set; }
public OptionItem SQLDBPath { get; set; }
public OptionItem SQLDBUri { get; set; }
public OptionItem SQLDBUpdateTime { get; set; }
public OptionItem SQLDBFilterMemberTag { get; set; }
public OptionItem WEBRootUri { get; set; }
public OptionItem WEBRootPort { get; set; }
public OptionItem WEBLANGDefault { get; set; }
public OptionItem WEBCacheEnable { get; set; }
public OptionItem WEBFrontEndEnable { get; set; }
public OptionItem WEBRequestDebugEnable { get; set; }
public OptionItem DOKUWikiAuthEnable { get; set; }
public OptionItem DOKUWikiRootUrl { get; set; }
public OptionItem DOKUWikiRootPath { get; set; }
public OptionItem DOKUWikiQuestLogin { get; set; }
public OptionItem DOKUWikiQuestPassword { get; set; }
public OptionItem DOKUWikiDefaultGroup { get; set; }
public OptionItem LOGRemoteServerAddress { get; set; }
public OptionItem LOGRemoteServerPort { get; set; }
public OptionItem LOGRemoteServerEnable { get; set; }
public OptionItem LOGDuplicateEntry { get; set; }
public OptionItem LOGDebug { get; set; }
public OptionItem PrnQuiet { get; set; }
public OptionItem IsRun { get; set; }
public List<OptionItem> IPFLocation { get; set; }
public List<OptionItem> IPFLocationEnable { get; set; }
public List<OptionItem> IPFType { get; set; }
public List<OptionItem> IPFIsIpBlackList { get; set; }
public List<OptionItem> IPFIsGeoAsnBlackList { get; set; }
public List<OptionItem> IPFIsGeoCountryBlackList { get; set; }
public List<OptionItem> IPFIpList { get; set; }
public List<OptionItem> IPFGeoListASN { get; set; }
public List<OptionItem> IPFGeoListCountry { get; set; }
}
public class OptionItem
{
public OptionItem()
{
}
public OptionItem(bool b, string t1, string t2)
{
this.bval = b;
this.tag1 = t1;
this.tag1 = t2;
}
public OptionItem(Int32 n, string t1, string t2)
{
this.num = n;
this.tag1 = t1;
this.tag1 = t2;
}
public OptionItem(string v, string t1, string t2)
{
this.value = v;
this.tag1 = t1;
this.tag1 = t2;
}
public bool bval = false;
public Int32 num = 0;
public string value = null;
public string tag1 = null;
public string tag2 = null;
public System.Collections.Specialized.StringCollection collection = null;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,291
|
Urban Decay ― американский косметический бренд со штаб-квартирой в Ньюпорт-Бич, Калифорния, является дочерней компанией французской косметической компании L'Oréal.
Современные продукты включают средства для губ, глаз, кожи лица и тела. Бренд наиболее известен своей коллекцией Naked, которая включает в себя двенадцать различных палитр теней для век. Его целевой рынок ― женщины, хотя он не ограничивается этим ассортиментом. Продукция продается в крупных универмагах в США, таких как Macy's, Sephora, Ulta, Nordstrom, и на официальном веб-сайте, а также в ряде других стран, таких как Мексика и Германия.
История
Розовые, красные и бежевые тона доминировали в палитре индустрии красоты до середины 1990-х годов. В 1995 году Сэнди Лернер, соучредитель Cisco, и Патриция Холмс находились в особняке Лернера за пределами Лондона, когда Холмс смешала малиновый и черный цвета, чтобы сформировать новый цвет. Затем они решили создать косметическую компанию, которую назвали Urban Decay. Запущенная в январе 1996 года, она предлагала линейку из 10 помад и 12 лаков для ногтей. Их цветовая палитра была вдохновлена городским пейзажем, поэтому оттенки помад назывались: Смог, Ржавчина, Нефтяное пятно, Кислотный дождь и так далее.
В 2000 году компания LVMH купила бренд Urban Decay . В 2002 году группа Falic выкупила Urban Decay. В 2009 году владельцем бренда стала компания Castanea Partners. 26 ноября 2012 года компания L'Oréal объявила о покупке Urban Decay. L'Oréal приобрела компанию в 2012 году. Она заплатила, по оценкам, 350 миллионов долларов за бренд.
Весной 2015 года Urban Decay расширила свое присутствие в социальных сетях с помощью сайта Tumblr, The Violet Underground. В нем представлены совместные работы с молодыми дизайнерами, такими как Барон фон Фэнси.
Амбассадоры
Лицом бренда были такие знаменитости, как Руби Роуз и Николь Ричи. В июне 2019 года Urban Decay объявил о своем новом девизе Pretty Different и назвал своими послами Эзру Миллера, Лиззо, Джоуи Кинг, Кароль Джи и CL . Год спустя они объявили еще о трех послах для продвижения своей палитры Naked Ultraviolet ― Нормани, G.E.M и Камилу Мендес. В январе 2021 года южнокорейская группа Monsta X была объявлена глобальным послом бренда.
Примечания
Ссылки
Производители косметики и парфюмерии США
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,470
|
title: DeploymentRequiresServiceAssociated
layout: analysis-message
owner: istio/wg-user-experience-maintainers
test: no
---
This message occurs when pods are not associated with any services.
A pod must belong to at least one Kubernetes service even if the pod does NOT expose any port.
See the [Istio Requirements](../../../../ops/deployment/requirements).
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,505
|
package org.hivesoft.confluence.utils;
import com.atlassian.confluence.core.ContentEntityObject;
import com.atlassian.user.User;
import java.util.List;
public interface PermissionEvaluator {
User getRemoteUser();
String getRemoteUsername();
User getUserByName(String userName);
boolean canAttachFile(ContentEntityObject contentEntityObject);
boolean canCreatePage(ContentEntityObject contentEntityObject);
boolean isPermissionListEmptyOrContainsGivenUser(List<String> listOfUsersOrGroups, User user);
boolean canSeeVoters(String visibleVoters, boolean canSeeResults);
List<User> getActiveUsersForGroupOrUser(String userOrGroupName);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,465
|
Professional Color Booster and Color Revival Mask by RichIt for Silky, Smooth, Moisture-Rich Hair and Color Refresh between Color Treatments!
Fight fading and give your color a pick-me-up with the salon-quality, professional Color Booster Mask by RichIt. This nourishing color conditioner allows you to achieve longer-lasting, better-looking color treatment with specially formulated semi-permanent color pigments for a variety of hair colors. Choose the mask that best matches your color and treat your tresses to a nourishing color refresh!
Of course, the color mask isn't just for color-treated hair. Everyone can benefit from the spa-quality Argan and Coconut oil enriched mask formula. The semi-permanent hues give new, subtle shades to natural colors and the nourishing formula prevents moisture loss for silky, smooth tresses. Enhance and refresh your color with the RI Color Booster Mask.
mei-ling montanez said: What's red?
This is the first color base conditioner I've tried before. I have pink hair and I wanted to see what happens if I use the sliver blonde one. Let's say my bright pink is now a dusty pretty pink. Will continue to buy for my clients with blonde hair.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,955
|
— город в Японии, находящийся в префектуре Окаяма.
Географическое положение
Город расположен на острове Хонсю в префектуре Окаяма региона Тюгоку. С ним граничат города Окаяма, Сетоути, Акаива, Мимасака, Ако и посёлки Ваке, Камигори, Саё.
Население
Население города составляет , а плотность — чел./км².
Символика
Деревом города считается Pistacia chinensis, цветком — Rhododendron indicum.
Примечания
Ссылки
Официальный сайт
Города префектуры Окаяма
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,284
|
\section{Introduction}
Among the great successes in describing interactions between colloidal particles in suspension is the Derjaguin, Landau, Verwey and Overbeek (DLVO) model ~\cite{verwey1948}. Apart from short ranged interactions, this model treats charged colloids as Yukawa particles. Thus colloids may be interpreted within a framework which encompasses a great many other systems from the mesons which Yukawa was originally interested in ~\cite{yukawanobel} to dusty, or complex plasmas ~\cite{ivlev,loewen2011}. Indeed, as shown in Fig. \ref{figYukawa}, charged colloids and complex plasmas both exhibit the same phase behaviour : under the right conditions both systems exhibit phases characteristic of the Yukawa system, that is to say fluid (F) along with body-centred and face-centred cubic crystals (BCC and FCC respectively).
In the case of colloidal dispersions, there are more components than just the colloids : the particles are immersed in a solvent, the electrostatic charge they carry is balanced by counter-ions (in colloidal dispersions, one typically assumes charge neutrality) and salt ions, not to mention the liquid solvent in which the system is immersed ~\cite{ivlev}. One can proceed to an \emph{effective one-component} system where only the colloids are considered by integrating out the degrees of freedom of the smaller species ~\cite{likos2001}. Here the liquid solvent has no impact on the equilibrium phase behaviour (it acts to damp the dynamics of the colloids, leaving the system as non-inertial on most reasonable timescales) ~\cite{ivlev}. However the effects of the ions do need to be integrated out, and this can be done using the approach pioneered by Derjagiun, Landau, Verwey and Overbeek ~\cite{verwey1948}. In addition to the effects of the electrostatics, DLVO theory also includes other interactions between the colloids, such as van der Waals forces. In our systems, these are not important, because the colloids are refractive index matched to the solvent which reduces the effects of the van der Waals forces to a fraction to the thermal energy $k_BT$. Any residual van der Waals effects are suppressed by a polymer layer at the surface of the particle. The polymer layer is much thinner ($\lesssim 10 \mu$m) than the particle size ~\cite{bryant2002} and thus the short ranged interactions may be treated as a hard core ~\cite{royall2013myth}.
In colloidal dispersions, the particle concentration is often high enough that steric interactions due to the finite particle size can come into play ~\cite{hynninen2003}, and it is appropriate to include the hard core $u_\mathrm{hc}(r)$. The hard core Yukawa interaction then reads
\begin{eqnarray}
u(r)&=&u_\mathrm{hc}(r)+ u_{y}(r),
\label{eqU}\\
u_{y}(r)& =& \epsilon_{y}\frac{\exp [-\kappa ( r-\sigma ) ] }{r/\sigma}.
\label{eqYukawa}
\end{eqnarray}
Here, the potential at contact (when the colloids touch) is given by
\begin{equation}
\beta\epsilon_{y}=\frac{Z^{2}}{(1+\kappa\sigma/2)^{2}}\frac{\lambda_{B}}{\sigma},
\label{eqEpsilonYukawa}
\end{equation}
\noindent where $Z$ is the colloid charge, and the inverse Debye screening length is given by $\kappa$=$\sqrt{4\pi\lambda_{B}\rho_{ion}}$, where
$\rho_{ion}$ is the number density of monovalent ions. The Bjerrum length
\begin{equation}
\lambda_{B}=\beta e^{2}/(4\pi\epsilon_{0}\epsilon_{r}),
\label{eqBjerrum}
\end{equation}
\noindent is the distance at which the interaction energy between two electronic charges is $k_B T$, where $e$ is the electronic charge, $\epsilon_{0}$ the permittivity of free space, and $\epsilon_{r}$ the dielectric constant.
\begin{figure*}[t]
\includegraphics[width=\linewidth]{figYukawa}
\caption{Phase diagram of Yukawa systems. The fluid-solid phase boundary (solid line) is the analytic approximation [Eq. \ref{eqGamma}], the dashed line at large $\kappa_p$ denotes that its validity is limited by $\kappa_p<10$. The approximate position of the BCC-FCC crystal boundary is indicated by the dotted line. Narrow coexistence regions are not shown here. Symbols represent various crystallization/melting experiments in colloidal dispersions ~\cite{royall2003,royall2006}(pink) and complex plasmas ~\cite{khrapak2011}(blue). Squares and bullets indicate, respectively, BCC and fluid phases (as observed), arrows show the direction in which parameters varied during the experiments. Triangles are FCC crystals. Characteristic snapshots of observed fluid and crystalline phases are also shown ~\cite{ivlev}.}
\label{figYukawa}
\end{figure*}
In their investigations of hard core Yukawa phase diagrams, Hynninen and Dijkstra ~\cite{hynninen2003} showed that the effect of the hard core was small in the case that freezing occurred at a colloid volume fraction $\phi \lesssim 0.3$, i.e. that the colloids are typically far enough apart from one another that short-range interactions are irrelevant. Example interaction parameters would be $(\beta \epsilon_y=20,\kappa\sigma=1.0)$. At lower concentrations, therefore, the system can be treated as a point Yukawa system, as can also be the case with complex plasmas ~\cite{ivlev}. A point Yukawa treatment where the hard core is neglected enables arbitrary Yukawa parameters to be represented in a 2d plot ~\cite{robbins1987,hamaguchi1997}. Here we shall use the Yukawa (screened Coulomb) coupling parameter $\Gamma_s$ and scaled inverse Debye length $\kappa_p$ which are defined below \cite{ivlev}.
In colloidal dispersions, the dominance of such long-ranged interactions means that there is no specific requirement that strongly attractive but short ranged van der Waals interactions be suppressed and thus we are not limited to sterically stabilised, refractive index matched systems. Water and ethanol are popular solvents and silica and polystyrene are popular materials for the colloids. As an aside, this indicates that the melamine particles often used in complex plasma experiments are in principle no different to particles used in colloidal experiments, moreover their size ranges overlap ~\cite{ivlev}.
While the DLVO theory is only valid in the range that linearised Poisson-Boltzmann theory holds (weak electrostatic interactions), higher charging can also be treated with a Yukawa interaction by using a \emph{renormalised} or \emph{effective} charge that is smaller than the physical charge on the particles ~\cite{alexander1984,trizac2002}. Thus, providing the effective colloid charge can be found, a Yukawa behaviour is recovered. Once the effective charge $Z_\mathrm{eff}$ and Debye length $\kappa^{-1}$ are determined, a number of studies have been made in aqueous based systems finding excellent agreement with the Yukawa model \cite{palberg1999,monovoukis1989,sirota1989,yoshizawa2012}.
Behaviour inconsistent with the DLVO model has also been seen. The DLVO model recasts interactions between multiple components into a one-component Yukawa treatment. Even after accounting for charge renormalisation, anomalous behaviour has been observed. In particular the observation of condensation like behaviour of the colloids into a ``colloidal liquid'' and ``colloidal gas'', with voids appearing in the system, was attributed to ``like charge attraction'' ~\cite{ito1994}. A variety of explanations have been put forward to explain this phenomenon, many arriving at the conclusion that the effective interaction between the colloids was attractive, a surprise for particles with like charge ~\cite{kepler1994,larsen1997,han2003}. However direct measurement with optical tweezers found no evidence of attraction, rather that some earlier measurements (though not the original observation by Ito \emph{et al.} ~\cite{ito1994}) may have been influenced by artefacts ~\cite{baumgartl2005,baumgartl2006}. Among the few theoretical explanations to have withstood the test of time is that of van Roij and coworkers who considered that the entropy of the salt ions might drive phase separation to a colloid-rich and colloid-poor phase ~\cite{vanroij1997,vanroij1998}. Crucially, in van Roij's treatment, at the two-body colloid-colloid level, a repulsion between the particles is maintained : the (repulsive) Yukawa form in Eq. \ref{eqYukawa} is satisfied. The entropic terms driving the phase separation do not feature in the (one-component) DLVO treatment because degrees of freedom of the small ions are \emph{integrated out} and captured in a one-body term. Quantitative agreement with van Roij's predictions was recently found in a phase separating binary system whose behaviour would similarly not be expected in a pure Yukawa picture ~\cite{yoshizawa2012}. Other deviations from DLVO behaviour include ion-colloid decoupling leading to a macroscopic electric field which results in extended sedimentation profiles ~\cite{piazza1993,rasa2004,royall2005s}.
Of particular interest here is a mechanism originally put forward to account for the condensation effects observed by Ito \emph{et al.} ~\cite{ito1994}. While the two-body term between the colloids (Eq. \ref{eqYukawa}) is repulsive, the three body interactions induced between three colloids are \emph{attractive} \cite{russ2002}. Such deviations from two-body behaviour have been observed in experiment in the form of a perturbation to the fluid structure \cite{brunner2002}, and indications of non-spherical interaction potentials in crystals at higher density ~\cite{reinke2007}. Furthermore simulation work indicates that including the three-body terms leads to an increase in the fluid region of the phase diagram at the expense of the BCC phase \cite{hynninen2003,hynninen2004}.
Now the self-dissociation of water means the ionic strength is $\gtrsim 10^{-7}$ Mol and additional contributions such as counter-ions mean that a Debye screening length of $\kappa^{-1} \lesssim 300$ nm is typical in experiment. Since particle resolved studies require colloid sizes of at least a micron, with aqueous solvents it is difficult to reach conditions where the Debye length is comparable to (or greater than) the particle size appropriate for the regime where the system behaves purely as a Yukawa system without significant contribution from the hard core $u_\mathrm{hc}$ and other e.g. van der Waals short range forces. However in solvents such as cycle hexyl bromide of interest here the ionic strength can reach $10^{-12}$ Mol so the Debye length can be sufficient that micron-sized particles are far apart and (point) Yukawa behaviour is found ~\cite{royall2003,royall2006}. In these systems, colloidal crystals of exceptionally low volume fraction, where the inter particle spacing can run to tens of microns have been observed ~\cite{yethiraj2003,royall2003,leunissenThesis}. These systems have also been observed to become dynamically arrested, and to fail to crystallise and thus form a glass at low colloid density ~\cite{klix2010}, as indeed have aqueous systems ~\cite{sirota1989}.
Here we consider the Yukawa parameters associated with such ``low-density crystals''. Our purpose is to make a quantitative comparison between the low-density crystals and the predictions of Yukawa theory in the form of the equilibrium phase diagram ~\cite{robbins1987,hamaguchi1997}. In particular we map our experimental data to generic Yukawa parameters $(\Gamma_s,\kappa_p)$ and compare the state observed in experiment to the theoretical prediction. We consider the crystal polymorph observed with that predicted. Our analysis indicates that while the Yukawa phase diagram predicts face-centred cubic crystals for some parameters, in our experiments we find body-centred cubic crystals only. This contribution is organised as follows : in Section \ref{sectionMapping} we discuss our mapping procedure and assumptions, in Section \ref{sectionExperimental} we outline our experimental technique. We present our results in Section \ref{sectionResults} and discuss these in section \ref{sectionDiscussion}.
\section{Mapping to Yukawa theory}
\label{sectionMapping}
The comparison between charged colloids and complex plasmas is illustrated in Fig. \ref{figYukawa}. The main panel in Fig. \ref{figYukawa} is the Yukawa phase diagram in the $\Gamma_s$, $\kappa_p$ plane. Here we show literature data for complex plasmas ~\cite{khrapak2011} and colloids ~\cite{royall2003,royall2006}. We emphasise that the colloidal particles illustrated in Fig. \ref{figYukawa} are two microns in size. They are thus effectively identical to particles used at the smaller end of complex plasma experiments \cite{ivlev}. The key difference is thus the immersing medium, a liquid solvent rather than a plasma.
The freezing line in Yukawa systems is given with reasonable accuracy for $\kappa_p<10$ \cite{ivlev} by
\begin{equation}
\Gamma_s(\kappa_p)=\frac{106}{1+\kappa_p + \frac{1}{2}\kappa_p^2}
\label{eqGamma}
\end{equation}
\noindent where the screened coupling parameter $\Gamma^{(s)}$ is the Yukawa interaction evaluated at the mean inter particle separation $\rho^{-\frac{1}{3}}$, $u_y(\rho^{-\frac{1}{3}})$ where $\rho$ is the particle number density and $\kappa_p$ is a scaled inverse Debye length, given by $\kappa_{p}$=$\kappa\sigma_{c}\rho^{-\frac{1}{3}}$ \cite{ivlev,robbins1987,hamaguchi1997}.
Here we assume the colloids take their \emph{saturated effective} charge $Z_\mathrm{sat}^\mathrm{eff}$. That is to say, the maximum charge given under charge renormalisation. An approximate value is given by:
\begin{equation}
Z_\mathrm{eff}^\mathrm{sat}=\frac{(2+\kappa\sigma)\sigma}{\lambda_{B}}
\label{eqZ}
\end{equation}
\noindent which represents the effective colloid charge ~\cite{ivlev}. Thus the number density of ions can be estimated as the effective charge number per colloidal particle due to the counter ions and that from salt and background ions $\rho_\mathrm{salt}$,
\begin{equation}
\rho_\mathrm{ion}=Z_\mathrm{eff}^\mathrm{sat}\rho + \rho_\mathrm{salt}.
\label{eqRhoIon}
\end{equation}
\noindent Now although no salt is added, some background ions are present, due for example to solvent self-dissociation. Here we treat this contribution $\rho_\mathrm{salt}$ as a free parameter and determine the scaled screening parameter as outlined above. We shall see below that assuming agreement with the Yukawa freezing line enables the value of $\rho_\mathrm{salt}=10^{-10}$ $m^{-3}$ which corresponds to 8.3 nMol. Finally, the evaluation of the Yukawa interaction at the mean interparticle separation can be obtained from Eq. \ref{eqYukawa}.
\section{Experimental}
\label{sectionExperimental}
We used poly(methyl methacrylate) (PMMA) colloids sterically stabilized with polyhydroxyl steric acid \cite{bosma2002,ohtsuka2008}. The colloids were labelled with fluorescent rhodamine dye to enable fluorescent imaging and had a diameter of 2000 nm and polydispersity of 5\% as determined with static light scattering. Different volume fractions of the particles were suspended in cyclohexyl bromide (CHB) whose dielectric constant $\epsilon_{r}$ = 7.92 and whose refractive index is closely matched to that of the colloids enabling bulk 3d imaging. A rectangular glass capillary with inner dimensions of 0.10 x 1.00 mm (Vitrocom) was filled with the suspensions and sealed on each end with epoxy glue. The samples were studied with confocal laser scanning microscopy, CLSM (Leica SP5 fitted with a resonant scanner), with 543 nm excitation using a NA 63x oil immersion objective. For qualitative imaging, 2D data set was recorded (512 x 512 pixels), whereas for particle tracking, a full scan of the capillary in the z direction was obtained, providing 3D data sets, where care was taken to ensure the pixel size was the same in the three planes.
\section{Results}
\label{sectionResults}
\subsection{Phase behaviour and comparison with Yukawa theory}
\begin{figure*}[t]
\includegraphics[width=0.68\linewidth]{figYukawaIoatzin}
\caption{Phase diagram of Yukawa system studied in this work in comparison with previous studies ~\cite{ivlev}.
Snapshots in (a) (b) (c) correspond to volume fraction $\phi$ of 0.0055, 0.02 and 0.23 respectively and the state points in $(\Gamma_s,\kappa_p)$ representation indicated in the main panel. Lines are as in Fig. \ref{figYukawa} : thick line is the freezing line [Eq. \ref{eqGamma}] and the thin dashed line approximately describes the BCC-FCC transition. Arrows denote increasing volume fraction. Scale bars = 25 $\mu$m.
}
\label{figYukawaIoatzin}
\end{figure*}
We show our comparison with Yukawa theory in Fig. \ref{figYukawaIoatzin} for the low-density crystals formed in this work. Compared to previous work where the experiments were mapped to Yukawa parameters ($\Gamma_s,\kappa_p$), we access a new region of the phase diagram. We fit the ionic strength such that our lowest concentrations $\phi$= 0.0055 and 0.01, identified as fluids are consistent with the Yukawa prediction. We note that the path the state points take in the ($\Gamma_s,\kappa_p$) space is not a straight line or even a smooth curve, nor even montonic. This is due to competing effects. Regadring the ionic strength, the added salt $\rho_\mathrm{salt}$ in Eq. \ref{eqRhoIon} is comparable to or smaller than the counter ion contribution $Z_\mathrm{eff}^\mathrm{sat}\rho$ which means that the Debye length $\kappa^{-1}$ drops as the volume fraction is increased so the screening is stronger. However the increase in $\phi$ means the range at which the screened coupling parameter $\Gamma_s$ is evaluated, $\rho^{-\frac{1}{3}}$, drops because the particles are (on average) closer together. The former acts to reduce $\Gamma_s$, the latter to increase it. We note that the highest three densities ($\kappa_p$) are predicted to the FCC. We describe how we determined the crystal structure in the following section.
We are now able to quantify the Debye screening length $\kappa^{-1}= 1.9 \mu$m and contact potential $\beta \epsilon_y=1110$ around freezing. This corresponds to a effective colloid $Z_\mathrm{eff}^\mathrm{sat}=850$. In particular the Debye length is bigger, compared to aqueous systems and indeed to previous particle-resolved studies where it was around 0.200-1 $\mu$m ~\cite{royall2003,royall2006}.
\subsection{Identification of crystal structures at low packing fractions}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.8\textwidth]{figBcc}
\end{center}
\caption{
(a,d) Three-dimensional rendering of the dense BCC crystal seen from the bottom of the sample for (a) volume fraction $\phi=0.015$ [corresponding to point b in Fig. \ref{figYukawaIoatzin}] and (d) volume fraction $\phi=0.16$ [point c in Fig. \ref{figYukawaIoatzin}], with arbitrarily scaled particles radii for better visualisation. In panel (d), a close-up of the ordered surface is highlighted. (b-c, e-f) Corresponding local bond order parameter diagrams for the bcc phase of the two considered volume fractions: in panel (b) the distribution $P(\bar{q}_4,\bar{q}_6)$ is centred at moderate values corresponding to a solid region whose nature is determined via the $P(\bar{q}_4,\bar{w}_4)$ distribution to be of BCC nature. The same procedure is applied in panels (e,f) for the higher volume fraction, showing a higher degree of order, particularly in the $P(\bar{q}_4,\bar{q}_6)$ distribution.}
\label{figBcc}
\end{figure*}
For the analysis of the crystal structures particle tracking of the 3d data sets based on \cite{dunleavy2015} was performed. Image manipulation techniques are used in order to enhance the contrast and remove the noise from the experimental datasets. The positions of the particles are identified through a maximization of the overlap between seeded local gaussian kernels and the intensity profile of the particles. For this algorithm, an estimation of the size of one particle is needed as input parameter, but no initial assumptions on the volume fraction are required. The resulting coordinates are obtained with a resolution of 1 pixel (200 nm) in each spatial direction.
The coordinates extracted via particle tracking are used in order to perform an analysis of the crystalline phases formed before the sedimentation of the sample. For this purpose, we employed the Steinhardt local rotational invariants, also known as bond-orientational order parameters. These discriminate between different possible crystal structures on the basis of spherical harmonics and have also been used for the detection of order in complex plasmas \cite{steinhardt1983,khrapak2011}. In particular, we consider the locally averaged order parameters $\bar{q}_4, \bar{w}_4,\bar{q}_6$ for square and hexagonal order, where the local average allows to take into account the effect of second nearest neighbours and to more sharply distinguish between different arrangements (see \cite{lechner2008} for a detailed discussion on the technique).
For each particle $i$, we perform a parameter-free detection of the nearest neighbours via a Voronoi tessellation of the sample volume. This provides a list of the $N_b (i)$ nearest neighbours over which the local order parameters are calculated:
\begin{equation}
q_{lm}(i)=\frac{1}{N_{b}(i)}\sum_{j=1}^{N_{b}(i)}Y_{lm}(\bm{r}_{ij}) \;,
\end{equation}
where $Y_{lm}(\bm{r}_{ij})$ are the spherical harmonics. Summing over the list of $\tilde{N}_b(i)$ particles identified by the neighbours and the particle $i$ itself one obtains the locally averaged order parameters
\begin{equation}
\bar{q}_{lm}(i)=\frac{1}{\tilde{N}_{b}(i)}\sum_{k=0}^{\tilde{N}_{b}(i)}q_{lm}(k),
\end{equation}
and summing over all the harmonics we finally get
\begin{equation}
\bar{q}_{l}(i)=\sqrt{\frac{4\pi}{2l+1}\sum_{m=-l}^{l}|\bar{q}_{lm}(i)|^{2}} \;.
\end{equation}
\begin{equation}
\bar{w}_l(i)=\dfrac{\sum\limits_{m_1 +m_2 + m_3=0} \left(\begin{array}{ccc}l & l & l \\m_1 & m_2 & m_3\end{array}\right) \bar{q}_{lm_1}(i)\bar{q}_{lm_2}(i)\bar{q}_{lm_3}(i) }{\left(\sum_{m=-l}^{l}|\bar{q}_{lm}(i)|^{2}\right)^{3/2}}
\end{equation}
where the term in brackets is the Wigner symbol.
We focus our analysis on a bulk region discarding top, bottom and lateral edges for a thickness of about 2.5 $\mu$m. In the lowest density samples, we can discriminate between a fluid and a solid phase, where the fluid presents the characteristics of a layered liquid along the vertical $z$ dimension. In Fig.~\ref{figBcc} we show the results of the local order analysis for a low density system (top row, volume fraction $\phi=0.015$) and a dense sample (bottom row, $\phi=0.16$). The very limited range of the $\bar{q}_4$ order parameter allows us to discard the hypothesis of a face centred cubic crystal. In order to asses the nature of the solid phase, we use an additional order parameter $\bar{w}_4$, particularly suitable for the distinction of hexagonal close packed structures (HCP) and FCC from BCC: in Fig.~\ref{figBcc}(b) we show that no peak is detected in the HCP or FCC region, leading to the identification of the solid phase as a BCC phase. We see that the state point at $\phi=0.16$ [(c) in Fig. \ref{figYukawaIoatzin}] is identified as BCC while the theory predicts that it should be FCC. Indeed we analysed all crystalline systems and found only BCC crystals. We speculate on the causes of this discrepancy below.
\section{Discussion}
\label{sectionDiscussion}
Our work extends the range of Yukawa parameters ($\Gamma_s,\kappa_p$) for particle-resolved studies of colloidal systems. We have shown that the formation of low-density crystals in these systems \cite{yethiraj2003,royall2003,leunissenThesis} is compatible with Yukawa theory, and that Debye screening lengths can run to many microns. The screened coupling parameter is found to be $\Gamma_s=67$, the largest value obtained previously was $\Gamma_s=16$. We further use our parameterisation to estimate the lowest freezing density attainable in the case of no salt ($\rho_\mathrm{salt}\rightarrow0$). This turns out to be around $\phi= 0.0004$ for our system and a Debye length of some $14$ $\mu$m. In principle, therefore it might be possible to produce crystals of very much lower density still.
We now consider possible reasons for the discrepancy of the polymorph we identified. One possibility might be three-body and higher order effects, which could be significant for such long-range interaction potentials. However as we have noted above simulations predict the opposite behaviour, of a suppression of the BCC phase relative to the two-body Yukawa system and the formation of FCC instead ~\cite{hynninen2003,hynninen2004}. Our primary speculation on the other hand is that the system has not yet fully equilibrated. Indeed in some samples, the region close to the wall appearing more crystalline, indicating a possible crystallisation front beginning at the wall of the same cell as has a been observed in the hard sphere suspensions ~\cite{sandomirski2011}. While homogenous crystallisation for our parameters has yet to be studied in detail, we note that in Yukawa systems, although FCC is the favoured polymorph, BCC can form first in the Ostwald rule of stages and indeed polymorph selection can even proceed through the metastable hexagonal close-packed polymorph ~\cite{desgranges2007}. We believe that the same could occur here, although we emphasise that the effect of the wall will be profound in the crystallisation mechanism and thus the homogenous crystallisation studies of Desgranges and Delhomme ~\cite{desgranges2007} may not hold for our case.
However we note that our system is confined in a capillary of height 100 $\mu$m. Confinement has been considered in comparable systems up to four layers ~\cite{oguz2012}. There some preference was found for square symmetry which might lead to BCC being favoured in our larger system. Thus it would be most interesting in the future to investigate the crystallisation kinetics to see if the system indeed showed signs of approaching its FCC bulk equilibrium state. However we emphasise that these systems are stable for around two days ~\cite{royall2003} which does place some limits on the experimental time window. While no change is observed on the timescale of two days (here limit our experiments to four hours), after a week the system was found to behave as of the contact potential $\beta \epsilon_y$ has dropped along with the Debye length. This behaviour is inferred from the shift in the freezing boundary to higher colloid volume fraction for older samples ~\cite{royall2003,royall2006}. Although desirable, an in-depth study of these ageing phenomena has yet to be carried out. Possible sources of the change in the system over time include ion dissolution from the glass capillary in which the system is confined.
\subsection*{Acknowledgements}
The authors would like to thank Alfons van Blaaderen, Marjolein Dijkstra, Bob Evans, Antti-Pekka Hynninen, Alexei Ivlev, Mirjam Leunissen and Hartmut L\"{o}wen for many helpful discussions. Andrew Dunleavy is particularly thanked for the generous provision of the particle tracking code. CPR would like to acknowledge the Royal Society for financial support. FT and CPR acknowledge the European Research Council under the FP7 / ERC Grant agreement n$^\circ$ 617266 ``NANOPRS''. IRdA was supported by a doctoral scholarship from CONACyT. AS acknowledges financial support by the Deutsche Forschungsgemeinschaft (grant No. VI237/4-3)
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{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,892
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from subprocess import call
import platform
def say(what):
command = {
"Darwin": "say",
"Linux": "espeak"
}[platform.system()]
call([command, what])
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,727
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Ardisia hornitoana är en viveväxtart som beskrevs av C.L. Lundell. Ardisia hornitoana ingår i släktet Ardisia och familjen viveväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Viveväxter
hornitoana
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,963
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Q: Create a widget to embed into QMainWindow I have this task that I couldn't solve yet. Working with PyQt and Qt Creator.
I want to embed a custom created widget created in QT Creator into another QMainWindow.
1) Steps I do:
Create a Widget file in QT creator:
2) Save it as *.ui and apply this line to convert it to a *.py file:
pyuic5 gen_settings.ui -o gen_settings.py
3) Open it and see that it starts with
from PyQt5 import QtCore, QtGui, QtWidgets
class Ui_gen_settings(object):
def setupUi(self, gen_settings):
gen_settings.setObjectName("gen_settings")
4) Which results in function call of course:
TypeError: arguments did not match any overloaded call:
addWidget(self, QWidget): argument 1 has unexpected type 'function'
when I call it in another QMainWindow file:
class Ui_MainWindow(object):
def setupUi(self, MainWindow, My_Custom_widget):
MainWindow.setObjectName("MainWindow")
self.gridLayout.addWidget(My_Custom_widget, 1, 4, 1, 1)
Any ideas how to solve it?
A: First of all I recommend you read the PyQt docs referring to Qt Designer.
Going to the problem, Qt Designer does not provide a widget but a class that serves as an interface to a widget, and that can be seen in his statement:
class Ui_gen_settings(object):
# ...
The class inherits from object and not from QWidget, QDialog, QMainWindow, etc.
In the docs that indicate initially it is recommended to create a widget and use the interface provided by Qt Designer. For this it is correct to use pyuic but I will change the gen_settings.py to gen_settings_ui.py so that the change is understood.
pyuic5 gen_settings.ui -o gen_settings_ui.py
So now we create a file called gen_settings.py that contains the widget and use the interface.
gen_settings.py
from gen_settings_ui import Ui_gen_settings
from PyQt5 import QtWidgets
class Gen_Settings(QtWidgets.QWidget, Ui_gen_settings):
def __init__(self, parent=None):
super(Gen_Settings, self).__init__(parent)
self.setupUi(self)
Then when you create the .ui corresponding to Ui_MainWindow you add a QWidget that is an empty container.
In the image, the Widget container is the one in the upper left, and now we will replace it with Gen_Settings, so we must promote it using the following procedure:
*
*Right click on the widget container and select the Promote To ... option.
*The following Dialog will appear and fill in the fields as shown in the image (I'm assuming that gen_settings_ui.py and gen_settings.py are in the same folder as the current .ui)
*You press the Add button and then the Promote button.
Then you convert the .ui to .py with pyuic and you will get the following:
from PyQt5 import QtCore, QtGui, QtWidgets
class Ui_MainWindow(object):
def setupUi(self, MainWindow):
MainWindow.setObjectName("MainWindow")
# ...
self.widget = Gen_Settings(self.centralwidget)
self.widget.setObjectName("widget")
self.gridLayout.addWidget(self.widget, 0, 0, 1, 1)
# ...
from gen_settings import Gen_Settings
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,805
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import { FC, lazy, LazyExoticComponent } from 'react';
import { addAction } from '../../../room/lib/Toolbox';
addAction('team-channels', {
groups: ['team'],
id: 'team-channels',
anonymous: true,
full: true,
title: 'Team_Channels',
icon: 'hash',
template: lazy(() => import('./index')) as LazyExoticComponent<FC>,
order: 2,
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,884
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Papyrus 90 (in the Gregory-Aland numbering), designated by 𝔓90, is a small fragment from the Gospel of John 18:36-19:7 dating palaeographically to the late 2nd century.
The Greek text of this codex is a representative of the Alexandrian text-type. Aland placed it in Category I (because of its date).
Philip W. Comfort says "𝔓90 has [close] textual affinity with 𝔓66 ... [and] some affinity with א (Aleph)."
It is currently housed at the Sackler Library (Papyrology Rooms, P. Oxy. 3523) in Oxford.
Greek text
The papyrus is written on both sides. The characters that are in bold style are the ones that can be seen in 𝔓90.
Gospel of John 18:36-19:1 (recto)
Gospel of John 19:1-7 (verso)
See also
List of New Testament papyri
Oxyrhynchus Papyri
References
Further reading
T. C. Skeat, Oxyrhynchus Papyri L (London: 1983), pp. 3–8.
Robinson, James M,Fragments from the Cartonnage of P75, Harvard Theological Review, 101:2, Apr 2008, p. 247.
Philip W. Comfort, Early Manuscripts & Modern Translations of the New Testament, pp. 68–69
Images
Leaf from 𝔓90
External links
P90/P.Oxy.L 3523
POxy – Oxyrhynchus Papyri online database
Robert B. Waltz. NT Manuscripts: Papyri, Papyri 𝔓90.
New Testament papyri
2nd-century biblical manuscripts
Early Greek manuscripts of the New Testament
Gospel of John papyri
Barabbas
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 447
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\section{Introduction}
Active galactic nuclei (AGN) are powered by accretion onto a central supermassive blackhole. According to the unified model (e.g. Antonucci 1993), an optically-thick torus of dust and gas surrounds this central engine. Hence orientation of the system plays a central role in determining the observable features of AGN. In Type 1 AGN, the system is oriented face-on, leaving an unobstructed view of the central engine and the broad line region. In contrast, Type 2 AGN are oriented edge-on, blocking the accretion disk and the broad line region. These obscured AGN can be identified by their narrow optical and mid-infrared emission lines which originate in gas photoionized by accretion disk photons. This narrow line region (NLR) extends hundreds of parsecs away from the central source and is therefore not significantly affected by torus obscuration.
The luminosity of emission lines formed in the narrow line region can therefore be used as isotropic indicators of intrinsic AGN luminosity. The flux of the [OIII]$\lambda$5007 line is commonly used as such a diagnostic (e.g. Bassani et al. 1999, Heckman et al. 2005) as it is one of the most prominent lines and suffers little contamination from star formation processes in the host galaxy. This line can be attenuated by dust in the host galaxy, though this effect can be somewhat remedied by applying a reddening correction using the observed Balmer decrement (i.e. the observed ratio of the narrow H$\alpha$/H$\beta$ emission-lines compared to the intrinsic ratio) and the extinction curve for galactic dust \citep{oster}.
Isotropic indicators of AGN luminosity also exist in the infrared band and are much less affected by dust extinction than the optical [OIII] line. Recently, the luminosity of the [OIV] 25.89$\mu$m line has been shown to be a robust proxy of AGN power (e.g. Mel\'{e}ndez et al. 2008, Rigby et al. 2009, Diamond-Stanic et al. 2009): it is formed in the NLR, so it is not affected by torus obscuration, and with an ionization potential of 54.9 eV, starburst activity does not significantly contribute to this line. AGN also emit over 20\% of their bolometric flux in the mid-infrared (MIR), where photons produced by the continuum are absorbed by the torus and re-radiated (e.g. Spinoglio \& Malkan 1989). This MIR emission from the dusty torus can also be a proxy for the intrinsic AGN luminosity. Two potential issues with the MIR continuum are contamination by emission from dust heated by stars (e.g. Buchanan et al. 2006; Deo et al. 2009) and possible anisotropy in the torus emission (e.g. Pier \& Krolik 1992, Buchanan et al. 2006, Elitzur \& Shlosman 2006, Nenkova et al. 2008).
Radio and hard X-ray (E $>$ 10 keV) flux can serve as proxies of the intrinsic AGN continuum. Radio emission has been shown to be similar between type 1 and type 2 AGN (e.g. Giuricin et al. 1990, Diamond-Stanic et al. 2009, Mel\'endez et al. 2010) and correlated with optical luminosity, in particularly the [OIII] flux \citep{Xu}, making high resolution radio observations that isolate emission from the nucleus another diagnostic of intrinsic AGN power. Hard X-rays can pierce through the obscuring torus, provided that the object is not heavily Compton thick (N$_H < 10^{25}$ cm$^{-2}$), and has therefore been used as a method to select AGN samples (e.g. Winter et al. 2008, Treister et al. 2009).
Connections have been observed between star formation activity in the host galaxy and the central AGN (e.g. Kauffmann et al. 2003). Starburst activity can be parametrized by various IR features, such as the equivalent width (EW) of polycyclic aromatic hydrocarbons (PAHs). IR and optical data, such as the ratio of fine structure lines and the shape of the spectral slope, can also reveal the relative amount of AGN to starburst activity. Samples of Seyfert 2 galaxies (the predominant local class of type 2 AGN) are useful in examining the relationships among these star formation vs. AGN-to-starburst indicators as the obscuration of the central engine allows detailed study of the host galaxy.
To address these issues, we will use two complete and homogeneous Sy2 samples selected based on isotropic indicators of AGN luminosity (one [OIII]-selected sample and one MIR-selected sample). We will compare the various diagnostics of intrinsic AGN luminosity and probe for biases resulting from sample selection criteria, starburst contamination, errors introduced from extinction correction, and scatter due to the various physical mechanisms producing these emission features. Such biases are likely minimized in the diagnostic ratios with the smallest dispersion. Where available, we compare these ratios with the Sy1 values to probe for differences due to the inclination of the system, thus testing to which extent these indicators of intrinsic AGN luminosity are truly ``isotropic.'' We will also test the agreement among mid-infrared and optical star formation indicators. Finally, we will examine the possibility that the fraction of the AGN ionizing luminosity that is converted into [OIII] and [OIV] emission is systematically higher in systems in which there is a copious supply of dense gas associated with starburst activity.
\section{The Data}
\subsection{Sample Selection}
The selection of the SDSS [OIII] sample is discussed in detail in LaMassa et al. (2009). In brief, Type 2 AGN were drawn from a parent sample of approximately 480,000 galaxies in SDSS Data Release 4 = DR4 (Adelman-McCarthy et al. 2006). The Type 2 AGN with an observed [OIII] flux greater than 4 $\times$ 10$^{-14}$ erg s$^{-1}$ cm$^{-2}$ were selected, providing a complete sample of 20 Sy2s (hereafter the ``[OIII] sample,'' listed in Table \ref{o3_sample}).
The mid-IR sample comprises the Seyfert 2 galaxies from the original {\it IRAS} 12$\mu$m survey \citep{12m}. This represents a complete sample of Sy2s down to a flux-density limit of 0.3 Jy at 12 $\mu$m, drawn from the {\it IRAS} Point Source Catalog (Version 2), with latitude $|b| > 25^o$ to avoid contamination from the Milky Way. We have dropped NGC 1097 from this original sample as it has since been classified as a Type 1 Seyfert (Storchi-Bergmann et al. 1993), leaving 31 mid-IR selected Sy2s (hereafter the ``12$\mu$m sample,'' listed in Table \ref{12m_sample}).
\subsection{Optical Data}
The optical data for the [OIII]-sample were drawn from SDSS DR4, whereas the optical data for the 12$\mu$m sample were collected from the literature or from SDSS Data Release 7 (DR7) where available. The reddening corrected [OIII] flux (F$_{[OIII],corr}$) was calculated using the observed H$\alpha$/H$\beta$ ratio and an intrinsic ratio of 3.1 with R=3.1 extinction curve for galactic dust \citep{oster}. Tables \ref{o3_opt} and \ref{12m_opt} list the optical emission line fluxes and ratios utilized for this study, as well as the relevant literature sources for the 12$\mu$m sample. The black hole masses (M$_{BH}$) were derived for the [OIII] sample by the SDSS velocity dispersion ($\sigma$) and the M-$\sigma$ relation (M$_{BH}$ = 10$^{8.13}$($\sigma$/200 km s$^{-1}$)$^{4.02}$ M$_{\sun}$, Tremaine et al. 2002). We used literature values for M$_{BH}$ for the 12$\mu$m sources, with most of the masses derived using the M-$\sigma$ relation cited above. For F04385-0828, F05189-2524 and TOLOLO 1238-364, the full width half max (FWHM) of the [OIII] line was used as a proxy for the velocity dispersion (Wang \& Ahang 2007, Greene \& Ho, 2005), and photometry of the host galaxy was used to estimate M$_{BH}$ for F08572+3915 (see Veilleux et al. 2009 for details).
\subsection{Infrared Data}
The infrared data presented here were obtained from the Infrared Spectrograph ({\it IRS}, Houck et al. 2004) on board the {\it Spitzer Space Telescope}. Low-resolution spectra were obtained using the Short-Low (SL, 3.6''$\times$57'' aperture size) and the Long-Low (LL, 10.5''$\times$168'' aperture size) modules and high-resolution spectra were provided by the Short-High (SH, 4.7''$\times$11.3'' aperture size) and the Long-High (LH, 11.1''$\times$22.3'') modules.
The Sy2s in the [OIII]-sample were observed in IRS staring mode in both high and low resolution under Program ID 30773. For the 12-$\mu$m sample, high resolution data existed for all 31 Sy2s but low resolution data were only available for 30 galaxies (IRAS 000198-7926 lacked low resolution data). The high resolution data were obtained in IRS staring mode for 30 of the Sy2s (NGC 5194 had only IRS spectral mapping mode high-resolution data in the archive). Several galaxies had multiple IRS observations: we analyzed these observations independently and compared our results between the two observations. For the low-resolution data, IRS staring mode was used when available with the remainder observed in IRS spectral mapping mode. The Spectral Modeling Analysis and Reduction Tool (SMART, Higdon et al. 2004) was used to reduce the staring mode observations for the 12$\mu$m-sample to be consistent with previous IRS analysis of the 12 $\mu$m sample (e.g. Tommasin et al. 2008, Wu et al. 2009, Buchanan et al. 2006), Spitzer IRS Custom Extraction (SPICE) was used to analyze the staring mode observations for the [OIII]-sample\footnote{Though the IRS staring data for the 12$\mu$m sample was reduced using SMART and the [OIII]-sample with SPICE, the effect of the different reduction software is expected to be negligible on the derived parameters used in this analysis.}, and the Cube Builder for IRS Spectra Maps (CUBISM, Smith et al. 2007a) was utilized to analyze the spectral mapping observations. Table \ref{Spitzer_info} lists the Program ID(s) for each galaxy, the IRS mode used, and spectral extraction area for low-resolution spectral mapping mode data (discussed below).
\subsubsection{High-Resolution IRS Staring Spectra}
We used the basic calibrated data (BCD) pipeline products as the starting point for our analysis. Rogue pixels were removed using the IDL routine IRSCLEAN\_MASK and the rogue pixel mask matching the campaign number of the observation. Dedicated off-source background observations were taken for all sources in the [OIII]-sample and for most of the Sy2s in the 12$\mu$m sample. Multiple background observations, if present, were coadded within each nod and subsequently subtracted from the source image. The background-subtracted source images were then coadded between the two nods. The galaxies in the 12$\mu$m sample that had dedicated background observations and thus were background subtracted are marked with a ``b'' in Table \ref{Spitzer_info}. For sources in the 12$\mu$m sample without dedicated off-source observations, no background subtraction was performed.
Spectra were extracted from these combined observations, using the full aperture extraction mode. The edges of each order were then inspected, removing any data points that fell outside of the calibrated range for that order (IRS Data Handbook, Version 3.1, Table 5.1). The orders were then combined using a 2.5-$\sigma$ clipping mean, resulting in a final cleaned spectrum.
\subsubsection{Low-Resolution IRS Staring Spectra}
The low-resolution data were processed in a similar fashion as the high-resolution data, i.e. we started with the BCD products and removed the rogue pixels with IRSCLEAN\_MASK. However, for these observations a background data set was built for each nod and order by coadding the off-source order and nod position. The background-subtracted nods (following the same procedure as above) were combined for each order and the spectra then extracted using tapered column extraction. The orders were combined using a 2.5-$\sigma$ clipping average. This procedure was executed separately for the SL and LL module. Fourteen of our galaxies had low resolution IRS staring mode data; the rest were acquired in spectral mapping mode. We note that IRAS 00198-7926 did not have archival low resolution spectral data.
\subsubsection{Spectral Mapping Spectra}
The IRS spectral mapping observations were analyzed with CUBISM \citep{CUBISM}, which uses the BCD data to create 3-D spectral cubes (one spectral dimension and 2 spatial dimensions). For the low resolution data, background observations were built from the other order of the on-source module (e.g. SL 2 was used as the background for SL 1, etc.). After the rogue pixels were removed, using the default ``autogen bad pixels'' option in CUBISM, a spectral cube was built. Spectra were then extracted using matched apertures among the detectors and centered on the nucleus. The aperture extraction size for these low-resolution spectral mapping observations are listed in Table \ref{Spitzer_info}. The low resolution spectral mapping data for NGC 1068 was saturated near the nucleus and consequently not included in this analysis.
For the IRS spectral mapping high resolution observation of NGC 5194, no background subtraction was performed. The spectrum was extracted over the full cube, corresponding to a size of 31.5''$\times$45'' in the LH module and 13.8''$\times$27.6'' in the SH module.
\subsection{Radio and Hard X-ray Data}
The radio and hard X-ray data were drawn from the literature; VLA radio data at 8.4 GHz were only available for the 12$\mu$m sample (Thean et al. 2000). In several cases, multiple radio components were analyzed; we included only the flux for the component that was nearest the published center of the galaxy. Twenty-six of the 31 12$\mu$m sources had radio data, with 3 additional sources having upper limits. The hard X-ray fluxes originated from the 22-month Swift-BAT Sky Survey \citep{Tueller} and from BeppoSax \citep{Dadina}. Only 11 out of the 31 12$\mu$m sources and one of the 20 [OIII] sources (IC 0486) have X-ray detections in the 14-195 keV range. We adopted an upper limit of 3.1$\times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$, the flux limit of BAT, for the remainder of the sample when an upper limit was not quoted in either Tueller et al. (2009) or Dadina et al. (2007).
\section{Measurements}
\subsection{IR Emission Line Fluxes}
The high resolution spectra were utilized to measure the emission line fluxes: a Gaussian profile was fit to the emission line feature, with the local continuum, centered on the line's rest-frame wavelength, fit by a zero- or first-order polynomial. The errors were estimated by calculating the root-mean-square (RMS) around this local continuum and measuring the flux values with the continuum shifted by $\pm$ the RMS. In the cases where an emission line was not present, a 3-$\sigma$ upper limit was estimated from the RMS around the best-fit local continuum (where the RMS is assumed to be the 1-$\sigma$ error). In the cases with multiple observations per galaxy, we measured the emission line fluxes independently and averaged the resulting values; these flux measurements agreed within several percent between most of the individual observations, with at most a factor of $\sim$1.5 discrepancy, which was only present in one of the sources.\footnote{We note that NGC 1143/4 has two high resolution archival observations, one centered at RA=43.8004, Dec=-0.1839, and the other at RA=43.7985 and Dec=-0.1807, a distance of $\sim$13.4''. We present the line fluxes from the first region as this corresponds to NGC 1144 which is classified as a Sy2 in SIMBAD. The optical data are for NGC 1144 as well.} Tables \ref{oiii_ir} and \ref{12m_ir} list the emission line flux values for the [OIII] and 12$\mu$m samples, respectively. Comparing our line flux values with Tommasin et al. (2008, 2010), we find that our [OIV] flux values largely agree within a factor of 1.5 (with the exception of NGC 1667 and NGC 7582 where their values are a greater than a factor of 2 higher than ours). However, their [NeII] flux values are generally systematically higher by a factor of $\sim$2.5 - $\sim$4.5, though we do obtain consistent values for NGC 424, NGC 5135 and NGC 5506. Despite these differences in the measured [NeII] line strength, we obtain similar results to Tommasin et al (2010), namely that as the relative contribution of the AGN to the ionization field increases (parameterized by [OIV]/[NeII]), the starburst strength (parameterized by the PAH equivalent width) decreases.
\subsection{IR Continuum Flux and PAHs}
The MIR continuum flux values (F$_{MIR}$) and PAH equivalent widths (EWs) were measured using the low resolution spectra. For the galaxies that had multiple observations, we utilized the observations that had consistent flux values in the overlap region between the SL and LL modules: Program ID 30572 for For NGC 1386, NGC 4388, NGC 5506 and NGC 7130; Program ID 0086 for NGC 5135; and Program IDs 00086 and 30572 for NGC 5347 (for this source, the analysis was done separately for each observation and the results averaged together). The MIR continuum flux was measured at 13.5 $\mu$m (rest-frame), averaged over a 3$\mu$m window; these flux values are listed in Table \ref{oiii_ir} for the [OIII]-selected sample and Table \ref{12m_ir} for the 12$\mu$m sample. This window was chosen as it is free from strong emission line and PAH features.\footnote{Though this range does include the [NeII] 12.81$\mu$m line, in most cases this comprises less than 1\% of the MIR flux, with the exception of NGC 7582 where the [NeII] line is $\sim$1.5\% of the MIR flux.} In LaMassa et al. (2009), we included the flux centered at 30$\mu$m as part of the MIR flux diagnostic. However, emission at this higher wavelength can be strongly affected by star formation processes in the host galaxy (e.g. Deo et al. 2009, Baum et al. 2010), so here we use F$_{13.5\mu m}$ as F$_{MIR}$.
We used PAHFIT \citep{PAHFIT} to measure the PAH EWs, a program which uses a model consisting of several components: a starlight component represented by blackbody emission at T = 5000 K, a thermal dust continuum constrained to default temperature bins (35, 40, 50, 65, 135, 200 and 300 K), IR emission lines, PAH (dust) features and extinction (we used a foreground extinction screen). As PAHFIT requires a single stitched spectrum, the SL spectrum was scaled to match the LL spectrum, with typical adjustments under 20\% (though several galaxies were adjusted by $\sim$40\% and NGC 7582 by greater than a factor of 6, indicating the presence of extended IR emission in this object). Here we utilize the EW of the PAH features at 11.3$\mu$m and 17$\mu$m, which consist of the features within the wavelength range 11.2-11.4$\mu$m and 16.4-17.9$\mu$m, respectively (Tables \ref{oiii_ir} and \ref{12m_ir}). However, we note that the current version of PAHFIT has a bug which assumes the PAH EW feature to be Gaussian rather than Drude, which could underestimate the PAH EW by a factor of 1.4. We report the EWs as reported from PAHFIT, with the caveat that these may be lower limits.
We compared our results with the 11.2$\mu$m feature from Wu et al. (2009) and the 11.3$\mu$m and 17$\mu$m features measured by Gallimore et al. (2010), where in the latter, we added their published 11.2$\mu$m and 11.3$\mu$m EW values. Wu et al. employed a spline fit between 10.8$\mu$m and 11.8$\mu$m to measure the EW. With this method, the results are widely influenced by the choice of anchor points for fitting the pseudo-continuum and can result in an underestimate of the EW compared to a method that utilizes spectral decomposition, such as PAHFIT \citep{PAHFIT}. Of the 28 sources we have in common with Wu et al. (2009), 12 of them had consistent 11.3$\mu$m EW values (within a factor of 2), 6 had lower values than we obtained (which would be expected from the disagreements between the spline vs. decomposition methods mentioned above) and 10 had higher values, where for 6 of these, PAHFIT had obtained an EW value of zero, yet the spline method yielded a measurement. Comparing our results with Gallimore et al. (2010) gave better results, though a discrepancy did still exist: of the 23 sources in common, we obtained consistent EW values (within a factor of 2) for 11 sources at 11.3$\mu$m and for 12 sources at 17$\mu$m. Though Gallimore used PAHFIT to measure these features, they modified the code to include more fine-structure lines, fit silicate emission features, and use the cold dust model from Ossenkopf et al. (1992); they also generated their own software to build spectral data cubes whereas we employed CUBISM. Such differences could account for the inconsistencies in our PAH EW measurements. Though the derived EWs are different from those reported by Wu et al. and Gallimore et al. for at least half the sources we have in common, our main conclusions based on PAH EWs agree qualitatively with Wu et al. and Gallimore et al.: PAH features are associated with other star formation activity indicators (Gallimore et al. 2010, Wu et al. 2009) and the EWs are inversely correlated to the strength of the ionization field (Wu et al. 2009 where they use the \textit{IRAS} colors to parameterize AGN strength).
In the discussion that follows, we divide the 12$\mu$m-sample into two classes, those with weak PAH emission (``PAH-weak'' sources) and those with strong PAH emission (``PAH-strong'' sources, galaxies with EW $>$ 1 $\mu$m in either the 11.3 $\mu$m or 17$\mu$m band, with PAH EWs detected in both bands); the strong PAH emission is likely due to starburst activity in the host galaxy (see $\S$5.2).
\section{Diagnostics of Intrinsic AGN Luminosity}
Our goal is to evaluate the relative efficacy of the five different proxies for the AGN intrinsic luminosity under consideration in this paper. We expect that these different proxies will not agree perfectly, due to the different physical mechanisms that produce and affect the emission features as well as biases resulting from sample selection, starburst contamination, statistical errors and in some cases, uncertain application of extinction corrections. To address this, we will undertake two kinds of comparions.
First, we will use our two Sy2 samples to inter-compare these proxies in a pair-wise fashion and measure the amount of scatter in the corresponding flux ratios. Which proxies agree best with one another? Second, we will compare these pairs of flux ratios to the corresponding values for unobscured Type 1 AGN to test which proxies are more ``isotropic,'' i.e. suffer the least AGN-viewing-angle dependence.
Figures \ref{hist_o4_o3} - \ref{xray_radio} show the histograms of a subset of ratios for the five proxies. In each plot, the solid black line represents both samples combined, the red dashed line and green dotted-dashed line delineate the 12$\mu$m sample (``PAH-weak'' and ``PAH-strong'' sources respectively) and the cyan filled histogram reflects the [OIII]-sample. Adjacent to these histograms are the luminosity vs. luminosity plots, showing the correlation between these indicators: the cyan asterisks represent the [OIII] sample, the red diamonds (green triangles) depict the ``PAH-weak'' (``PAH-strong'') 12$\mu$m sources, and the dashed black line represents the best fit from multiple linear regression analysis (i.e. the REGRESS routine in IDL), where in the figure captions, $\rho$ is the linear regression coefficient and P$_{uncorr}$ is the probability that the two quantities are uncorrelated. Though the distance dependence in luminosity vs. luminosity plots enhances the correlation compared to flux vs. flux plots, we employed this method as the 12$\mu$m sample lies at a systematically lower redshift, and thus have higher flux values, than the [OIII] sample. One of our main goals is to examine the dispersion in the flux ratios, where this distance dependence cancels out. In $\S$4.3, we test if these ratios are affected by luminosity.
Where available, the values for Sy1s are included in these plots. The results are summarized in Tables \ref{diag_results1} and \ref{diag_results2} which lists the mean and sigma of each ratio for the combined sample and the sub-samples separately. In the histograms and luminosity plots, the upper limits are plotted but not included in the analysis of the mean and sigma (except for the ratios involving the hard X-ray flux). Since only 12 of the 51 AGN were detected in hard X-rays, we have employed survival analysis to quantify the correlations among the proxies and to calculate the mean of the ratios. This approach takes the upper-limits into account (ASURV Rev 1.2, Isobe and Feigelson 1990; LaValley, Isobe and Feigelson 1992; for univariate problems using the Kaplan-Meier estimator, Feigelson and Nelson 1985; for bivariate problems, Isobe et al. 1986).
\subsection{Inter-Comparison of Proxies}
The isotropic luminosity diagnostics that agree best, and therefore may be least subject to the uncertainties and errors discussed above, are F$_{[OIII],obs}$, F$_{[OIV]}$ and F$_{MIR}$. A wider spread is present between the radio and hard X-ray fluxes compared with the optical and MIR values.
In all cases, a wider dispersion is present between all the flux ratios in the 12$\mu$m sample as compared to the [OIII] sample. Below, we examine whether such a scatter could be due to aperture effects, extinction corrections applied to the [OIII] flux, starburst contamination to the the MIR flux or if it represents a real difference between AGN selected on the basis of [OIII] flux versus MIR flux.
Since the 12$\mu$m Sy2s are typically more nearby than the [OIII]-selected galaxies, aperture effects can potentially play a significant role when comparing flux values by either missing NLR flux or obtaining too much host galaxy contamination. However, we find no evidence in our data for such an effect (see Appendix). Another possible explanation for this wider dispersion is that the optical data for the 12$\mu$m sample are drawn from the literature, which can introduce scatter into the optical diagnostics as such data are not taken and reduced in a uniform matter. The most striking example of this is the comparison of the [OIV] flux with the observed and extinction corrected [OIII]-flux: de-reddening the [OIII]-flux widens the dispersion in the 12$\mu$m-sample (see Figure \ref{hist_o4_o3}). This can be an artifact of using literature H$\alpha$ and H$\beta$ values and to a lesser extent can be due to large amounts of dust in the host galaxies of the 12$\mu$m sample as evidenced by the wide range of Balmer decrements. Goulding and Alexander (2009) note that galaxies that would not be identified optically as AGN (i.e. have a low ``D'' value, see $\S$5.1) tend to have similar Balmer decrements yet higher F$_{[OIV]}$/F$_{[OIII],obs}$ ratios than optically identified AGN. This result suggests that applying a reddening correction using the Balmer decrement still under-represents the intrinsic [OIII] flux. However, our 12$\mu$m sources with lower ``D'' values ($<$1.2) do not show systematically higher F$_{[OIV]}$/F$_{[OIII],obs}$ ratios (see Figure \ref{balmer}), indicating that ``extra'' extinction that Goulding and Alexander observe in their sources is not present in ours. As expected, the ratio of F$_{[OIV]}$/F$_{[OIII],obs}$ increases with H$\alpha$/H$\beta$, denoting that both quantities trace host galaxy extinction, though with wide scatter. Comparison with the locus of points for the [OIII] sample shows that the Balmer decrement is systematically higher for similar 12$\mu$m F$_{[OIV]}$/F$_{[OIII],obs}$ values, suggesting that the the 12$\mu$m Balmer decrements are over-estimating the amount of dust present rather than under-estimating. This result indicates that the literature Balmer decrements, which are not analyzed in a systematic and homogenous way, are introducing uncertainties that bias the results and do not better recover the truly intrinsic AGN luminosity.
However, this can not be the only cause of the greater scatter in the 12$\mu$m sample, since the ratio of MIR/[OIV] fluxes shows more scatter in the 12 $\mu$m sample (Figure \ref{mir2_oiv}), though these data were analyzed homogeneously. Could the presence of Sy2s in the 12$\mu$m sample that have significant contributions from starburst activity create the wider dispersion in these diagnostics? To address this issue, we isolated the ``PAH-strong'' sources, which have greater amounts of star formation activity (discussed in detail below). The distributions between the ``PAH-strong'' and ``PAH-weak'' sub-samples are similar, suggesting that starburst processes are not responsible for the wider dispersion. We also focused on sources with a limited ``D'' value, which as noted above indicates the relative contribution of AGN to starburst activity. Repeating the calculation of mean and standard deviations on the flux ratios for the 12$\mu$m sources with 1.2$\leq$ D $\leq$ 1.7 did not result in a significant decrease (a factor of 2 or more) in the dispersion with the exception of log (F$_{MIR}$/F$_{[OIII],obs}$) ($\sigma$=0.42 dex), log (F$_{[OIII],obs}$/F$_{8.4GHz}$) ($\sigma$=0.60 dex) and log (F$_{[OIII],corr}$/F$_{8.4GHz}$) ($\sigma$=0.51 dex). For these first two ratios, this is largely due to the removal of the 3 outliers (F04385-0828, F08572+3914 and Arp 220) with systematically higher (lower) F$_{MIR}$/F$_{[OIII],obs}$ (F$_{[OIII],obs}$/F$_{8.4GHz}$) values from the full sample. The dispersions for the other ratios were still systematically higher than the [OIII]-sample. We conclude that there is a real difference between the AGN selected on the basis of [OIII] emission-lines and MIR continuum.
We also compared our results with two other samples of Seyfert 2 galaxies: one a complete sample down to a flux limit of (1-3) $\times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$ at 14 - 195 keV drawn from the 3 and 9 month Swift-BAT survey (Mel\'endez et al. 2008 and references therein) and the other drawn from the revised Shapley-Ames catalog (Shapley \& Ames 1932; Sandage \& Tammann 1987), consisting of galaxies with $B_T \leq$ 13 \citep{DS09}. Here we include those radio quiet Seyfert types 1.8 - 2 that have measured [OIII] and [OIV] fluxes, giving 12 and 56 Sy2s, respectively. The log of the ratios of [OIV] to [OIII],obs for both samples are higher than our combined sample, 0.60 $\pm$ 0.74 dex (Mel\'{e}ndez et al. 2008) and 0.57 $\pm$ 0.67 dex (Diamond-Stanic et al. 2009) vs. 0.08 $\pm$ 0.41 dex, but the differences are not statistically significant. A wider dispersion is present in these comparison samples than the [OIII]-selected sample (as was the case for the 12$\mu$m sample). This may indicate that selection based on [OIII] leads to better agreement between between the [OIII] and [OIV] flux rather than selection based on other methods. This effect could be due to the [OIII]-bright sources having less extinction in the NLR than Sys selected in other ways.
As the samples in Weaver et al. (2010) and Winter et al. (2010) samples were selected based on their hard (14-195 keV) X-ray flux from the Swift-BAT 9 month (Winter et al. (2010) and Weaver et al.(2010)) and 22 month (Weaver et al. (2010)) catalog, we can compare our Sy2 X-ray flux ratios. Using the values in Winter et al. (2010), we find log (F$_{14-195 keV}$/F$_{[OIII],obs}$) = 2.76$\pm$0.59 dex and log (F$_{14-195 keV}$/F$_{[OIII],corr}$)\footnote{We note that for the cases where the Winter et al. 2010 Balmer decrement was less than the assumed intrinsic value (3.1), we did not apply a redenning correction, but rather used the observed value, both here and in $\S$4.2.} = 2.34$\pm$ 0.69 dex for Sy2 galaxies. The log (F$_{14-195 keV}$/F$_{[OIV]}$) ratio from Weaver et al. (2010) for Sy2s is 2.38$\pm$0.45 dex. All three values are systematically higher than what we obtain for our samples of Sy2 galaxies by roughly an order of magnitude (see Table \ref{diag_results2}). This is depicted graphically in Figure \ref{bat_hist}. Employing survival analysis, we compared these ratios between the BAT-selected Sy2s and the [OIII] and 12$\mu$m samples separately and find that they differ significantly (i.e. p $<$0.05, corresponding to the 2$\sigma$ level, that they are drawn from the same parent population), with the caveat that with only one [OIII] Sy2 detected by BAT, the comparison between BAT and [OIII] selected Sy2s may be less robust. These differences suggest that the samples selected in hard X-rays do not fairly sample the population of Type 2 AGN selected in the MIR and possibly the optical, however comparisons with BAT-selected Sy1s reveal mixed results (see $\S$4.2).
\subsection{Comparison with Sy1s}
In order to determine if the proxies we are considering are affected by the orientation of the AGN, and evaluate the extent to which they may be considered truly ``isotropic,'' we compared our results for Sy2 with the corresponding values for Sy1, using data taken from the literature. The Sy1 MIR fluxes were calculated from the 14.7$\mu$m flux densitites reported in Deo et al. (2009), where they analyzed a heterogeneous sample of Sy1 and Sy2 galaxies available in the \textit{Spitzer} archive, ranging in redshift 0.002 $< z <$ 0.82. The radio flux values were derived from the high-resolution 8.4-GHz flux density values from Thean et al.(2000), which presented analysis of the extended 12$\mu$m sample.\footnote{The extended 12$\mu$m sample probes to a lower flux density limit than the original 12$\mu$m sample: 0.22 Jy vs. 0.30 Jy, giving a total of 118 detected Sys over the 59 detected in the original sample \citep{Rush}.} The hard X-ray data (14-195 keV) are drawn from Mel\'endez et al. (2008, sample selection described above), the 22-month Swift-BAT Catalog \citep{Tueller} and Rigby et al. (2009, same parent sample as Diamond-Stanic et al. 2009, with X-ray data derived from the 22-month Swift-BAT Catlog, BeppoSAX (Dadina 2007) and Integral (Krivonos et al. 2007)). The comparison Sy1 [OIII] and [OIV] flux values are derived from Mel\'endez et al. (2008) and Tommasin et al. (2008, 2010), which presents high resolution \textit{Spitzer} spectroscopy of the extended 12$\mu$m sample. As only Winter et al. (2010) quote Balmer decrements, we only have comparison Sy1 F$_{[OIII],corr}$ values for the samples selected from the BAT catalog (i.e. F$_{14-195 keV}$ and F$_{[OIV]}$ from Weaver et al. 2010). We utilize both the Kolmogorov-Smirnov test (``K-S'' test) and Kuiper's test on the detected data points (excluding the three [OIV] and three radio upper limits in the 12$\mu$m data) to determine to which extent the flux ratios are significantly inconsistent between the Sy1 and Sy2 galaxies: a lower Kuiper ``D'' value indicates that these two populations are drawn from the same parent population, suggesting that such fluxes are independent of viewing angle. The Kuiper test is similar to the more often used K-S test but with the following modification: the ``D'' statistic of the K-S test represents the maximum deviation between the cumulative distribution functions (CDFs) of the two samples, whereas the ``D'' statistic in Kuiper's test is the sum of the maximum and minimum deviations between the CDFs of the two samples, so that this statistic is as sensitive to the tails as to the median of the distribution. The results of the K-S test and Kuiper test agree in that they do not lead us to reject the null hypothesis that the two samples are drawn from the same parent population, with the exception of the F$_{[OIV]}$/F$_{[OIII],obs}$ ratio, where the tests lead to conflicting results. We note that two-sample tests work better for larger data sets, so the probabilities quoted in Table \ref{Kuiper} should be interpreted as approximate.
The comparisons of the Sy2 and Sy1 samples are shown in Figures \ref{hist_o4_o3} through \ref{xray_radio}. In each, the dotted-dashed and (in the cases of more than one comparison sample) dashed line(s) on these plots indicate the mean values for the Sy1 diagnostic ratios and the correlations from linear regression we calculated from the literature values.
A mild disagreement between the average Sy1 and Sy2 F$_{[OIV]}$/F$_{[OIII],obs}$ ratio (up to a factor of $\sim$2) is evident. Here the comparison Sy1 data come from the Diamond-Stanic et al. (2009) and the Mel\'endez et al. (2008) samples. We obtain mixed results as to the significance of this difference based on the statistical test used: according to the K-S test, the F$_{[OIV]}$/F$_{[OIII],obs}$ ratio for the Sy1 and Sy2 populations are statistically significantly different (D=0.301, p=0.042), but not according to the Kuiper test (D=0.310, p=0.223). We find similar disparate results when we run these tests on the F$_{[OIV]}$/F$_{[OIII],obs}$ ratio for the detected points between the Sy1s and Sy2s in the Mel\'endez et al. (2008) and Diamond-Stanic et al (2009) samples, namely that the K-S test implies different parent populations (p=0.005 and p=0.004, respectively) but not the Kuiper test (p=0.097 and p=0.126, respectively). Mel\'endez et al. (2008) and Diamond-Stanic et al. (2009) (as well has Haas et al. 2005, who compared seven quasars with seven Fanaroff-Riley II (FRII) radio galaxies) have reported significant differences between the observed [OIII] and [OIV] flux between Sy1s (quasars for Haas et al. 2005) and Sy2s (FRIIs for Haas et al. 2005), with the type 2 sources having higher F$_{[OIV]}$/F$_{[OIII],obs}$ ratios. These authors have attributed the diminumtion of [OIII] in type 2 AGN to extinction in the NLR. Baum et al. (2010) suggests that such [OIII] obscuration results from the AGN torus: using the 12 $\mu$m sample, they find a correlation between the F$_{[OIV]}$/ F$_{[OIII],obs}$ ratio and the Sil 10$\mu$m feature, which probes torus obscuration.\footnote{They define the Sil strength by the natural logarithm of the observed flux of the feature divided by the interopolated continuum flux at 10$\mu$m.} In type 1 Sy1s, this silicate feature is in emission, whereas Sy2s exhibit Sil absorption, making the Sil strength a probe of system orientation. The ratio of F$_{[OIV]}$ to F$_{[OIII],obs}$ increases with Sil absorption (parameterized by negative values of the Sil strength) which could suggest that the torus is extincting part of the [OIII] emission. Our results may confirm these previous studies as we find that Sy2s tend to have lower observed [OIII] emission as compared to Sy1s and this may be due to NLR extinction. We note, however, that such extinction affects the [OIII] line only up to a factor of 2 on average between our Sy2 and comparison Sy1 samples, albeit with a wide dispersion, and this difference between the two populations may not be significant.
The average log (F$_{MIR}$/F$_{[OIII],obs}$) ratio is consistent between Sy1s (2.56 $\pm$ 0.50 dex) and Sy2s (2.62 $\pm$ 0.62 dex), which could seemingly contradict the results cited above where NLR extinction causes attenuation of the [OIII] flux in Sy2s but not in Sy1s. The clumpy torus model of Nenkova et al. (2008) and smooth torus model of Pier \& Krolik (1992) predicts a slight anistropy in emission at 12$\mu$m depending on viewing angle: as the viewing angle increases from 0$^{\circ}$ (Sy1) to 90$^{\circ}$ (Sy2), the torus flux decreases by a factor of $\sim$2. The effects of depressed MIR emission in Sy2s and enhanced MIR emission in Sy1s, assuming [OIII] is more extincted in the former than the latter, would therefore result in F$_{MIR}$/F$_{[OIII],obs}$ ratios that are more consistent than F$_{[OIV]}$/F$_{[OIII],obs}$, which is indeed what we observe. However, the average differences between F$_{MIR}$/F$_{[OIII],obs}$ and F$_{[OIV]}$/F$_{[OIII],obs}$ are within the scatter of these ratio values, and we are unable to rule this out as the main driver for the disagreement, rather than invoking anisotropies in torus emission.
Interestingly, the relationship between L$_{MIR}$ and L$_{[OIV]}$ is nearly identical for Sy1 and Sy2 galaxies (Figure \ref{mir2_oiv}). Though the MIR flux is not corrected for starburst contamination (see Appendix), and the Sy2s in the 12$\mu$m sample are thought to harbor more star formation activity than Sy1s (e.g. Buchanan et al. 2006), we see no evidence that star formation activity is contributing significantly to the MIR emission. As the F$_{MIR}$/F$_{[OIV]}$ diagnostic ratio shows the smallest dispersion for the combined sample, a similar relationship to Sy1 galaxies and no evidence for luminosity bias (see Sectin $\S$4.3), the MIR and [OIV] flux may be the most robust proxies for the intrinsic AGN luminosity in Type 2 AGN. However, the KS test and Kuiper's test indicates a lower probability that the Sy1 and Sy2 samples agree than the F$_{MIR}$/F$_{[OIII],obs}$ ratio, though this could be driven by the lower N$_{eff}$ value for F$_{MIR}$/F$_{[OIII],obs}$ rather than a robust statistical agreement.
The different slopes between Sy1s and Sy2s in the luminosity plots of the radio data against other intrinsic AGN flux proxies (Figures \ref{o3_radio}, \ref{oiv_radio} and \ref{mir2_radio}) suggest disagreements between these samples. However, the F$_{[OIV]}$/F$_{8.4GHz}$ and F$_{MIR}$/F$_{8.4GHz}$ flux ratios are consistent between Sy1 and Sy2 galaxies, indicating that the disparate slopes are perhaps influenced by scatter due to the wide range of radio loudness in AGN. Results of the KS test and Kuiper's test (Table \ref{Kuiper}) also indicate that the differences in the radio flux ratios between Sy1s and Sy2s are not statistically significant. Diamond-Stanic et al. (2009) compared the 6 cm radio data between Sy1s and Sy2s and found that for the Sy2s with a measured X-ray column density, these two samples show no statistically significantly differences, though they find a higher probability that they are drawn from the same distribution ($\sim$68 - 78\%) than we do ($\sim$55\%). Mel\'endez et al. (2010) also found that the 8.4 GHz and [OIV] fluxes between Sy1 and Sy2 galaxies are not significantly different, though sources dominated by star formation (i.e. less than 50\% of the [NeII] line attributable to AGN ionization) had statistically different F$_{[OIV]}$/F$_{8.4GHz}$ values than AGN dominated sources, indicating that radio emission may not accurately trace intrinsic AGN power. This latter result may agree qualitatively with our Figure \ref{oiv_radio}, where the ``PAH-strong'' sources lie at or below the best-fit line between L$_{[OIV]}$ and L$_{8.4GHz}$.
The hard X-ray proxy performs much more poorly (Figures 7 - \ref{xray_radio}), based on both the wider dispersion in the diagnostic flux ratios and the larger disagreement between the Sy1 and Sy2 flux ratios. The mean hard X-ray emission (normalized by other isotropic indicators) in Sy2s tends to be about an order of magnitude weaker than in Sy1s, though this is driven largely by the 12$\mu$m sample as only one source was detected by BAT in the [OIII] sample. This disagreement agrees with the results of Rigby et al. (2008) and Weaver et al. (2010), where the X-ray flux was normalized by the [OIV] emission. Indeed, using survival analysis, we find that F$_{14-195 keV}$/F$_{[OIV]}$ disagrees significantly between BAT-selected Sy1s and both the [OIII] and 12$\mu$m sub-samples. Such a large disagreement is not found between the Sy1s and Sy2s in the Winter et al. (2010) sample (see Table \ref{diag_results2}), which is driven by the lower [OIII] flux observed in their Sy2s as compared to their Sy1s. The hard X-ray to [OIII] flux ratios, both observed and reddening corrected, do not differ significantly between the BAT-selected Sy1s and the [OIII]-selected Sy2s, but do for the 12$\mu$m sample.\footnote{F$_{14-195 keV}$/F$_{[OIII],obs}$ has mixed results for the 12$\mu$m sample: according to two of the tests (Logrank and Peto \& Prentice Generalized Wilcoxon Test), the difference is significant though with other tests, the null hypothesis that the two samples are drawn from the same parent distribution can only be discarded at the $\sim$12\% confidence level.} According to the Logrank and Peto \& Prentice Generalized Wilcoxon tests, the hard X-ray flux normalized by the MIR flux differs signficantly for both Sy2 subsamples and the BAT-selected Sy1s. Consistent with the results from $\S$4.1, hard X-ray selected AGN do not represent the population of those selected in the MIR, and there may be some evidence that they do not fully sample the optically selected sources. Compton scattering may be responsible for weakening the observed hard X-ray emission in Sy2s, as suggested by Weaver et al. (2010), which indicates that the 14-195 keV emission is not truly isotropic.
\subsection{Luminosity Dependence}
As we have seen above, there is significant scatter in the flux ratios of the different proxies for AGN intrinsic luminosity. Here we examine the possibility that some of this scatter is caused by systematic differences that correlate with the accretion rate of the black hole (in units of the Eddington limit).
To make this test, for any pair of luminosity proxies we parameterized $L_{AGN}/L_{Edd}$ by the square root of the product of the luminosities of the two proxies divided by the mass of the central black hole (M$_{BH}$, listed in Tables \ref{o3_sample} and \ref{12m_sample}). Linear regression fits were performed, with the correlation coefficients and probability of uncorrelation listed in Table \ref{lum_dep}.
We find three statistically significant anti-correlations (Figure \ref{ledd}): F$_{[OIV]}$/F$_{[OIII],obs}$, F$_{MIR}$/F$_{[OIII],obs}$ and F$_{MIR}$/F$_{8.4GHz}$. The anti-correlations for the ratios involving F$_{[OIII],obs}$ are largely driven by those galaxies with a high Balmer decrement. When we exclude the 6 sources with H$\alpha$/H$\beta \geq$ 9, which may be those systems with the most NLR etinction, these anti-correlations are no longer statistically significant. This may indicate that the bolometric correction to the observed [OIII] luminosity might have a weak dependence on the Eddington ratio. However, the observed [OIII] luminosity, which partly parameterizes the Eddington ratio, does not as accurately trace intrinsic AGN flux for these more dust-obscured sources. If the Eddington ratio is defined as just $L_{OIV}/M_{BH}$ and L$_{MIR}/M_{BH}$ in these relationships, the anti-correlations are no longer significant. Hence, the weak trends in Figure \ref{ledd} a) and b) is likely driven more by NLR extinction bias on the [OIII] flux rather than the accretion rate of the black hole. This latter mechanisms could be affecting the F$_{MIR}$/F$_{8.4GH}$ ratio, albeit with wide scatter.
\section{Starburst Activity in the Host Galaxy}
\subsection{Comparison of Different Diagnostics}
Given the strong connection between Type 2 AGN and star-formation (e.g. Kauffmann et al. 2003) we expect that the signature of both processes will be present in optical and MIR spectra of AGN. By analogy to the previous section (where we compared different proxies for the intrinsic AGN luminosity) we now undertake a comparison of different diagnostics of the relative energetic significance of a starburst vs. the AGN.
One such diagnostic involves the use of the MIR polycyclic aromatic hydrocarbon (PAH) features. These have been shown to be correlated with star formation activity (e.g. Smith et al. 2007) and possibly anti-correlated with the presence of an AGN (O'Dowd et al. 2009; Voit 1992). More specifically, we used the equivalent width (EW) of the PAH features to assess the relative amount of starburst activity in the host galaxy (e.g. Genzel et al. 1998). Another empirical diagnostic of the relative contribution of the starburst in the MIR can be parametrized by the MIR spectral index: $\alpha_{20-30\mu m}$\footnote{$\alpha_{\lambda_1 - \lambda_2}$ = log($f_{\lambda_1}$/$f_{\lambda_2}$)/log($\lambda_1$/$\lambda_2$)}. Larger values of $\alpha_{20-30\mu m}$ indicate the presence of cold dust from starburst activity (Deo et al. 2009 and references therein). The ratio of the [OIV] to [NeII] 12.81$\mu$m MIR emission-lines probes the hardness of ionizing spectrum and hence the relative importance of the AGN and starburst. A larger ratio ($\sim$1) implies the dominance of AGN activity whereas a lower ratio ($\sim$0.02) is pure starburst activity \citep{Genzel}. The analogous diagnostic from optical spectra is the distance a galaxy spectrum lies from the locus of star forming galaxies in the Baldwin, Phillips \& Televich BPT (1981, BPT) diagram (D = $\sqrt{([NII]/H\alpha + 0.45)^2 + ([OIII]/H\beta + 0.5)^2}$, Kauffmann et al. 2003). A larger ``D'' parameter indicates pure AGN activity while a smaller value implies a mixture of starburst and AGN processes in the host galaxy.
Figures \ref{alpha_ew} and \ref{oiv_neii_ew} illustrate the relationship between these AGN and star formation activity indicators for the Sy2s in our combined sample. The color coding is the same as in Figures \ref{hist_o4_o3} - \ref{xray_radio}. According to linear regression analysis, $\alpha_{20-30\mu m}$ and the PAH EWs are correlated and [OIV]/[NeII] and the PAH EWs are anti-correlated at greater than the 3.5$\sigma$ level: PAH EW 11.3 $\mu$m vs. $\alpha_{20-30\mu m}$ has $\rho$=0.609, P$_{uncorr}$=1.47$\times 10^{-5}$; PAH EW 17 $\mu$m vs. $\alpha_{20-30\mu m}$ has $\rho$=0.600, P$_{uncorr}$=5.26$\times 10^{-6}$; PAH EW 11.3$mu$m vs. [OIV]/[NeII] has $\rho$=-0.677, P$_{uncorr}$=2.26$\times$10$^{-6}$; PAH EW 17$\mu$m vs. [OIV]/[NeII] has $\rho$=-0.515, P$_{uncorr}$=4.89$\times10^{-4}$. We also note that the Sy2s with strong PAH emission mostly lie at systematically higher PAH EW values than the relation found between $\alpha_{20-30\mu m}$ and the PAH EWs. The majority of the Sy2s have D values between $\sim$1.2-2.0, with five of the strong PAH sources at systematically lower D values, $\sim$0.5-1.1 (see Figure \ref{d_figs}\footnote{The two ``PAH-weak'' sources with low D values are Arp 220 and F08572+3915, which have [OIV] upper limits and no measureable PAH EW at 11.3$\mu$m.}). The Sy2s with lower D values have higher PAH EW values and lower F$_{[OIV]}$/F$_{[NeII]}$ ratios, though they exhibit a weaker trend with the IR spectral index.
These results indicate that the various MIR and optical indicators of starburst activity agree both qualitatively and quantitatively.
\subsection{The Spatial Scale of the MIR Emission and the Role of the Host Galaxy}
The results above refer to measurements of the central region enclosed by the IRS aperture. However, the 12$\mu$m sample was drawn from the \textit{IRAS} Point Source Catalog, where the aperture size (0.75 $\times$ 4.5' at 12$\mu$m) is much larger and will encompass contributions from the host galaxy. To quantify the extendedness of the MIR emission in the 12$\mu$m sample, we calculate the ratio of the \textit{IRAS} flux (at 12$\mu$m, Spinoglio \& Malkan 1989) to the \textit{IRS} flux. A ratio of $\sim$1 indicates the MIR emission is dominated by the galactic center, whereas higher ratios imply a greater amount of extended emission. In Figure \ref{iras}, we plot this ratio against the PAH EW at 11.3 $\mu$m and F$_{[OIV]}$/F$_{[NeII]}$. As expected, the relative amount of extended MIR emission decreases as the relative energetic importance of the AGN increases.
Comparison of the ratio by which the SL module was rescaled to match the LL module (see $\S$ 4.2) reveals the presence of extended emission on smaller spatial scales. Inspection of this extendedness factor does not show any significant differences between the ``PAH-weak'' and ``PAH-strong'' sources (with the exception of NGC 7582).
\subsection{Are [OIII] and [OIV] Biased by Star Formation?}
In this section, we investigate whether the relative fraction of the AGN bolometric luminosity that emerges in [OIII] and [OIV] line emission is preferentially higher in galaxies with more star formation. This might be expected if the gas clouds in the NLR that are photoionized by the AGN and produce these lines are directly related to the gas clouds responsible for star-formation. If true, this would imply that AGN selected using [OIII] or [OIV] would be biased towards galaxies with higher star formation rates.
To test this, we have plotted the ratio of both F$_{[OIII],obs}$ and F$_{[OIV]}$ to F$_{MIR}$ versus the star formation indicators analyzed above (PAH EWs, IR spectral index and the ``D'' parameter). We find no strong trends between the star formation indicators and [OIV] and [OIII] emission, as illustrated in Figures \ref{sb_bias} and \ref{sb_bias2}.
We conclude that there is no convincing evidence that host galaxies with a large star formation rate have preferentially higher relative luminosities of [OIII] and [OIV] at the luminosities represented in this sample, where the bolometric luminosity (L$_{bol}$) ranges from L$_{bol}$ $\approx$ 10$^{9}$ - 8 $\times$ 10$^{11}$ L$_{sun}$, which is $\sim$3$\times 10^{-5}$ to 0.5 of the Eddington luminosity (L$_{edd}$)\footnote{We estimated L$_{bol}$ by assuming the observed mid-infrared flux constitutes 20\% of the bolometric flux \citep{12m}.}. Thus, these proxies of intrinsic AGN power are not biased by star formation activity at these Eddington ratios.
\section{Conclusions}
We have taken an empirical approach in analyzing the agreement among the various indicators of isotropic AGN luminosity for two complete and homogeneously selected samples of Sy2s, one selected based on observed [OIII] flux and the other on MIR flux. The diagnostic ratios with the smallest spread are likely those where such biases from sample selection, starburst contamination, statistical errors, and scatter due to the various physical mechanisms that produce these emission features, are minimized. Such indicators, as well as those that agree the most with Sy1 relations, may be the most robust tracers of AGN activity. Our results on these indicators are summarized below.
- \textit{Sample Selection} The optically selected sample, picked on the basis of high [OIII] flux, shows tighter correlations among these diagnostics than the MIR selected sample. We investigated whether the inclusion of active star forming galaxies in the 12$\mu$m sample introduced the spread in these ratios by dividing the sample into galaxies that have large amounts of starburst activity (``PAH-strong'') and those that do not (``PAH-weak''). The distribution of the diagnostic ratios for the two sub-samples are similar. Isolating the 12$\mu$m sources with a limited D range (1.2$\leq$D$\leq$1.7) also results in large scatter that is systematically higher than that observed in the [OIII] sample for all but three of the flux ratios. A similarly wide spread between F$_{[OIV]}$/F$_{[OIII],obs}$ is present in other samples (i.e. Mel\'endez et al. 2008 and Diamond-Stanic et al. 2009), indicating that sample selection based on [OIII] is primarily responsible for the tighter correlations we observe. This may be due to less extinction in the NLR region which would be expected in sources that have high observed [OIII] flux.
- \textit{Extinction Correction} Applying an extinction correction to the [OIII] flux tightens the correlations with the other luminosity indicators for the [OIII]-selected sample, yet broadens the dispersion for the 12$\mu$m sample. It is not clear whether this difference is primarily due to the different sources of the emission-line data (homogenous SDSS data for the [OIII] sample and heterogeneous data for the 12$\mu$m sample), or whether it simply points out the limitations of extinction corrections in very dusty AGN. Comparison of the optical vs. MIR properties of the most dusty AGN in the SDSS suggest that the former effect is important (Wild et al. 2010).
- \textit{Agreement Among Sy2s} The observed [OIII], [OIV] and MIR luminosities agree the best in the combined Sy2 sample. The widest spread among the various proxies is seen in the radio emission. The X-ray data are dominated by upper limits, but also show a significantly larger dispersion than the optical and IR isotropic flux indicators.
- \textit{Comparison with Sy1s} The mean ratio of the observed [OIII] flux to the [OIV] flux is lower in Sy2 than in Sy1s by a factor of 1.5-2, while the mean ratio of the observed [OIII] flux to MIR is consistent between Sy2s and Sy1s. The former result, which represents a statistically significant difference between Sy1s and Sy2s according to the KS test but not Kuiper's test, agrees with previous findings (e.g. Haas et al. 2005, Mel\'endez et al. 2008, Diamond-Stanic et al. 2009) and has been interpreted as extinction affecting [OIII] in the NLR, or even torus obscuration attenuating the [OIII] emission (Baum et al. 2010). However, the latter result can not be simply explained by a larger amount of dust extinction of [OIII] in the Sy2s, but it could be due to a slight anisotropy in the MIR emission as predicted by Pier \& Krolik (1992) and Nenkova et al. (2008) where Sy1s could have up to a factor of two higher MIR flux as compared to Sy2s. The wide scatter in these ratios can also be responsible for the discrepancy between the F$_{[OIV]}$/F$_{[OIII],obs}$ and F$_{MIR}$/F$_{[OIII],obs}$ values, which may be the main driver for the mild disagreement rather than torus emission anisotropy. The F$_{[OIV]}$/F$_{8.4GHz}$ and F$_{MIR}$/F$_{8.4GHz}$ mean values are consistent between Sy1 and Sy2 galaxies (in agreement with Diamond-Stanic et al. 2009 and Mel\'endez et al. 2010 for the [OIV] and radio comparison), though the slopes of the luminosity plots show disagreements and there is wide scatter which is expected due to the wide range of radio loudness observed in AGNs.
- \textit{Hard X-ray Selected Samples} We find that the hard X-ray flux (relative to the [OIII], [OIV], and MIR fluxes) is suppressed by about an order-of-magnitude in MIR selected Sy2s compared to both Sy1s (consistent with Rigby et al. 2008) and to hard X-ray selected Type 2 AGN (Winter et al. 2010 and Weaver et al. 2010). The comparison with the [OIII] sample is mixed, with statistically significant differences between the Sy2s and Sy1s when the X-ray flux is normalized by the [OIV] and MIR flux, but not when normalized by the [OIII] flux. However, hard X-ray selected Sy2s differ significantly from [OIII]-selected Sy2s (though with only one [OIII] Sy2 detected by BAT, this analysis may be less robust than the 12$\mu$m comparison). These results indicate that hard X-ray emission (E $>$ 10 keV) is anistropic and hard X-ray selected samples are biased against the more heavily obscured type 2 AGN that are present in MIR and possibly [OIII] selected samples. As Weaver et al. (2010) note for sources detected in hard X-rays, Compton scattering could be responsible for the hard X-ray attenuation observed in Type 2 AGN as compared to Type 1. In more obscured sources, Compton scattering may be pushing them below the flux sensitivity of BAT.
F$_{MIR}$ and F$_{[OIV]}$ agree the best, both in comparison with the other indicators in the combined Sy2 sample (having the least scatter) and in having a nearly identical flux ratio in Sy2s and Sy1s as well as not suffering from a luminosity bias. Among the indicators we have considered, they are the best proxies for truly isotropic AGN emission.
We tested the level of agreement of various optical and infrared indicators of star formation activity compared with proxies of AGN activity. Similar to previous works, we find statistically significant relations among the various indicators of the relative energetic significance of star formation and an AGN. These include the MIR spectral slope (parametrized by $\alpha_{20-30\mu m}$), the PAH EWs at 11.3$\mu$m and 17$\mu$m, the ratio of [OIV]/[NeII] fluxes, and the location of the galaxy in the commonly used diagnostic diagram based on optical emission-lines. We note that the last two diagnostics are a measure of the relative contribution of AGN vs. starburst activity to the incident ionizing radiation. As the incident radiation field becomes more dominated by the AGN activity, the PAHs become weaker relative to the MIR continuum. In part this is expected because of the``dilution'' of the MIR continuum by AGN-heated dust, but the AGN may also be directly affecting the PAH emission (e.g. O'Dowd et al. 2009).
We also found that the Sy2s that were clearly starburst/AGN composites based on the above indicators were systematically those cases in which large-scale MIR emission from the host galaxy greatly exceeded that from the AGN. We quantified this by comparing the ratio of the 12$\mu$m flux from the large \textit{IRAS} aperture with the 15.5$\mu$m flux from the small \textit{IRS} aperture. A smaller aperture is therefore necessary to isolate AGN emission in systems with high rates of star formation on large scales and/or low AGN luminosities.
The ratios of the [OIII]/MIR and [OIV]/MIR fluxes show little if any evidence for a correlation with any of the above measures of the relative amount of star formation. This lack of a relationship suggests that the [OIII] and [OIV] lines are not biased to be a preferentially higher fraction of the AGN bolometric luminosity in host galaxies with more star formation activity (more dense gas).
\acknowledgments{This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. The authors thank the anonymous referee whose insightful comments improved the quality of this manuscript.}
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"redpajama_set_name": "RedPajamaArXiv"
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Tintern
Tintern (Welsh: Tyndyrn) is a village on the west bank of the River Wye in Monmouthshire, Wales, about 5 miles north of Chepstow. It has been popular with tourists since the late 18th Century, who visit for the natural scenery and the ruined Tintern Abbey.
The settlement of Tintern has been formed through the coalescence of two historic villages, previously separate parishes - Tintern Parva, forming the northern end of the village and Chapel Hill which forms the southern end.
Tintern, Monmouthshire, Wales - October 30, 2014
View down river towards the Abbey which can just be seen.
Autumnal colours at Tintern.
River Wye at Tintern.
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"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Japanologi är vetenskapen om det japanska språket och japansk kultur.
Se även
Japanologer
Japanofil
Vetenskap i Japan
Orientalistik
WP:Projekt Japan
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Kamel reached into the sack and pulled out some coins. I'll see if I can find something to eat. The road goes straight for about a kilometer and then turns. Please send details of these courses. Hillel wrapped the ice pack in a damp towel and placed it on his thigh. This is what must happen. Linley scratched the back of his neck. Herbert has been busy in his office. There has been a large increase in trade between the United States and China. He lay in agony until the doctor arrived.
Sridhar isn't a racist. Can you touch your toes without bending your legs? He paid no attention to my advice. He is the one who took care of his wound. Julia said that he'd rather come tomorrow. Cotton absorbs water. I want you to know that I believe you. Pitawas passed away several years ago. I'll give you Dr. Shiegal's telephone number. Helge shouldn't even be there.
I'm definitely ready. I forgot to put sugar in your tea. Juan thought he'd be safe there. What's the big deal anyway? It was noticed after more than a month that that promise had not been carried out. I wonder what they're planning. The book is written in Spanish. A firewall will guarantee Internet security. I wanted to run away with them. Yesterday we walked to school but later rode the bus home.
We used to compete furiously in college. Are there any other possibilities? Sandip looked toward Bob. I want you to meet Moore. You can't be at two places at once. They lost the Scotland match in the last five minutes. He has something to do with the matter. I don't know where to buy imported cheese. He's a bread addict. Erik sat down on the bench.
There must be a mistake. I'm not arresting them. I'm certain that he'll come. Wolfgang is pretty sure that Sanche won't be there. Why did you come back from Germany? She made an effort at joking but it fell quite flat. The cat was sitting on the window and the dog was barking outside. Are you questioning my judgment? I am very glad to be out of high school. Could you speak more slowly?
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Тетяна Пилипівна Вільчинська ( 10 червня 1962, Берізки, УРСР) — українська мовознавиця, докторка філологічних наук (2001), професорка (2012). Дружина Олександра Вільчинського.
Життєпис
Тетяна Вільчинська народилася 10 червня 1962 року у селі Березках, нині Кривоозерської громади Первомайського району Миколаївської области України.
Закінчила філологічний факультет Одеського державного університету ім. І. І. Мечникова (1984). Доцентка (2001—2012), від 2012 — професорка катедри української мови, від 2014 — декан філологічного факультету (нині факультет філології і журналістики) Тернопільського національного педагогічного університету імені Володимира Гнатюка.
Наукова діяльність
У 1996 році захистила кандидатську дисертацію на тему «Семантико-словотвірна характеристика оцінних назв осіб в українській мові» (Львівський державний університет ім. Івана Франка). У 2009 — докторську дисертацію на тему «Розвиток концептосфери сакрального в українській поетичній мові XVII-XX ст.» (Інститут філології Київського національного університету імені Тараса Шевченка).
Сфера наукових інтересів: Лексикологія і граматика української мови, когнітивна лінгвістика, лінгвоконцептологія, етнолінгвістика, лінгвокультурологія, прагмалінгвістика, аксіологічна лінгвістика, лінгвопоетика.
Головна редакторка «Наукових записок ТНПУ. Серія Мовознавство», член редакційних колегій журналів «Наукові записки ТНПУ. Серія Літературознавство», «Медіапростір» (ТНПУ імені Володимира Гнатюка), «Філологічний дискурс» (Хмельницька гуманітарно-педагогічна академія); рецензентом інших фахових видань.
Авторка понад 170 публікацій, а також численних підручників, посібників, наукових статей і навчально-методичних матеріалів з історії мови та сучасної української літературної мови.
Праці:
Концептуалізація сакрального в українській поетичній мові XVII-XVIII ст. (2008),
Когнітологія та концептологія в лінгвістичному висвітленні (2011, у співавторстві),
Сучасна українська літературна мова. Морфологія (2016, у співавторстві),
Фразеологізми у творах Богдана Лепкого. Словник (2010),
Фразеологія творів Бориса Харчука (2012),
Індивідуально-авторська картина світу: становлення поняття (2008),
Лінгвалізація сакрально-хтонічного в індивідуально-авторській картині світу М. Коцюбинського (на прикладі концепту «чорт»; 2015).
Примітки
Джерела
Вільчинська Тетяна Пилипівна // ТНПУ.
Сердечні вітання! // ТНПУ.
Лінгвістична творчість Вільчинської Тетяни Пилипівни // Наукова бібліотека ТНПУ.
Посилання
Тетяна Вільчинська // Чтиво.
доктори філологічних наук
українські професори
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The Mexican-American War
James A Watkins
James A. Watkins is an entrepreneur, musician, and a writer with three non-fiction books and hundreds of magazine articles read by millions.
American President John Quincy Adams tried to buy Texas from Spain for one million dollars in 1819. Texas is as large as France but was only inhabited by 4,000 Spanish subjects.
When Mexico won its independence from Spain in 1821, it claimed Texas as part of its new empire. Since Texas was nearly devoid of people, and decades of efforts to entice people from Spain or Mexico to settle there had failed, Mexico invited Yankees (Anglo-Americans) to buy land and reside in Texas, with the condition that they agree to embrace Mexican citizenship, and convert to Roman Catholicism.
By 1831, there were 30,000 Yankees living in Texas along with 1,000 African slaves. The family of Stephen Austin alone had purchased 15,000 acres of Texas land from the Mexican government. The population of Mexicans in Texas totaled 5,000—they called themselves Tejanos.
The people of Texas were allowed to govern themselves until 1834, when the new government of Mexico revoked their autonomy. Stephen Austin went to Mexico City to plead the case of Texans to direct their own affairs, but instead he was imprisoned for a year. He came home a rebel who favored independence for Texas.
American President Andrew Jackson attempted to buy Texas from Mexico for five million dollars, and might have succeeded had he not sent the arrogant, Mexican-hating Anthony Butler to make the offer, who offended everybody in Mexico City.
Texas declared its independence in 1836, after the President of Mexico, Antonio Lopez de Santa Anna, had become a dictator.
Texas was not alone: eleven Mexican states rebelled against Santa Anna. Texas was one of three Mexican states that declared independence, but the only state that was successful at winning independence from Mexico. Texas was far different from the other Mexican states in that 20,000 Anglo-Americans owned ranches there.
President Santa Anna led an army north to punish Texas. He arrived at San Antonio with perhaps 4,000 soldiers, and surrounded 200 or so Yankee settlers inside a mission called the Alamo. In the fighting, all but one of the Texans was slain. The Mexicans counted 1,544 dead.
Word circulated around Texas that Santa Anna had executed prisoners at the Alamo. It is doubtful there were any prisoners, as the men inside the Alamo fought to the death. Santa Anna did execute 350 Texan prisoners three weeks later, who had surrendered after the Battle of Goliad, and the two events may have been confused. In any event, "Remember the Alamo" became the battle cry of Texas. The United States remained neutral in this conflict, and even refused to loan money to Texas for armaments.
One month later, the Texans defeated the army of Santa Anna in the fifteen minute Battle of San Jacinto, and captured him. In exchange for his freedom, Santa Anna acknowledged Texas as an independent republic, and skulked back to Mexico City in disgrace. Sam Houston was elected the first president of the Republic of Texas.
In the meantime, Anastasio Bustamante was elected president of Mexico (again) and his new government refused to accept the independence of Texas. Texas would remain an independent nation for the next nine years.
In 1850, a census of Texas revealed a population of 212,000 Americans; 58,000 African slaves; and 11,000 people of Mexican descent. 90 percent of the Americans had come to Texas from the southern United States. They put on aristocratic airs, but they were proud and violent men who loved to gamble, drink, duel, and race horses. Many of them were Scots-Irish.
Texans were the loneliest people in America. Few of them had access to schools, churches, or courts. Their education came from the family Bible; their justice from the Colt 45. In legend Texas is western; in reality it is southern.
In 1861, Texas voted to secede from the United States. By 1865, King Cotton had become King Cattle.
STEPHEN AUSTIN
BATTLE OF THE ALAMO
BATTLE OF SAN JACINTO
After his disgrace, General Santa Anna redeemed himself in 1838, when he defeated French troops that had invaded Mexico. He lost his leg to a French cannon during the battle, and he once again became the national hero of Mexico.
Santa Anna was installed as dictator of Mexico for the fifth time, and put down a rebellion led by two Mexican generals. In 1842, he attacked Texas again, which gained him nothing but intense anti-Mexican feelings by Texans, who from then on sought to become part of the United States for protection. In 1844, Santa Anna was deposed, and went into exile in Cuba.
The United States did not think highly of Mexico. In its view, Mexico started out with an equal land mass and 2/3 of the population of the United States, as well as incredible natural resources, when it became a nation. But it was not a success story, as was the United States.
Mexico was a basket case that was full of impoverished people in a stagnant economy, and rife with banditos. The only commodity of Mexico was silver. Mexico did not encourage foreign investment and in fact was known to declare "Death to foreigners!"
Liberal Mexicans wanted war against the Yankees, thinking this would help forge a national identity. In 1844, Mexican President Jose Joaquin Herrera accepted that Texas was lost, and declared Mexico's intention to make peace with Texas as long as it remained an independent country.
In 1845, Mexico severed diplomatic relations with the United States because the people of Texas had made it known they now wanted to join the United States.
The United States annexed Texas in 1845, because Texans begged it to do so. Texas sought to join the United States because Santa Anna—back from exile and presidente yet again—had publicly sworn to drown Texans in their own blood. The terms of the annexation included five million dollars of American money given to Texas, which it needed to defend itself against the feared coming onslaught by Mexico. But the annexation outraged Mexico.
In 1846, upon rumors that Mexico was planning an attack to retake Texas, the United States declared war on Mexico. The Mexican government soon returned the favor.
Indeed the Mexican government was under pressure to stand up to the hated Yankees. 63 American soldiers were ambushed and killed by Mexicans on the north side of the Rio Grande prior to the declaration of war.
Many Americans opposed the war. Those in favor viewed the territories of New Mexico and California as only nominally Mexican possessions with very tenuous ties to Mexico, and as actually unsettled, ungoverned, and unprotected frontier lands.
SANTA ANNA SURRENDERS TO THE BEDRIDDEN SAM HOUSTON
Mexico and the United States approached the Mexican-American War unprepared. 70 percent of the American forces were volunteer militias of raunchy frontier toughs, devoid of uniforms, equipment, and discipline. They wore dirty, torn shirts, had uncombed hair and unwashed faces, they hollered and cursed like fiends. Some of them were not above plunder, rape, and murder.
The Mexican army lacked training, discipline, and munitions. Most Mexican troops were forced into the army, many from Mexican prisons, which made them less than enthusiastic fighters. Mexican artillery pieces were obsolete and faulty. Their cannonballs traveled so slowly that there were reports of American soldiers dodging them as if it were a game of dodge ball.
Mexico had an empty treasury, a corrupt bureaucracy, no navy, a demoralized, poorly equipped, unpaid army, and no arms industry. But the Mexican army was battle-hardened, featured a magnificent cavalry, and was far larger than the American army. Mexico hoped the Tejanos and African slaves in Texas would join their side in the conflict.
The U.S. Army under General Zachary Taylor moved into northern Mexico with 2200 soldiers, won two quick victories, and occupied Matamoros. In the battle for Matamoros the Mexicans had 1500 killed; the Americans 35.
General Taylor then marched on Monterrey, the metropolis of northern Mexico, which was guarded by mountains, a river, and 10,000 troops. Military manuals dictated it would take 20,000 soldiers to take such a fortification, but Taylor had only 6,645. Taylor won anyway, thanks to his skilled artillerymen and engineers.
The Mexicans responded by making Santa Anna their fourth president in two years. He marched north with 20,000 men to vanquish the 4,700 soldiers Taylor had left. Santa Anna was met by the "Mississippi Rifles" of Jefferson Davis, who sounded a banshee howl later called a "Rebel Yell." Santa Anna withdrew from the fight and beat a hasty retreat.
General Winfield Scott invaded Mexico with a 2nd U.S. Army, and the first amphibious craft—shallow-draft gunboats. General Scott safely landed 8,600 men at Veracruz without a single casualty—an incredible military feat.
Winfield Scott was an honorable man who forbade the mistreatment of civilians, and insisted that his men purchase, rather than simply take, provisions. Among his officers were Robert E. Lee and Ulysses S. Grant, as well as Thomas "Stonewall" Jackson, George McClellan, George Pickett, and George Meade.
General Scott quickly captured Veracruz despite its three formidable forts manned by 135 cannon and 3500 men. The Americans lost 73 men in the assault. Scott then marched toward Mexico City, 250 mountainous miles away.
While Scott made his way to Mexico City, Santa Anna offered to surrender for a $10,000 bribe. General Scott paid him but Santa Anna reneged on the surrender part.
Scott descended out of the mountains to Mexico City with 10,000 men. The capital was defended by 30,000 soldiers. Scott overcame heroic resistance, even from desperate civilians.
Mexico City fell in 1847. 10,000 Mexican soldiers were killed in action defending their capital. The United States lost 1,000 men.
Again Santa Anna went into exile. After this victory, the Duke of Wellington, a man not given to hyperbole, called Winfield Scott "the greatest living soldier."
The unsung hero of the Mexican-American War was Quartermaster General Thomas Sidney Jesup. He was in charge of purchasing for the army; managed 23 federal arsenals that produced tens of thousands of weapons, uniforms, boots, and tents; and transported those supplies to distant, primitive locales.
The United States suffered 1,548 men killed in action in the Mexican-American War. In every battle they were outnumbered, but not outgunned. 10,970 American troops succumbed to dysentery, influenza, smallpox, measles, venereal disease, snakebites, and tarantula bites. It was the first war led by graduates of West Point; and the first reported by modern war correspondents.
MEXICAN AMERICAN WAR MAP
BATTLE OF MONTEREY
GENERAL ZACHARY TAYLOR
BATTLE OF CHURUBUSCO (NEAR MEXICO CITY)
GENERAL WINFIELD SCOTT
AMERICANS CONQUER MEXICO CITY
The Peace Treaty
In 1848, a peace treaty was signed by which Mexico agreed to sell the United States its northernmost territories for just over eighteen million dollars ($500,000,000 in today's dollars). This land included the future states of New Mexico, Arizona, Nevada, Colorado, and California.
Spain had utterly failed in its attempt to colonize these lands as it had done successfully in Mexico proper. The largest town north of the present border of Mexico was Santa Fe, with only 6,000 people, followed by San Antonio with 1500. The Spanish were not that interested in this territory because it appeared to lack precious metals, and it had a far larger population of fierce Amerindians than they wanted to handle.
Mexico was in no position to argue. The previous decade had seen it wracked by numerous and massive revolts by its Indian population, especially the Mayans in the Yucatan, who were determined to turn their backs on the "white world," and throw out white sugar farmers that threatened their corn culture.
The Indians of Mexico demanded that land be confiscated by the Mexican government from landowners and farmers to be redistributed to peasants. Mining states were also facing revolts; bandits ravaged estates in central states; peasants pillaged towns and haciendas in the north at will. Mexico was disintegrating from within.
There were plenty of voices in America that wanted to annex all of Mexico—for the sake of the miserable Mexicans. They thought Mexico would benefit from Yankee law, religion, and enterprise. They did not win the day.
MEXICO BEFORE THE MEXICAN AMERICAN WAR
LAND SOLD BY MEXICO TO AMERICA
Republic of California
When Mexico became independent in 1821, California was a remote and nearly uninhabited land. It had no schools or industry; life was lived in chaos and anarchy. The Spanish only occupied California with soldiers, farmers, and missionaries in the late 18th century because they had received reports that the Russians had their eye on it.
Spanish missionaries established 21 bucolic adobe plantations along the El Camino Real (the "royal road") from San Diego to Sonoma, one day's journey apart. Spanish soldiers established presidios at eight of these locations. Spain awarded huge tracts of land to ex-soldiers in California, which had been made into successful cattle ranches.
Disease carried by the Spaniards killed off 75% of the 75,000 Native Americans in California. By 1800, only 2,000 Hispanics lived in what is now the American state of California, which the Spanish called Alta (upper) California (as opposed to Baja [lower] California, which is a state in Mexico today).
Andrew Jackson unsuccessfully tried to buy northern California. The fledgling Republic of Mexico had neither the resources nor the inclination to do anything with this immense piece of land. This was the era that inspired the "Legend of Zorro."
What Mexico did do was pass a colonization act in 1824 that granted 700 Mexicans vast estates of 4,500 to 50,000 acres of California land. These men became known as rancheros. They treated Native Americans like slaves—in fact the death rate for Indians working for rancheros was double that of African slaves in the American South. The rancheros lived the high life, gambling, horse-racing, bull-baiting, riding the range, and dancing.
In 1834, the rancheros convinced the Mexican government to confiscate the lands of the California missions, expel the Franciscan friars, and divide the land up amongst the rancheros.
In 1845, U.S. President James K. Polk offered to buy California and New Mexico for what is today over $800,000,000. Britain was also interested and Polk did not want the British to have them. Many Americans believed it was the Manifest Destiny of America to stretch from the Atlantic to the Pacific Oceans.
Everybody knew Mexico was strapped for cash, and had no real interest in the land itself. But the offer was refused, in large part because the government of Mexico was in total chaos. In 1846 alone, the presidency changed hands four times, the war ministry six times, and the finance ministry sixteen times.
In 1848, the entire population of California was 40,000—10,000 Americans; 10,000 Mexicans; 20,000 Native Americans. An independent California Republic was declared that year. Californios of substance saw no advantage to any connection with Mexico. On the other hand, they greatly admired the United States.
No one foresaw the discovery of gold in 1848 at Sutter's Mill. Sutter had purchased 49,000 acres at the confluence of the Sacramento and American rivers, with a dream of building a little Switzerland. After the discovery of gold, 100,000 Americans rushed out to live in California in just two years. Then, California applied to become a state of the United States, and was accepted in 1850.
CALIFORNIA MISSIONS BUILT BY THE ROMAN CATHOLIC CHURCH
CALIFORNIA RANCHEROS
SAN FRANCISCO CALIFORNIA IN 1851
SANTE FE NEW MEXICO WAS THE LARGEST CITY IN ALL THE TERRITORY SOLD BY MEXICO TO AMERICA (1846)
Aftermath of the Mexican American War
Some Americans look back at these events as dark days in American history. Their voices may have been summed up best by Ulysses S. Grant in his memoirs:
"I was bitterly opposed to the measure, and to this day regard the war, which resulted, as one of the most unjust ever waged by a stronger against a weaker nation. . . . The Southern rebellion was largely the outgrowth of the Mexican war. Nations, like individuals, are punished for their transgressions. We got our punishment in the most sanguinary and expensive war of modern times."
American veterans of the Mexican–American War suffered from debilitating diseases contracted during war, long after the war itself was over. If the post-war deaths of these soldiers are considered, this was proportionately the most deadly in American military history.
Mexico lost half its territory after this war. The lost territories were essentially unsettled and ungoverned. The 30,000 Hispanics who lived there mostly stayed on, though some moved south into Mexico. The tiny populations of these immense areas had become substantially American.
A quarter of the population of America lives in these lands today. Due to massive legal and illegal immigration from Mexico, over a third of these people are of Hispanic origin, and of those 33 percent were born in Mexico. Perhaps the land is being reconquered through immigration. It is infinitely more valuable today than it was in 1848.
This article is a companion piece to my article The History of Mexico. These articles were preceded by a look at Mexico before it became an independent nation: Colonial Mexico. Next time, I am going to write about Modern Mexico, and explore why Mexicans are leaving their homes and towns to pour into the United States.
My sources include: The Penguin History of Latin America by Edwin Williamson; Latin America: A Concise Interpretive History by E. Bradford Burns and Julie A. Charlip; Throes of Democracy by Walter A. McDougall; and America by George Brown Tindall and David E. Shi.
THE UNITED STATES OF AMERICA AFTER THE MEXICAN AMERICAN WAR
AMERICAN PRESIDENT JAMES K POLK
by James Kenny4
The Colonization of California - France, Prussia, Russia, and England
by Ryan Thomas0
Mexico in Modern Times
by James A Watkins70
American Revolution Lesson Plans for 8th Grade American History
by Shannon35
The History of Mexico: Independence to Modern Times
Lost Roanoke Colony is Found: Evidence in Maps, Artifacts and DNA Tracking
What Native Americans Knew About Medicine Long Before There Were Drugstores
by Mike and Dorothy McKenney6
Thirteen Colonies Lesson Plans for 8th Grade American History
by Shannon106
3 months ago from Chicago
Brad Masters ~ You are welcome. Thank you.
Brad NOYFB
You are welcome and an inspiration to the use of facts over fantasy.
I did write an article on the democrat plan to take the White House in 2020. And it started its development in 2008 and it was no coincidence that the democrats dumped Hillary and floated up Obama as the great left hope.
Thank You and Have a great weekend
Brad Masters ~ God Bless You Brother!
Brad Masters ~ You're welcome. What is happening in Central America is that leftists are encouraging those people to go to America, the Land of Milk and Honey. The plan is to overwhelm our borders, get them inside our country, keep them here, and make them voters for the Democrat Party. Ten or twenty million new voters might sway our elections forever and give the Left permanent power.
Thanks, and that should be a sobering thought to realize that Mexico is the problem, Then if we pursue the states they want to take back as Mexico, they would then be knocking the rest of the US states because they can make a living in Mexico. I thought of it more like Texas as you also mentioned. That is why people aren't running across the border to Mexico.
The separation of Mexico from Central America is another conundrum as people from Guatemala and other Central American countries can't really use that take back, but it didn't stop them from demanding to get into the country.
It is going to be quite the political ride under 2021 because of the border and immigration issues that won't be resolved, but will take up the entire presidential campaign, in my opinion. .
What about making it like Puerto Rico? Just kidding, one Puerto Rico is more than we can handle.
Brad Masters ~ You are quite welcome. I appreciate you coming over to read it. Thank you for your comments and kind compliments.
What would have happened if America had annexed all of Mexico? I don't know. I'd like to think Mexico would be like Texas, Arizona, California, or at least similar to New Mexico. But we've got to remember Mexico got messed up by Mexicans.
I can tell you this, the open borders Latinos who say they are taking back the Southwest are so slow they do not realize that had those states remained in Mexico they would be like the rest of Mexico today - not like America - not like they are. They want to take back what we made into fantastic places. It is not as if Mexicans would have made states like Texas and California. If they could have they would have.
Thanks for letting me know about this article. It should be printed in Spanish and given to AOC. The Rancheros was an interesting and unknown to me background in history.
Santa Anna seemed to be like Jason in the horror movies, he just kept coming back.
"Mexico was a basket case that was full of impoverished people in a stagnant economy, and rife with banditos. The only commodity of Mexico was silver. Mexico did not encourage foreign investment and in fact was known to declare "Death to foreigners!""
Today, it seems that little has changed in over 150 years. Today, we have the Drug Cartel, illegal drugs, and illegal immigration from Mexico. The presidents in Mexico come and go, but the Mexicans keep trying to sneak into the US. At the same time, the US keeps giving aide to Mexico, with little given in return.
The Mexican American War should have been a win for the US, but as it victory is still under attack after more than 150 years, that win is questionable. My definition of winning of a war is that it doesn't keep coming back. It may be a military win but politicians seem to give away the farm at the end of a military victory.
That is why WWI became WWII and that became the Cold War and followed by the Korean War, Vietnam War and on. Anyway, based on your article Santa Anna and Mexico won their victories initially by having their large numbers overcome their opponent. But, when the US encountered those same odds, they won.
It is interesting to see the two main generals in the US Civil War working on the same side for the US. Then they faced each other, what a shame, can you imagine how they could have worked together for the US.
As, I have said before you are the historian, and this article was not only well written, it was an easy and interesting read. I as most other people had no idea that there was anything after the Alamo.
Finally, what do you think would have been the result today, if the US had annexed Mexico as proposed?
10 months ago from Chicago
Susie Mann! Thank you for reading my article and leaving such an encouraging note. I really appreciate it.
Susie Mann
Excellent and very informative. I wanted to read more!
4 years ago from Chicago
qq~ I had to shake my head with sadness upon reading your comments, which show a complete lack of historical knowledge and show you to be totally devoid of wisdom or even the awareness of what wisdom might be. No nation in the history of this planet has done 1/10th as much GOOD for humankind as America has. The way you paint it, you might be surprised to learn that over 100 million people have willingly immigrated here and they still come by the millions, some across shark-infested waters on rickety rafts. Your mind is clouded by evil spirits. I will pray for you.
Here is a part of history that u cannot find in schools or see at TV.This is america propaganda at its finest...There are lots of historians that told the truth about this..."a lie told often enough becomes the truth meaning" this is the main weapon of american right now. Basically "merica" stoled 2000.000 km2 from Mexico,an independent and sovereign state that was weaken by the war against Spain.
Lots of historicans only worked with facts that "mericans" cant deny. Most of them knew how big american propaganda is nowdays that's why they only referred to facts skipping stories about crimes,rape and such even they were part of the american behaviour. They did the same with the indians,but they don't talk ... about them since they took their lands and killed most of them. Killing indians was the national sport of America. Watch most of the movies from 50s and u will understand. Indians had more common sense,respect for humans or nature then americans.
In fact,americans are europeans,even if most of 'mericans'talk ... about Europe. The problem is that a big procent of the europeans fled to USA because they had problems with the law in their countries. Maybe that's why USA influence is so harmfull overall.
By the way,you don't start a war because you believe that someone might attack you. =))They did the same with Iraq,history repeats itself. Mass distruction weapons,y right:) Nice joke folks. Its so funny,someone remember WEAPONS OF PS 2 DESTRUCTION before the Iraq war? Lol,its not a funny joke,CNN propaganda said on TV that Hussein might try a chemical attack in US by using...play station consoles:) Rofl:) Lets put this way,Hussein was bad,a terrorist or whatever u might say,i agree,but George Bush is worse...
And btw,back to history again,after USA took Texas,Mexico admitted that they lost Texas BUT THEY REFUSED TO SELL CALIFORNIA so USA started a second war.I repeat,u don't start a war because u believe that some1 might attack you. In this way,Russia can conquer all the little countries around its borders. If u a democracy,ofc USA is a fake one,u WAIT UNTIL THAT country actually start a conflict with u and then u go to war:) Lol. Mexicans had no plans for another war,they had no resources after the war with Spain. And these are facts. Historians had official statements from american politicians like Abraham Lincoln statement,he said something like this: wait until Mexico herself became the aggressor and we all know the rest of the story:) Mericans went to war regardless of the reality or other people misery,because MERICA! Read history,not propaganda,u will find out that USA was build out of people misery, takeing advantages of wars. In ww1, ONLY 50k US troops died DURING THE WAR AND NO CIVILIANS,a small amount of troops put foot in Europe,but somehow,i got few ideas why, american public think that they saved EU in WW1. Facts,numbers,who care? You can listen to CNN now:)
WW2, only 400k US troops,2k civilians,died during the war but MERICA again saved Europe;) 25millions from Russia died in ww2. FACTS! Biggest fights were fought on eastern front,9 out of 10 germans were killed by the russians,BUT...US saved EUROPE...=)) History repeats itself. Now USA fight wars in name of democracy:)) I got a question,why no wars in Sudan or Congo if u guys care about that?:)) Why 1% of the resources for those areas and 90% for Iraq? And btw,what are the results of 10 years of democracy there? And u think that's how u fight vs terrorism? Do u believe that US troops fought for democracy,defended local civilians? Well,i don't even want to talk about that or have such facts in my PC. Stories about USA natzi soldiers are all over the internet:)When they get back to US they talk with an army of psychologists and they still doing dumb things. How many kids lost legs,hands,how many little grils,talking about those under the age of 14 only were raped? Do u heard something about this? Do u heard about the "OILF FOR FOOD PROGRAM? Ofc,u don't see such things at CNN. Youtube,google,yahoo,delete them pretty fast also:)Even Facebook data base is used by NASA:) Why?Easy,u folks are control freaks:) And btw,american public don't know how its to actually have tanks in your city,to live in fear...Then u guys wont be so
enthusiastic about going to wars...Europes knows how its,we learned our leasson.And i speak facts only. The indoctrination and hate in US is huge. Im don't HATE AMERICA,i don't hate anybody,but im here to tell the truth. Facts. This world is obviously is not going in tot he right direction under the influence of USA. Canada influence would be much better but there are only 40mils of canadians:P :((
There is an experiment,involving kids from all over the planet. An easy one. I can elaborate if someone wants to figure out. Pretty much someone tested the level of hate and racism on random kids from all over the world: Iran,Iraq,France,China,USA and so on...and i can elaborate and talk about this for years...The experiment shows that the level of hate,racism,indoctrination,lack of respect was huge @american kids...He started this experiment asking himself why americans kids kill themselves in school if they got no wars in USA and a 58k pib per capita.
You expect such things to happen in other places,not in the great democracy of USA=)))
USA democracy is fake,USA influence over the world is bad,either we talk about politics,fiancial crisis,culture and so on...And this modern life style from US is so harmful.
And i can talk on and on,America history is fake most of the time...
And Europe should keep away USA influence for the love of humanity...but CNN propaganda and media is hiding all this things.Like i said,"a lie told often enough becomes the truth meaning".
If i watch a news channel from Canada i get informations,if i watch a news channel from USA i get a headache.
And i addmit that terrorism,China or Russia influence is bad overall and all that,i addmit everything that u guys might say,BUT,im here to say that USA influence is worse then all these bad things combined.
The real problem is that u can see the bad meaning in a terrorist attack or whatever,but u cant see the bad things from USA influence over the world as clear. Maybe Europe will once again pull something out will have a big impact over the world...This time not with the use of guns,but with a modern life style. Its obiously to anyone with common sense that the american one is wrong and harmful. I believe that wars were part of our development as a species but now aint cool at all.Europe,China have 5000years culture so this is my hope.
Look whos making moneys out of wars and such...If u believe that Microsoft,Google and such,are the biggest coorporates,well think again...
Shell Gas is bigger then all the cars manufacturers combined,but the ones related to guns are much bigger...Most of the countries have huge debts,including USA,WHERE ARE ALL THE MONEYS? hmm,good question:) Maybe we should ask those greedy pricks who started the financial crisis?Also i don't understand why CNN is so concerned over the China poluation so much...
Well,i have some news,no,u wont find such things by watching tv,
USA alone=40% of the global poluation. More then China,Russia,Germany or France combined...
Again,other facts: USA government is responsible for killing more innocent people then all the terrorist groups combined. I only talk about kids and womans that had no connection with the wars whatsoever.So figure that out...
And i don't want to talk more about Wall Street or NASA and all their bullshit...
USA history is fake,USA democracy is fake right now,and most important,USA INFLUENCE IS SO HARMFULL OVERALL. Unfortunately I don't see anyone powerfull enough to say such things because Europe is just a USA colony right now:( . NASA can listen to Merkel or another one of US 'allies" and no1 will dare to question that...USA=control freak. Is Merkel a terrorist to or another Eu lider? What if Poland will do such things?Well,there will be bombs over Warsaw in no time,that's for sure...In the name of democracy ofc.
George Shirey— You are welcome. Thank you for reading my work and leaving your comments.
As I noted in my article: "In 1850, a census of Texas revealed a population of 212,000 Americans; 58,000 African slaves; and 11,000 people of Mexican descent."
As far as President Polk goes . . . let me see . . . two days before Polk took office Mexico broke off diplomatic relations with the U.S. Polk ordered General Zachary Taylor to take up positions along the Rio Grande in case of invasion by the Mexican Army. Mexcican regulars killed eleven Americas May 9th in Texas. Then Polk told Congress: "Mexico has passed the boundary of the United States, has invaded our territory, and shed American blood on American soil." Congress approved Polk's plans for war but there was much opposition among Whigs, 67 of whom voted against the war resolution.
George Shirey
Thanks for this concise piece on the Mexican American Relations in the mid-19th C. Do you have an estimate on how many total Mexicans were living in the area sold prior to the treaty sale? Also, do you know of any good articles pointing to how Polk influenced Congress to approve a war resolution. I have heard that it was through deception and lies.
Thanks, again.
tracykarl99— How nice to hear from you again! :)
I had to take a break from writing my book and get away from it for a while. After the 1st of the new year, I'll stop writing Hubs and finish the book. I have a clear plan now on how to wrap it up. It was too long.
Thanks for your kind compliments. I'm glad you like this Hub.
I was impressed by this hub and your historical knowledge of Mexico as related to California and the war which left the missions in devastation. You are a passionate writer, James! Hope the book publishing project is coming along to fruition. Great to read your hubs:)
magnoliazz— What a pleasure to hear from you again! Thank you for coming by to visit, and for your gracious compliments.
I agree about that wall. There must be people in the government, or behind the scenes, that just don't want that wall built.
I'm going to write one more Hub soon about Mexico. How it got where it is, and why tens of millions left there to come here. Stay tuned! :D
magnoliazz
8 years ago from Wisconsin
A great history lesson, your writing skills are above excellent.
When I was reading this, I thought to myself that history is repeating itself in so many ways. Mexico to this day has more natural resources than the US, but they fail to drag themselves out of 3rd world status.
Right now there is a border war going on, but the mass media never really tells us much about it. How can these borders continue to be so wide open?
We need a Berlin type wall on the border. Where is it?
Polly— There is a lot in history that is sad, to be sure. I am glad you loved this little story, though. Thank you for reading it and leaving your comments. And you are surely welcome. It's always a good day when I hear from you. :-)
Pollyannalana
8 years ago from US
Wow that was a lot of money, especially that far back, but I guess they printed up their own however much they wanted just like today. You know all through school I was an honor roll student and I made high grades in history too but it went in one eye and out the other, I couldn't stand it and all those dead soldiers, but I like catching up on it now, well maybe it started a little when I was sixteen and went to Massachusetts. Paul Revere, shot heard round the world, Old North Church and Witches grave yard and seeing those things they were bound in, head and arms. All history does seen sad though. I loved your story. Thx
lone77star— Thank you for your gracious compliments, kind sir. I have read that about Texas having the right to split into five states. I think it also is the only state that reserved the right to secede from the union?
I appreciate your keen comments. It is good to hear from you on this subject, my Texan friend.
Rod Martin Jr
8 years ago from Cebu, Philippines
James, this is another awesome work. Kept me reading hungrily to the end.
I'm from Texas and I thought I knew the whole story, but you've educated this Texan with a few things I hadn't known. Nicely done!
One tidbit I learned in Texas History lessons in grade school: Texas, when it joined the Union, apparently reserved the right to split into 5 states because of its great size. Today, Texans with big egos might find such an idea more than a little fantastic. Split Texas? Hogwash!
Writeme ASmile— I am glad you enjoyed it. I'll be writing about Mexico of the present in the next week or so. Thank you very much for taking the time to read my article. I do appreciate it.
Writeme ASmile
8 years ago from Writemesmiles@yahoo.com
I enjoyed the history lesson. History was my least favorite subject in school. However, I enjoyed your article, very much. I cannnot wait to read your views and research on why Mexicans are leaving their homes and towns to pour into the United States.
tonymac04— You are welcome, my friend. Yes I suppose history is largely written by the victors. The view of the losers might be summed up this way:
"The border remains a military zone. We remain a hunted people. Now you think you have a destiny to fulfill in the land that historically has been ours for forty thousand years. And we're a new Mestizo nation. And they want us to discuss civil rights. Civil rights. What law made by white men to oppress all of us of color, female and male. This is our homeland. We cannot - we will not- and we must not be made illegal in our own homeland. We are not immigrants that came from another country to another country. We are migrants, free to travel the length and breadth of the Americas because we belong here. We are millions. We just have to survive. We have an aging white America. They are not making babies. They are dying. It's a matter of time. The explosion is in our population."
Jose Angel Gutierrez, Prof. Univ. Texas at Arlington, founder La Raza Unida Party at UC Riverside 1/1995
Robert— Thank you, my brother, for coming to see me. I enjoyed reading your warm words. You are a true friend. Thank you for being you.
Tony McGregor
8 years ago from South Africa
Thanks for an interesting history lesson. Reading it I was reminded of the dictum (I'm not sure who first said it?) that "History is always written by the victors."
Thanks for an interesting and informative read.
Your dedication is humbling and your ability to draw in people even those with different views is extraordinary. Keep up the good work. I never liked History this much.
akirchner— I am glad you liked the pics. I spend a lot of time on them and rarely are they recognized. Thank you for visiting and commenting. :)
Rod Marsden— It is amazing the lineup of officers in this war who became famous in the American Civil War. I appreciate your readership, my Aussie friend. Thank you for your always excellent comments.
drbj— You are quite welcome. I surely enjoy your Hubs very much, too. You are a first class wit. Thank you for visiting and commenting. :D
Audrey Kirchner
8 years ago from Washington
Always wonderful taking a history walk with you, James. Well done and as always the pics are delightfully informative.
Rod Marsden
8 years ago from Wollongong, NSW, Australia
A good summing up. I like the way you touched upon the big movers and shakers of the American Civil War. Yes, Grant did not approve of the war with Mexico but went anyway. He knew Lee from the Mexican war.
quietnessandtrust— Ecce Homo! Thank you brother! That is priceless.
always exploring— "When will they ever learn?" Reminds me of the song "Where have all the flowers gone?"
Thank you for visiting me. I appreciate your comments.
quietnessandtrust— Thank you, my brother! I still need to show the rest of the story. I'll be doing that soon, as I bring the history of Mexico up to present day. There is a reason why tens of millions wanted to get out of there. And we need to understand it.
Hello, hello,— You are welcome! Thank you for coming!
I always enjoy hearing from you. :-)
davidseeger— You are welcome. Thank you. Baja-Oklahoma!? That's funny! :)
Thank you for taking the time to read my work. It is good to hear from you again.
drbj and sherry
8 years ago from south Florida
Thanks again, James, for tremendously augmenting and enlarging my education about this subject. My previous knowledge was limited to inattention in History class and seeing "The Alamo" film - twice.
You performed your accustomed magic with the subject, my friend. :)
quietnessandtrust
http://www.youtube.com/watch?v=qD1WlJesC6Q
deepthought— You are quite welcome. Glad to provide the fix. I hope I am not viewed as a pusher! :-)
Thank you for visiting and commenting.
Cmerritt— De nada, my friend. I have always thought history can be exciting if presented right. I am doing my best. Thank you for the affirmation. :)
aguasilver— You are most welcome, John. I always look forward to hearing your voice. Thank you for checking in.
Stan Fletcher— You are welcome, brother. It is great to hear from you again. You are one witty writer. Thanks for visiting! :D
Ruby Jean Richert
8 years ago from Southern Illinois
Very interesting and well written.War is terrible.Will we never learn?.
Fantastic work here brother!!!
I have often wondered why to this day, Mexico is still in poverty, ignorance and despair.
I have my opinions on the matter as well.
It is strange to me that some here in California and Mexico want to retake California.
I say BRING IT man!!!
Hello, hello,
8 years ago from London, UK
Thank you, James, for a great historical hub. I have learned a lot form it.
b. Malin— You are most welcome. I love history, too. I thank you for visiting my Hub, and for your gracious compliments.
Tamarajo— Thank you for taking the time to read my article. I appreciate your kind comments.
Tom Whitworth— Good for Captain Whitmore!! One thing I can say for sure: it has been a blessing for Texas, California, Nevada, and Arizona to be part of the United States, rather than part of Mexico. Thanks for the visit, my good friend.
Micky Dee— You are welcome, sir. It is a pleasure to see that you've been by to visit and left your words here for me.
davidseeger
8 years ago from Bethany, OK
Good hub, James. Thanks. There is no way that this history can be related with agreement by all readers. There are people who dispute "facts" because of family stories which are irrrefutable. But I think what you have written here is as nearly unbiased history as is possible. Well done. One quible. I've always refered to Texas as Baja-Oklhoma. I am of course unbiased. However I think this is more appropriate. Especially during football season.
onegoodwoman— It is always a pleasure to hear from you. I appreciate your expression of admiration for my "talent." If I have any, it is a gift for which I have done nothing to deserve. Therefore, I am grateful.
ehern33— Thank you for the commendation. I sincerely appreciate you reading my Hub and leaving your kind comments.
8 years ago from In the middle of nowhere and worldwide but still that T.O.kid from da north of America
Well James my friend here I am once again to get my hitory fix once more, and as usual you have enlighten my head with knowledge and history so long forgotten
so thank you
Chris Merritt
8 years ago from Pendleton, Indiana
Thanks.......AGAIN!!....for sharing James...
seriously, I never tire of your history lessons. Subjects I had NO interest about 15 minutes ago, I walk away, so glad I read, and learned. Gracias
8 years ago from Malaga, Spain
Learned a lot form this one James, many thanks.
Stan Fletcher
8 years ago from Nashville, TN
well done! I haven't thought about a lot of this since taking Texas history in Jr. High. Thanks for the accurate portrayal of Santa Anna....
Vladimir Uhri— You are welcome, Brother. Thank you for your kind comments.
J D Murrah— Few people would know this history better than you, my friend. For the sake of postmodern attention spans, I had to present a truncated history here. You filled in important gaps in my presentation with these words:
"Santa Anna intentionally butchered the bodies of the defenders of the Alamo. There is something about bayoneting dead bodies which is distasteful. The religious freedom issue was a major item in Texas Independence as well. The Tejanos objected to the lack of religious freedom and the political oppression. Santa Anna nationalized the churches and wanted one single religion, which riled up a lot of the people."
I was aware of the next paragraph where you wrote about Polk, Santa Anna, and Cuba. I should have included that.
I didn't mean to rile you up, my friend. Thank you very much for helping complete the picture for me.
b. Malin
I have to agree with Tamarajo. Thank you so much for sharing such a wonderful Hub, History Lesson with us. I always loved History...such a learning experience.
Tamarajo
a strange but fascinating piece of history I did not know. Well presented.
Tom Whitworth
8 years ago from Moundsville, WV
My freshman year at WVU I had a liberal history professor who climed America was an Imperialistic Country and cited the Mexican-American war as an example. I told my ROTC instructor Capt. Whitmore and he came to history class and set Professor Bagby straight in quick order!!!!!!!!!!!!!!
Micky Dee
Looks like an unbiased account of the history. There are arguments from all sides. Thanks for the history.
onegoodwoman
8 years ago from A small southern town
I always learn something from you.
And I admire your ability to put
history into a 'conversational'
piece....it is a talent not
given to me.
eovery— Thank you for the high marks, brother! I appreciate you!
ehern33
Great write-up and refresher course for me as I haven't read the history of this area for a very long time. Interesting read and commend you.
Vladimir Uhri
8 years ago from HubPages, FB
Great information my friend. Thanks
J D Murrah
8 years ago from Refugee from Shoreacres, Texas
You took on a large task on this one. I enjoyed it although as a Texas historian, I had some issues with the portrayal of the militia, which consisted of Texas Rangers and a few other items. Some of the things are that Santa Anna intentionally butchered the bodies of the defenders of the Alamo. There is something about bayoneting dead bodies which is distasteful. The religious freedom issue was a major item in Texas Independence as well. The Tejanos objected to the lack of religious freedom and the political oppression. Santa Anna nationalized the churches and wanted one single religion, which riled up a lot of the people.
Another item is that at the start of the Mexican American War, Santa Anna was in Cuba. President Polk made him an offer to come to Mexico and end the war. Santa Anna took the money, returned to Mexico and then pressed the war more vigorously. He was often stunned at how his forces were beat by inferior numbers. he commented "God must be a Yankee" as his forces were beaten repeatedly.
I know doing an overview of so much history is often a challenge. It was a good overview, and I, being a Texan tend to look at the events with a more critical eye than most. I enjoyed it, even if I did get riled up about a few items.
eovery
8 years ago from MIddle of the Boondocks of Iowa
Marked up as always.
Keep on hubbing!
stars439— Thank you for being my first visitor!! I appreciate your gracious laudations, my brother. God Bless You!
stars439
8 years ago from Louisiana, The Magnolia and Pelican State.
Fantastic Hub. Your work is always impecably perfect to the letter. GBY.And Awesome photographs.
|
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JumpStart is a computer network installation tool set used by the Solaris operating system.
Usage
JumpStart is used to manage operating system installation in many Information technology environments (corporate and otherwise) where Solaris operating system computers are widely used. It can provide easier installation (minor setup on central server, then one command on an installation "client" system to start it installing). It also allows completely consistent system installation on many systems over time - each install can have exactly the same system configuration and software tools. Alternatively, different types of systems can be installed for different purposes, in each case with consistent installations for a given defined type. Tools used to manipulate JumpStart include JET, the JumpStart Enterprise Toolkit.
Created by: Thomas Fritz in 1994, at Sun.
Structure
JumpStart consists of two main parts: network booting of a system, and then network installation.
Network booting proceeds similarly to Solaris' standard network booting capabilities. A JumpStart and network booting server is set up on the same local network as the system(s) to be installed. Technically, the network boot and install servers can be separate functions, but they are typically the same system.
Once a client system begins the JumpStart process, it then accesses the operating system component software packages stored on the JumpStart server, usually but not exclusively using Network File System.
Those packages, and optionally additional tools or applications, are automatically installed, and then the system is rebooted. Some additional configuration may be manually performed, or the system's configuration may be set up completely automatically.
See also
Kickstart (Linux)
Fully Automatic Installation
System Installer
References
Solaris 10 Installation Guide: Network-Based Installations
Unix package management-related software
Sun Microsystems software
Booting
Network booting
Provisioning
|
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\section{Basic definitions}
\subsection{Gluing of abelian categories}
Let $({\cal C}_i)$, $i=1,\ldots, n$ be a collection of abelian
categories.
\begin{defi}
A {\it (left) gluing data} for $({\cal C}_i)$ is a
collection of right-exact (covariant) functors
\begin{equation}\label{functors}
\Phi_{i,j}:{\cal C}_j\rightarrow{\cal C}_i
\end{equation}
for all pairs $(i,j)$ such that $\Phi_{i,i}=\operatorname{Id}_{{\cal C}_i}$, and a
collection of morphisms of functors
\begin{equation}\label{morphisms}
\nu_{i,j,k}:\Phi_{i,j}\circ\Phi_{j,k}\rightarrow\Phi_{i,k}
\end{equation}
for all triples $(i,j,k)$
such that $\nu_{i,i,k}=\operatorname{id}$, $\nu_{i,j,j}=\operatorname{id}$ and the
following associativity equation holds:
\begin{equation}\label{asso}
\nu_{i,j,l}\circ(\Phi_{i,j}\nu_{j,k,l})=
\nu_{i,k,l}\circ(\nu_{i,j,k}\Phi_{k,l})
\end{equation}
for all quadruples $(i,j,k,l)$.
\end{defi}
One defines {\it right gluing data} similarly by inverting
arrows and requiring functors to be left-exact.
Henceforth, the term ``gluing data" by default will refer
to the ``left gluing data".
For a gluing data $\Phi=(\Phi_{i,j};\nu_{i,j,k})$ we define
the category ${\cal C}(\Phi)$ as follows. The objects of
${\cal C}(\Phi)$ are collections $(A_i;\a_{ij})$ where $A_i$ is an
object of ${\cal C}_i$ ($i=1,\ldots, n$),
$\a_{ij}:\Phi_{i,j}A_j\rightarrow A_i$ is a morphism in ${\cal C}_i$
(for every pair $(i,j)$) such that the following diagram
is commutative:
\begin{equation}\label{comp}
\setlength{\unitlength}{0.25mm}
\begin{array}{ccccc}
\Phi_{i,j}\Phi_{j,k}A_k &
\setlength{\unitlength}{0.50mm}
\lrar{\Phi_{i,j}\a_{jk}}
& \Phi_{i,j}A_j\\
\ldar{\nu_{i,j,k}} & & \ldar{\a_{ij}} \\
\Phi_{i,k}A_k &
\setlength{\unitlength}{0.50mm}
\lrar{\a_{ik}} & A_i
\end{array}
\end{equation}
for every triple $(i,j,k)$.
A morphism $f:(A_i;\a_{ij})\rightarrow (A'_i;\a'_{ij})$ is a
collection of morphisms $f_i:A_i\rightarrow A'_i$ such that
$f_i\circ\a_{ij}=\a'_{ij}\circ\Phi_{i,j}(f_j)$ for all $(i,j)$.
\begin{lem} The category ${\cal C}(\Phi)$ is abelian.
\end{lem}
\noindent {\it Proof} . For a morphism $f:(A_i;\a_{ij})\rightarrow(A'_i,\a'_{ij})$ in
${\cal C}(\Phi)$ there are natural objects of
${\cal C}(\Phi)$ extending the collections $(\ker(f_i))$ and
$(\operatorname{coker}(f_i))$, which constitute the kernel and the cokernel of
$f$. Indeed, the composition of the natural morphism
$$\Phi_{i,j}\ker(f_j)\rightarrow\Phi_{i,j}A_j\rightarrow A_i$$
with $f_i:A_i\rightarrow A'_i$ is zero; hence, it factors through
a morphism $\Phi_{i,j}\ker(f_j)\rightarrow\ker(f_i)$, and we get
the structure of an object of ${\cal C}(\Phi)$ on $(\ker(f_i))$.
Similarly, the composition of the natural morphism
$$\Phi_{i,j}A'_j\rightarrow A'_i\rightarrow\operatorname{coker}(f_i)$$
with $\Phi_{i,j}f_j:\Phi_{i,j}A_j\rightarrow\Phi_{i,j}A'_j$ is zero;
hence, it factors through a morphism
$\operatorname{coker}(\Phi_{i,j}f_j)\rightarrow\operatorname{coker}(f_i)$. However,
$\operatorname{coker}(\Phi_{i,j}f_j)\simeq\Phi_{i,j}\operatorname{coker}(f_j)$ since
the functor $\Phi_{i,j}$ is right-exact. Thus, we get
a structure of an object of ${\cal C}(\Phi)$ on the collection
$(\operatorname{coker}(f_i))$.
It follows that both the cokernel of kernel and kernel of cokernel of $f$
coincide with the natural object extending the collection
$(\operatorname{im}(f_i))$, and therefore, ${\cal C}(\Phi)$ is abelian.
\qed\vspace{3mm}
\begin{rems}
\noindent 1. Dualizing the above construction we obtain
the glued category for right gluing data.
\noindent 2. The more general ``gluing" procedure is obtained
by considering an abelian category ${\cal C}$ with
a right-exact functor $\Phi:{\cal C}\rightarrow{\cal C}$ together with
morphisms of functors $\operatorname{Id}\rightarrow\Phi$ and $\Phi^2\rightarrow\Phi$
such that $\Phi$ is a monoid
object in the category of functors from ${\cal C}$ to itself.
In our situation ${\cal C}=\oplus_i {\cal C}_i$ and $\Phi$ has
components $\Phi_{i,j}$. Note that if ${\cal C}$ is the category
of vector spaces over a field $k$, then any $k$-algebra $A$
induces a gluing data on ${\cal C}$ in the above generalized sense.
The functor $\Phi$ in this case is tensoring with $A$ and
the glued category is just the category of $A$-modules.
\end{rems}
\subsection{Adjunctions}
Let $\Phi$ be a left gluing data for $({\cal C}_i)$.
The functor $j_k^*:{\cal C}(\Phi)\rightarrow{\cal C}_k:
(A_i;\a_{ij})\mapsto A_k$ has the left adjoint
$j_{k,!}:{\cal C}_k\rightarrow{\cal C}(\Phi)$ such that $j_k^*\circ
j_{k,!}=\operatorname{Id}_{{\cal C}_k}$.
Namely,
$$j_{k,!}(A)=(\Phi_{i,k}(A);\nu_{i,j,k})$$
where
$$\nu_{i,j,k}:\Phi_{i,j}\Phi_{j,k}(A)\rightarrow\Phi_{i,k}(A)$$
is the structural morphism of the gluing data.
The following theorem states that
the existence of such adjoint functors essentially characterizes
the glued category.
\begin{thm}\label{braverman}
Let ${\cal C}_k$, $k=1,\ldots,n$ be a collection of abelian
categories, and ${\cal C}$ an abelian category equipped with exact
functors $j_k^*:{\cal C}\rightarrow{\cal C}_k$ for $k=1,\ldots, n$. Assume that
for every $k$ there exists the left adjoint functor $j_{k,!}:{\cal C}_k\rightarrow{\cal C}$
such that $j_k^*\circ j_{k,!}=\operatorname{Id}_{{\cal C}_k}$. Assume also that
for an object $A\in{\cal C}$ the condition $j_k^*A=0$ for all $k$ implies
that $A=0$. Then ${\cal C}$ is equivalent to ${\cal C}(\Phi)$ where $\Phi$
is the gluing data with $\Phi_{ij}=j_i^*\circ j_{k,!}$.
\end{thm}
The proof is not difficult and we leave it to the reader.
A more general statement of this kind is Theorem 2.6 of \cite{BBP}.
Assume that the functor $\Phi_{i,j}$ has the right adjoint
$\Psi_{j,i}:{\cal C}_i\rightarrow{\cal C}_j$ for every $(i,j)$. Then the
functors $\Psi_{j,i}$ give rise to a right gluing data for
$({\cal C}_i)$. Namely, by adjunction we have a natural morphism of
functors $\operatorname{Id}_{{\cal C}_k}\rightarrow\Psi_{k,j}\circ\Psi_{j,i}\circ\Phi_{i,k}$
induced by $\nu_{i,j,k}$, so we can form the composition
$$\lambda_{k,j,i}:\Psi_{k,i}\rightarrow\Psi_{k,j}\circ\Psi_{j,i}\circ
\Phi_{i,k}\circ\Psi_{k,i}\rightarrow\Psi_{k,j}\circ\Psi_{j,i}$$
where $\Phi_{i,k}\circ\Psi_{k,i}\rightarrow\operatorname{Id}_{{\cal C}_i}$
is the canonical adjunction morphism. One can see that the
associativity condition analogous to (\ref{asso}) holds for
$\lambda_{i,j,k}$.
On the other hand, the morphism of functors
$\nu_{j,i,j}:\Phi_{j,i}\circ\Phi_{i,j}\rightarrow\operatorname{Id}_{{\cal C}_j}$
by adjunction gives rise to a morphism
\begin{equation}\label{mu}
\mu_{i,j}:\Phi_{i,j}\rightarrow\Psi_{i,j}
\end{equation}
for every $(i,j)$.
By construction the glued categories ${\cal C}(\Phi)$ and
${\cal C}(\Psi)$ are canonically equivalent.
In particular, by duality we have right adjoint functors
$j_{k,*}:{\cal C}_k\rightarrow{\cal C}(\Phi)$ to the restriction functor
$j_k^*:{\cal C}(\Phi)\rightarrow{\cal C}_k$:
$$j_{k,*}(A_k)=(\Psi_{i,k}(A_k);\a'_{ij})$$
where the morphisms
$\a'_{ij}:\Phi_{i,j}\Psi_{j,k}A_l\rightarrow\Psi_{i,k}A_k$ are deduced by
adjunction from the right gluing data $(\Psi_{i,j},\lambda_{i,j,k})$.
One has $j_k^*\circ j_{k,*}=\operatorname{Id}_{{\cal C}_k}$.
\subsection{Middle extensions and simple objects}
The morphism (\ref{mu}) gives rise to a morphism of
functors $\mu_k:j_{k,!}\rightarrow j_{k,*}$, such that for every object
$A\in{\cal C}(\Phi)$ the composition of the adjunction morphisms
$$j_{k,!}j^*_k A\rightarrow A\rightarrow j_{k,*}j^*_k A$$
coincides with $\mu_l(j^*_k A)$. Thus, we can define
the middle extension functor
$j_{k,!*}:{\cal C}_k\rightarrow{\cal C}(\Phi)$
$$j_{k,!*}(A_k)=\operatorname{im}(j_{k,!}(A)\rightarrow j_{k,*}(A)).$$
\begin{lem}\label{ressim}
With the above assumption
for every simple object $A\in{\cal C}(\Phi)$ and any
$l$ the restriction $j^*_lA\in{\cal C}_l$ is either simple or zero.
\end{lem}
\noindent {\it Proof} . Assume that $j^*_lA\neq 0$ and that there exist an exact sequence
$$0\rightarrow B_l\rightarrow j^*_lA\rightarrow C_l\rightarrow 0$$
with non-zero $B_l$ and $C_l$. Then by adjunction we have
non-zero morphisms $f:j_{l,!}(B_l)\rightarrow A$ and $g:A\rightarrow j_{l,*}(C_l)$ such that
$g\circ f=0$. Since $A$ is simple, $f$ should be surjective,
hence, $g=0$ --- a contradiction. Therefore, $j^*_lA$ is simple.
\qed\vspace{3mm}
\begin{lem}\label{gormac}
For every simple object $A_l\in{\cal C}_l$ there is a unique
(up to an isomorphism)
simple object $A\in{\cal C}(\Phi)$ such that $j_l^*A\simeq A_l$.
Namely, $A=j_{l,!*}A_l$.
\end{lem}
\noindent {\it Proof} . The uniqueness is clear: if $A\in{\cal C}(\Phi)$ is simple
and $j^*_lA\neq 0$,
then the adjunction morphisms $j_{l,!}j^*_lA\rightarrow A$ and $A\rightarrow
j_{l,*}j^*_lA$ are surjective and injective, respectively, and hence
$$A\simeq\operatorname{im}(\mu_l(j^*_lA):j_{l,!}j^*_lA\rightarrow j_{l,*}j^*_lA)
=j_{l,!*}(j_l^*A).$$
It remains to check that if $A_l\in{\cal C}_l$ is simple, then
$j_{l,!*}(A_l)$ is simple. Let
$B\subset j_{l,!*}(A_l)$ be a simple subobject. Since
we have an inclusion $B\subset j_{l,*} A_l$ it follows from
adjunction that $j^*_lB$ is a non-zero subobject of $A_l$.
Therefore, $j^*_lB=A_l$ and $B=j_{l,!*}(j^*_lB)=j_{l,!*}(A_l)$.
\qed\vspace{3mm}
\subsection{Gluing data for triangulated categories}
We refer to \cite{BBD} for definitions concerning $t$-structures
on triangulated categories.
\begin{defi} Let $({\cal D}_i)$ be a collection of triangulated
categories with $t$-structures.
A {\it gluing data} for $({\cal D}_i)$ is a
collection of exact functors
\begin{equation}\label{functors2}
{\cal D}\Phi_{i,j}:{\cal D}_j\rightarrow{\cal D}_i
\end{equation}
for all pairs $(i,j)$ that are $t$-exact from the right
(with respect to the given $t$-structures),
such that ${\cal D}\Phi_{i,i}=\operatorname{Id}_{{\cal D}_i}$, and a
collection of morphisms of functors
\begin{equation}\label{morphisms2}
{\cal D}\nu_{i,j,k}:{\cal D}\Phi_{i,j}\circ {\cal D}\Phi_{j,k}\rightarrow{\cal D}\Phi_{i,k}
\end{equation}
for all triples $(i,j,k)$
such that ${\cal D}\nu_{i,i,k}=\operatorname{id}$, ${\cal D}\nu_{i,j,j}=\operatorname{id}$ and the
analogue of the associativity equation (\ref{asso}) holds.
\end{defi}
Let ${\cal C}_i$ be the heart of the $t$-structure on ${\cal D}_i$.
It is easy to see that the functors
$$H^0{\cal D}\Phi_{i,j}|_{{\cal C}_i}=\tau_{\ge 0}{\cal D}\Phi_{i,j}|_{{\cal C}_j}$$
(where $\tau_{\ge 0}$ is the truncation with respect to the $t$-structure)
extend to a gluing data for ${\cal C}_i$. Indeed, since
${\cal D}\Phi_{i,j}$ commutes with $\tau_{\ge 0}$ we have natural morphisms
$$\tau_{\ge 0}\circ{\cal D}\Phi_{i,j}\circ
\tau_{\ge 0}\circ{\cal D}_{\Phi_{j,k}}|_{{\cal C}_k}\simeq
\tau_{\ge 0}\circ{\cal D}\Phi_{i,j}\circ{\cal D}_{\Phi_{j,k}}|_{{\cal C}_k}
\lrar{\tau_{\ge 0}{\cal D}\nu_{i,j,k}}
\tau_{\ge 0}\circ{\cal D}\Phi_{i,k}|_{{\cal C}_k}.
$$
\section{Grothendieck groups}\label{groth}
\subsection{Formulation of the theorem}
Let ${\cal D}\Phi=({\cal D}\Phi_{i,j},{\cal D}\nu_{i,j,k})$
be the gluing data for the derived categories
$({\cal D}^b({\cal C}_i))$ of the abelian categories
$({\cal C}_i)_{1\le i\le n}$, where ${\cal D}^b({\cal C}_i)$ are
equipped with standard $t$-structures.
Let $(\Phi_{i,j}=H^0{\cal D}\Phi_{i,j}|_{{\cal C}_i},\nu_{i,j,k})$ be
the induced gluing data for $({\cal C}_i)$.
Then for every $(i,j)$
there is an induced homomorphism of Grothendieck groups
$$\phi_{ij}=K_0(\Phi_{i,j}):K_0({\cal C}_j)\simeq K_0({\cal D}_j)\rightarrow
K_0({\cal D}_i)\simeq K_0({\cal C}_i).$$
Let us denote by ${\cal C}_{i,j}$ the full subcategory in ${\cal C}_i$
consisting of objects $A$ such that the morphism
$\nu_{i,j,i,A}:\Phi_{i,j}\Phi_{j,i}(A)\rightarrow A$ is zero.
\begin{lem}\label{subc}
${\cal C}_{i,j}$ is an abelian subcategory in ${\cal C}_i$
closed under passing to quotients and subobjects.
\end{lem}
\noindent {\it Proof} . It is clear that ${\cal C}_{i,j}$ is closed under passing
to subobjects. The statement about quotients follows
from the fact that the functor
$\Phi_{i,j}\Phi_{j,i}$ is right-exact.
\qed\vspace{3mm}
Let us denote by $K_{i,j}\subset K_0({\cal C}_i)$ the image of the
natural homomorphism $K_0({\cal C}_{i,j})\rightarrow K_0({\cal C}_i)$.
\begin{thm}\label{K0}
Let ${\cal D}\Phi$ be a gluing data for $({\cal D}^b({\cal C}_i))$, and
$\Phi$ the corresponding gluing data for $({\cal C}_i)$.
Assume that all categories ${\cal C}_i$ are artinian and
noetherian. Then the image of the natural map
$K_0({\cal C}(\Phi))\rightarrow\oplus_{i=1}^n K_0({\cal C}_i):
[(A_i,\a_{ij})]\mapsto ([A_i])$
coincides with the subgroup
$$K(\Phi):=\{ (c_i)\in\oplus_i K_0({\cal C}_i) \ | \
\phi_{i,j}c_j-c_i\in K_{i,j} \}.$$
\end{thm}
We need several lemmas for the proof.
\subsection{Homological lemmas}
\begin{lem}\label{inters}
Assume that the category ${\cal C}_i$ is artinian
and noetherian. Then for every collection of indices
$j_1$,..., $j_k$ the image of the natural map
\begin{equation}\label{inthom}
K_0({\cal C}_{i,j_1}\cap\ldots\cap{\cal C}_{i,j_k})\rightarrow K_0({\cal C}_i)
\end{equation}
coincides with $K_{i,j_1}\cap\ldots\cap K_{i,j_k}$.
\end{lem}
\noindent {\it Proof} . Since ${\cal C}_i$ is artinian and noetherian,
by the Jordan---H\"older theorem $K_0({\cal C}_i)$
is a free abelian group with the natural basis corresponding
to isomorphism classes of simple objects in ${\cal C}_i$.
Now the categories ${\cal C}_{i,j}$ are closed under passing to sub- and
quotient-objects by Lemma \ref{subc}.
Hence $K_{i,j}$ is spanned by the classes of simple objects
that belong to ${\cal C}_{i,j}$, while
the image of the homomorphism
(\ref{inthom}) is spanned by the classes of simple objects
lying in ${\cal C}_{i,j_1}\cap\ldots\cap{\cal C}_{i,j_k}$ and the
assertion follows.
\qed\vspace{3mm}
\begin{lem} Let ${\cal D}$ be a triangulated category with a
$t$-structure such that ${\cal D}=\cup_{n} {\cal D}^{\le n}$,
$F:{\cal D}\rightarrow{\cal D}$ is an exact functor which is
$t$-exact from the right, i.e., $F({\cal D}^{\le0})\subset{\cal D}^{\le0}$.
Let $\nu:F\rightarrow\operatorname{Id}$ be a morphism of exact functors. Then for any
object $X\in{\cal D}$ such that the morphism $\nu_X:FX\rightarrow X$ is zero the
natural morphism $\tau_{\ge 0}F(H^n(X))\rightarrow H^n(X)$ is zero
for any $n$, where $H^n(X)=\tau_{\le n}\tau_{\ge n}(X)[n]$,
$\tau_{\cdot}$ are the truncation functors associated with
the $t$-structure.
\end{lem}
\noindent {\it Proof} . We may assume that $X\in{\cal D}^{\le k}$ for some $k$.
Consider the following morphism of exact triangles:
\begin{equation}
\begin{array}{cccccc}
F(\tau_{\le k-1}X) &\lrar{} & FX &\lrar{} & F(\tau_{\ge k}X)
&\lrar{}\ldots\\
\ldar{\nu} & & \ldar{0} & & \ldar{\nu} \\
\tau_{\le k-1}X &\lrar{} & X &\lrar{} & \tau_{\ge k}X
&\lrar{}\ldots
\end{array}
\end{equation}
Since $F$ is $t$-exact from the right we have
$F(\tau_{\le k-1} X)\in{\cal D}^{\le k-1}$, hence
$\operatorname{Hom}^{-1}(F(\tau_{\le k-1}X),\tau_{\ge k}X)=0$.
This implies that all vertical arrows in the above diagram
are zero, hence the conclusion holds for $n=k$ and the
assumption holds for $\tau_{\le k-1}X$, so we may proceed by
induction.
\qed\vspace{3mm}
\begin{lem}\label{higherder}
For any object $A_j$ of ${\cal C}_j$ we have
$[H^n{\cal D}\Phi_{i,j}A_j]\in K_{i,j}$ for $n\le -1$.
In paricular,
$[\Phi_{i,j}A_j]-\phi_{i,j}[A_j]\in K_{i,j}$.
\end{lem}
\noindent {\it Proof} . Let us denote $A_i=\Phi_{i,j}A_j\in{\cal C}_i$,
so that we have the morphism $\b:{\cal D}\Phi_{j,i}A_i\rightarrow A_j$
induced by ${\cal D}\nu_{j,i,j}$.
Consider the following morphism of exact triangles:
\begin{equation}
\setlength{\unitlength}{0.15mm}
\begin{array}{cccccc}
{\cal D}\Phi_{i,j}{\cal D}\Phi_{j,i}(\tau_{\le -1}{\cal D}\Phi_{i,j}A_j)
&\lrar{} &
{\cal D}\Phi_{i,j}{\cal D}\Phi_{j,i}({\cal D}\Phi_{i,j}A_j) &\lrar{\pi} &
{\cal D}\Phi_{i,j}{\cal D}\Phi_{j,i}(A_i) &
\setlength{\unitlength}{0.12mm}
\lrar{}\ldots\\
\ldar{\gamma'} & & \ldar{\gamma} & & \ldar{} \\
\tau_{\le -1}{\cal D}\Phi_{i,j}A_j &\lrar{} & {\cal D}\Phi_{i,j}A_j &\lrar{} &
A_i &
\setlength{\unitlength}{0.12mm}
\lrar{}\ldots
\end{array}
\end{equation}
where the vertical arrows are induced by ${\cal D}\nu_{i,j,i}$.
The associativity equation for ${\cal D}\nu$ implies that
$\gamma=\delta\circ\pi$ where $\delta$ is the morphism
$$\delta={\cal D}\Phi_{i,j}\b:{\cal D}\Phi_{i,j}{\cal D}\Phi_{j,i}A_i\rightarrow
{\cal D}\Phi_{i,j}A_j.$$
It follows that the composition
$${\cal D}\Phi_{i,j}{\cal D}\Phi_{j,i}\tau_{\le -1}{\cal D}\Phi_{i,j}A_j
\stackrel{\gamma'}{\rightarrow}\tau_{\le -1}{\cal D}\Phi_{i,j}A_j\rightarrow
{\cal D}\Phi_{i,j}A_j$$
is zero. Since ${\cal D}\Phi_{i,j}{\cal D}\Phi_{j,i}\tau_{\le
-1}{\cal D}\Phi_{i,j}A_j\in{\cal D}^{\le -1}$ we get that $\gamma'=0$.
By the previous Lemma this implies that
$[H^n{\cal D}\Phi_{i,j}A_j]\in K_{i,j}$ for $n\le -1$ as required.
\qed\vspace{3mm}
\begin{lem}\label{devis}
Let ${\cal C}^n(\Phi)$ be the full
subcategory of ${\cal C}(\Phi)$ consisting of objects
$(A_i;\a_{ij})$ with $A_n=0$. Then ${\cal C}^n(\Phi)$
is equivalent to ${\cal C}(\Phi')$ for some new gluing data $\Phi'$ for $n-1$
categories ${\cal C}_{1,n}$,..., ${\cal C}_{n-1,n}$.
\end{lem}
\noindent {\it Proof} . If $(A_i;\a_{ij})$ is an object of ${\cal C}^n(\Phi)$,
then the condition (\ref{comp}) implies that
the composition
$$\Phi_{i,n}\Phi_{n,j}A_j\stackrel{\nu_{i,n,j}}{\rightarrow}
\Phi_{i,j}A_j\stackrel{\a_{ij}}{\rightarrow} A_i$$
is zero for every pair $(i,j)$, in particular,
$A_i\in{\cal C}_{i,n}$ for every $i$. For every pair $(i,j)$
such that $i,j\le n-1$ and $i\neq j$,
let us denote by $\Phi'_{i,j}$ the cokernel of the morphism
of functors $\nu_{i,n,j}:\Phi_{i,n}\Phi_{n,j}\rightarrow\Phi_{i,j}$.
Then we have a canonical morphism of functors $\Phi_{i,j}\rightarrow
\Phi'_{i,j}$ and
the above condition means that $\a_{ij}$ factors through a
morphism $\a'_{ij}:\Phi'_{i,j}A_j\rightarrow A_i$. It is easy to
see that the functor $\Phi'_{i,j}$ is right-exact (as the
cokernel of right-exact functors). Let us check that
$\Phi'_{i,j}$ sends ${\cal C}_j$ to ${\cal C}_{i,n}$.
Indeed, for any $A_j\in{\cal C}_j$ the morphism
\begin{equation}\label{phi'}
\nu_{i,n,i}:\Phi_{i,n}\Phi_{n,i}\Phi_{i,j}A_j\rightarrow\Phi_{i,j}A_j
\end{equation}
factors through a morphism
$\Phi_{i,n}\Phi_{n,i}\Phi_{i,j}A_j\rightarrow\Phi_{i,n}\Phi_{n,j}A_j$.
Hence, the composition of (\ref{phi'}) with the projection
$\Phi_{i,j}A_j\rightarrow\Phi'_{i,j}A_j$ is zero which implies that
$\Phi'_{i,j}A_j\in{\cal C}_{i,n}$.
Furthermore, there are unique
morphisms of functors $\Phi'_{i,j}\Phi'_{j,k}\rightarrow\Phi'_{i,k}$
compatible with $\nu_{i,j,k}$, so that we get a gluing data
$\Phi'$ for ${\cal C}_{1,n}$,...,${\cal C}_{n-1,n}$ such that
$(A_i,i=1,\ldots,n-1;\a'_{ij})$ is an object of ${\cal C}(\Phi')$.
Clearly, this gives an equivalence of the category ${\cal C}^n(\Phi)$
with ${\cal C}(\Phi')$.
\qed\vspace{3mm}
\subsection{Proof of theorem \ref{K0}}
Let us check first that the image of $K_0({\cal C}(\Phi))$ is
contained in $K(\Phi)$. It suffices to check that for every $i$
and every $A\in{\cal C}(\Phi)$ one has $\phi_{i,n}[A_n]-[A_i]\in K_{i,n}$.
We note that for every $A\in{\cal C}(\Phi)$
the kernel and the cokernel
of the natural morphism $j_{n,!}j^*_nA\rightarrow A$ belong
to ${\cal C}^n(\Phi)$. Hence, $K_0({\cal C}(\Phi))$ is generated by
the image of the map $K_0({\cal C}^n(\Phi))\rightarrow K_0({\cal C}(\Phi))$ and
by the classes of $j_{n,!}(A_n)$, $A_n\in{\cal C}_n$.
Let us check the above condition for these two classes of elements
separately. If
$(A_i;\a_{ij})\in{\cal C}^n(\Phi)$, then by definition
$A_i\in{\cal C}_{i,n}$ for any $i$, while $A_n=0$; hence
$\phi_{i,n}[A_n]-[A_i]=-[A_i]\in K_{i,n}$ for all $i$. On
the other hand, if $(A_i;\a_{ij})=j_{n,!}(A_n)$, then
$A_i=\Phi_{i,n}A_n$ and
$$\phi_{i,n}[A_n]-[A_i]=\phi_{i,n}[A_n]-[\Phi_{i,n}A_n]\in
K_{i,n}$$
by Lemma \ref{higherder}. Thus, the condition
$\phi_{i,n}[A_n]-[A_i]\in K_{i,n}$ is satisfied for all
objects of ${\cal C}(\Phi)$.
It remains to check that the map $K_0({\cal C}(\Phi))\rightarrow K(\Phi)$
is surjective.
Let us denote by ${\cal C}^l(\Phi)$ the full subcategory of
${\cal C}(\Phi)$ consisting of all objects $(A_i;\a_{ij})$ with
$A_i=0$ for $i\ge l$. Similarly, let
$K^l(\Phi)$ be the subgroup of elements $(c_i)$
in $K(\Phi)$ with $c_i=0$ for $i\ge l$. Note that the image
of $K_0({\cal C}^l(\Phi))$ is contained in $K^l(\Phi)$.
Let $p_j$ be the projection of $\oplus_i K_0({\cal C}_i)$ on its $j$-th
factor. Then $p_{l-1}(K^l(\Phi))\subset \cap_{j\ge l}K_{l-1,j}$
since for $(c_i)\in K^l(\Phi)$ and $j\ge l$ we have
$c_{l-1}=c_{l-1}-\phi_{l-1,j}c_j\in K_{l-1,j}$. Thus,
we have an exact sequence
$$0\rightarrow K^{l-1}(\Phi)\rightarrow K^l(\Phi)\rightarrow \cap_{j\ge l} K_{l-1,j}.$$
Hence, to
prove the surjectivity of the map $K_0({\cal C}(\Phi))\rightarrow K(\Phi)$
it is sufficient to check that the map
\begin{equation}\label{proj}
K_0({\cal C}^l(\Phi))\rightarrow \cap_{j\ge l} K_{l-1,j}:
[(A_i;\a_{ij})]\mapsto [A_{l-1}]
\end{equation}
is surjective for each $l$.
Note that for every
gluing data $\Phi$ (not necessarily extending to the derived
categories) the natural homomorphism $K_0({\cal C}(\Phi))\rightarrow
K_0({\cal C}_n)$ is surjective ($[j_{n,!}(A_n)]$ maps to $[A_n]$).
Now the iterated application of Lemma \ref{devis} gives an
equivalence of ${\cal C}^l(\Phi)$ with ${\cal C}(\Phi')$ for some
gluing data $\Phi'$ on $l-1$ categories
$\cap_{j\ge l}{\cal C}_{1,j}$,..., $\cap_{j\ge l}{\cal C}_{l-1,j}$,
such that the map (\ref{proj}) is identified with the
homomorphism
$$K_0({\cal C}(\Phi'))\rightarrow K_0(\cap_{j\ge l}{\cal C}_{l-1,j})\rightarrow
\cap_{j\ge l} K_{l-1,j}$$
which is surjective by Lemma \ref{inters}.
\qed\vspace{3mm}
\subsection{Gluing of finite type}
\begin{prop}\label{inj}
Assume that every functor $\Phi_{i,j}$ has
the right adjoint $\Psi_{j,i}$ and that the categories ${\cal C}_i$ are
artinian and noetherian. Then the natural homomorphism
$K_0({\cal C}(\Phi))\rightarrow\oplus_i K_0({\cal C}_i)$ is
injective.
\end{prop}
\noindent {\it Proof} . Since the category ${\cal C}(\Phi)$ is artinian and
noetherian, it follows that $K_0({\cal C}(\Phi))$ is the free
abelian group with the basis $[A]$, where $A$ runs through
the isomorphism classes of simple objects in ${\cal C}(\Phi)$.
Now the assertion follows immediately from Lemmas
\ref{ressim} and \ref{gormac}.
\qed\vspace{3mm}
\begin{defi} We say that $\Phi$ is a gluing data of {\it finite
type} if all the categories ${\cal C}_i$ are
artinian and noetherian, and
every functor $\Phi_{ij}$ has the right adjoint
$\Psi_{ji}$.
\end{defi}
Combining Proposition \ref{inj} with Theorem \ref{K0} we
obtain the following result.
\begin{thm}\label{main}
Let ${\cal D}\Phi$ be a gluing data for $({\cal D}^b({\cal C}_i))$,
and $\Phi$ the corresponding gluing data for $({\cal C}_i)$.
Assume that $\Phi$ is of finite type.
Then the natural map
$K_0({\cal C}(\Phi))\rightarrow\oplus_{i=1}^n K_0({\cal C}_i):
[(A_i,\a_{ij})]\mapsto ([A_i])$
induces an isomorphism $K_0({\cal C}(\Phi))\simeq K(\Phi)$.
\end{thm}
\section{Gluing for Coxeter groups}
\subsection{$W$-gluing}
Let $(W,S)$ be a finite Coxeter group,
and $\ell:W\rightarrow{\Bbb Z}_{\ge 0}$ be the length function.
Let $({\cal C}_w; w\in W)$ be a collection of abelian categories.
\begin{defi} A $W$-{\it gluing data} is a gluing data
$\Phi_w:{\cal C}_{w'}\rightarrow{\cal C}_{ww'}$, $\nu_{w,w'}:\Phi_w\circ\Phi_{w'}\rightarrow
\Phi_{ww'}$ for $({\cal C}_w)$ such that
$\nu_{w,w'}$ is an isomorphism for every pair
$w,w'\in W$ such that $\ell(ww')=\ell(w)+\ell(w')$.
\end{defi}
Note that the condition of finiteness of $W$ is imposed only
because we usually consider the gluing of a finite number
of categories. One can similarly treat the case of infinite number of
categories and define $W$-gluing data in this context.
\subsection{Quasi-actions of Coxeter groups}
\label{quasiactcox}
\begin{defi} A {\it quasi-action} of a monoid $M$ on a
category ${\cal C}$ is a collection of functors
$T(f)$, $f\in M$, from ${\cal C}$ to itself and of morphisms of functors
$c_{f,g}:T(f)\circ T(g)\rightarrow T(fg)$, $f,g\in M$ satisfying the
associativity condition.
\end{defi}
An {\it action} of a monoid is a quasi-action such that all
morphisms $c_{f,g}$ are isomorphisms.
Let $(W,S)$ be a Coxeter system. The above definition of
$W$-gluing data can be reformulated as follows: there is
a quasi-action of $W$ on $\oplus_{w\in W}{\cal C}_w$ given by
functors $\Phi_w:{\cal C}_{w'}\rightarrow{\cal C}_{ww'}$ such that
$\Phi_1=\operatorname{Id}$ and $c_{w,w'}$ are isomorphisms when
$\ell(ww')=\ell(w)+\ell(w')$.
Let us denote by $B$ the generalized
braid group corresponding to $(W,S)$ and by
$B^+\subset B$ the positive braids submonoid (see \cite{D}).
Let us denote by $b\mapsto\overline{b}$ the natural homomorphism
$B\rightarrow W$ and by $\tau:W\rightarrow B^+$ its canonical section
such that $\tau$ is the identity on $S$ and
$\tau(ww')=\tau(w)\tau(w')$ whenever
$\ell(ww')=\ell(w)+\ell(w')$.
According to the main theorem of \cite{D}, to give an action
of $B^+$ on a category ${\cal C}$, it is sufficient to have functors
$T(w)$ corresponding to elements $\tau(w)\in B^+$
and isomorphisms $c_{w,w'}:T(w)\circ T(w')\widetilde{\rightarrow} T(ww')$
for pairs $w,w'\in W$ such that $\ell(ww')=\ell(w)+\ell(w')$,
satisfying the associativity condition for
triples $w,w',w''\in W$, such that
$\ell(ww'w'')=\ell(w)+\ell(w')+\ell(w'')$.
It follows that a $W$-gluing data induces an action of $B^+$ on
$\oplus_{w}{\cal C}_w$. Conversely, assume that we are given the
functors ${\cal C}_{w}$ and isomorphisms $c_{w,w'}$ only for
$(w,w')$ with $\ell(ww')=\ell(w)+\ell(w')$ so that we have an action of
$B^+$ on $\oplus_w{\cal C}_w$. To get a quasi-action of $W$ we have
to give in addition some morphisms
$c_{s,s}:\Phi_s\Phi_s\rightarrow\operatorname{Id}$.
The natural question arises as to what compatibility conditions
relating $c_{s,s}$ and the action of $B^+$ one should impose
to obtain a quasi-action of $W$. The answer is given by
the following theorem.
\begin{thm}\label{quasiact}
Assume that we have an action of $B^+$ on a
category ${\cal C}$ given by the collection of functors $T(w)$,
$w\in W$ (such that $T(1)=\operatorname{Id}$),
and isomorphisms $c_{w,w'}:T(w)T(w')\widetilde{\rightarrow}T(ww')$
for $w,w'\in W$ such that $\ell(ww')=\ell(w)+\ell(w')$. Let
$c_{s,s}:T(s)T(s)\rightarrow\operatorname{Id}$, $s\in S$, be a collection of morphisms
satisfying the following two conditions:
\begin{enumerate}
\item
For every $s\in S$ the following associativity equation holds:
\begin{equation}\label{asso_simple}
T(s) c_{s,s,X}=c_{s,s, T(s)X}:T(s)^3X\rightarrow T(s)X.
\end{equation}
\item
For every $w\in W$ and $s,s'\in S$ such that $sw=ws'$ and
$\ell(sw)=\ell(w)+1$ the following diagram is
commutative:
\begin{equation}\label{sizig}
\begin{array}{ccccc}
T(s)T(w)T(s') &\lrar{} & T(w)T(s')^2\\
\ldar{} & & \ldar{c_{s',s'}}\\
T(s)^2T(w) &\lrar{c_{s,s}} & T(w)
\end{array}
\end{equation}
where the unmarked arrows are induced by $c_{s,w}$ and
$c_{w,s'}$.
\end{enumerate}
Then there is a canonical quasi-action of $W$ on ${\cal C}$.
\end{thm}
\noindent {\it Proof} . Following \cite{B} we denote by $P_s$ the
set of $w\in W$ such that $\ell(sw)=\ell(w)+1$.
Let us first construct the canonical morphisms
$c_{s,w}:T(s)T(w)\rightarrow T(sw)$
for every $s\in S$, $w\in W$.
When $w\in P_s$ they
are given by the structure of $B^+$-action. Otherwise
$sw\in P_s$ and we have the morphism
$c_{s,w}:T(s)T(w)\widetilde{\rightarrow} T(s)T(s)T(sw)\rightarrow T(sw)$
induced by $c_{s,sw}^{-1}$ and
$c_{s,s}$. It is easy to check using (\ref{asso_simple})
that the following triangle
is commutative for every $s\in S$, $w\in W$:
\begin{equation}\label{asso0}
\begin{array}{ccc}
T(s)^2T(w) & &\\
\ldar{c_{s,w}} & \ldrar{c_{s,s}} &\\
T(s)T(sw) &\lrar{c_{s,ws}}& T_{w}
\end{array}
\end{equation}
Similarly, one defines morphisms
$c_{w,s}:T(w)T(s)\rightarrow T(ws)$ for every $s\in S$, $w\in W$.
We claim that these morphisms satisfy the
following associativity condition: for any $w\in W$,
$s,s'\in S$ the diagram
\begin{equation}\label{asso1}
\begin{array}{ccccc}
T(s)T(w)T(s')&\lrar{c_{w,s'}}& T(s)T(ws')\\
\ldar{c_{s,w}}& &\ldar{c_{s,ws'}}\\
T(sw)T(s')&\lrar{c_{sw,s'}}& T(sws')
\end{array}
\end{equation}
is commutative. Assume at first that $w\in P_s$.
Consider two cases.
\begin{enumerate}
\item $ws'\in P_s$.
If $\ell(ws')=\ell(w)+1$ then $\ell(sws')=\ell(w)+2$ and the required
associativity holds by definition of the $B^+$-action.
Otherwise, $\ell(w)=\ell(ws')+1$, and hence
$\ell(sws')=\ell(ws')+1=\ell(w)=\ell(sw)-1$, $T(w)\simeq T(ws')T(s')$,
$T(sw)\simeq T(sws')T(s')$, and we are reduced to the
commutativity of the diagram
\begin{equation}\label{asso1_1}
\begin{array}{ccccc}
T(s)T(ws')T(s')^2&\lrar{c_{s',s'}}& T(s)T(ws')\\
\ldar{c_{s,ws'}}& &\ldar{c_{s,ws'}}\\
T(sws')T(s')^2&\lrar{c_{s',s'}}& T(sws')
\end{array}
\end{equation}
which is clear.
\item $w\in P_s$, $ws'\not\in P_s$. In this case
according to \cite{B}, IV, 1.7 we have $sw=ws'$, so the
required associativity follows from (\ref{sizig}).
\end{enumerate}
Thus, the diagram (\ref{asso1}) is commutative for $w\in P_s$.
Now assume that $w=sw'$ with $w'\in P_s$.
Consider the following diagram:
\begin{equation}\label{asso1_2}
\setlength{\unitlength}{0.17mm}
\begin{array}{ccccccc}
T(s)T(sw')T(s')&\lrar{c_{s,w'}^{-1}}&T(s)^2T(w')T(s')&
\lrar{c_{w',s'}}& T(s)^2T(w's') &
\lrar{c_{s,w's'}}&T(s)T(sw's')\\
&\ldrar{c_{s,sw'}}&\ldar{c_{s,s}}&
&\ldar{c_{s,s}}&\ldlar{c_{s,sw's'}}&\\
&&T(w')T(s')&\lrar{c_{w',s'}}& T(w's')
\end{array}
\end{equation}
In this diagram the square is commutative,
the left triangle is commutative by
definition of $c{s,sw'}$ and the right triangle
is commutative by (\ref{asso0}). Also the commutativity
of (\ref{asso1}) for $w'$ implies that the composition
of the top arrows coincides with the top arrow in
the diagram (\ref{asso1}) for $w$ and hence, the commutativity
of (\ref{asso1}) for $w$.
Now for any $w\in W$ with a reduced decomposition
$w=s_{i_1}s_{i_2}\ldots s_{i_l}$ which we denote by $\ss$
and any $w'\in W$,
we define the morphism $c_{\ss,w'}:T(w)T(w')\rightarrow T(ww')$
inductively as the composition
$$c_{\ss,w'}:T(w)T(w')\simeq
T(s_{i_1})T(s_{i_2}\ldots s_{i_l})T(w')\rightarrow
T(s_{i_1})T(s_{i_2}\ldots s_{i_l}w')\rightarrow T(ww')$$
where the latter arrow is $c_{s_{i_1},s_{i_2}\ldots
s_{i_l}w'}$. We are going to prove that $c_{\ss,w'}$ does not
depend on a choice of the reduced decomposition $\ss$ of $w$.
First we note that the following diagram
is commutative for every reduced decomposition $\ss$ of $w$
and every $s\in S$, $w'\in W$:
\begin{equation}\label{asso2}
\begin{array}{ccccc}
T(w)T(w')T(s)&\lrar{c_{w,s}}& T(w)T(w's)\\
\ldar{c_{\ss,w'}}& &\ldar{c_{\ss,w's}}\\
T(ww')T(s)&\lrar{c_{ww',s}}& T(ww's)
\end{array}
\end{equation}
Indeed, this follows from (\ref{asso1}) by induction in the
length of $w$. Now the required independence of $c_{\ss,w'}$
on a choice of $\ss$ follows from (\ref{asso2}) by induction
in the length of $w'$ (the base of the induction is the case
$w'=1$ when the assertion is obvious). Let us denote
$c_{w,w'}=c_{\ss,w'}$ for any reduced decomposition $\ss$ of
$w$. Using (\ref{asso2}) it is easy to show that one would
obtain the same morphism starting with a reduced
decomposition of $w'$. It remains to check that the following
diagram is commutative for any $w,w_1,w_2\in W$:
\begin{equation}\label{asso3}
\begin{array}{ccccc}
T(w_1)T(w)T(w_2)&\lrar{c_{w,w_2}}& T(w_1)T(ww_2)\\
\ldar{c_{w_1,w}}& &\ldar{c_{w_1,ww_2}}\\
T(w_1w)T(w_2)&\lrar{c_{w_1w,w_2}}& T(w_1ww_2)
\end{array}
\end{equation}
When $\ell(w_2)=1$ this reduces to (\ref{asso2}).
The general case follows easily by induction in $\ell(w_2)$.
\qed\vspace{3mm}
\subsection{Reduction to rank-2 subgroups}
The condition (2) of Theorem \ref{quasiact}
can be reduced to rank-2 subgroups of $W$ using the following result.
For every pair $(s_1,s_2)$ of elements of $S$, such that
the order of $s_1s_2$ is $2m+\varepsilon$ with $\varepsilon\in\{0,1\}$,
let us denote $s=s(s_1,s_2)=s_1$, $s'=s'(s_1,s_2)=s_1$ if $\varepsilon=0$,
and $s'=s_2$ if $\varepsilon=1$.
Let also $w=w(s_1,s_2)=(s_2s_1)^{n-\varepsilon}s_2^{\varepsilon}$.
Then $sw=ws'$ and $\ell(sw)=\ell(w)+1$.
\begin{prop}\label{sizig2} We keep the notation and assumptions
of Theorem \ref{quasiact}.
Assume that the diagram (\ref{sizig}) is commutative
for all the triples $s(s_1,s_2),s'(s_1,s_2),w(s_1,s_2)$
associated with pairs $(s_1,s_2)$ of elements of $S$ as above.
Then it is commutative for all $(w,s,s')$ such that
$sw=ws'$ and $\ell(sw)=\ell(w)+1$.
\end{prop}
\begin{lem}\label{sizig3} Let $(W,S)$ be a Coxeter system.
Assume that $sw=ws'$, where
$w\in W$, $s,s'\in S$, $\ell(sw)=\ell(w)+1$, $w\neq 1$.
Then there exists an element $s_2\in S$, such that
for the element $w(s_1,s_2)$ associated with the pair
$(s_1=s,s_2)$, one has
$\ell(w)=\ell(w(s_1,s_2))+\ell(w(s_1,s_2)^{-1}w)$.
\end{lem}
\noindent {\it Proof}. Consider a reduced decomposition $(s_2,\ldots, s_n)$ of
$w$. Then we have two reduced decompositions of $sw=ws'$:
${\bf s}=(s_1=s,s_2,\ldots,s_n)$ and ${\bf s'}=(s_2,\ldots,s_n,s')$. Now
the second sequence is obtained from the first one by a series
of standard moves associated with braid relations for couples of elements of
$S$. If the first move in such a series
does not touch the first member of ${\bf s}$, then it just
changes a reduced decomposition of $w$. Thus, we can choose
an initial reduced decomposition for $W$ in such a way that
the first move does touch $s_1$. Then we have $w=w(s_1,s_2)w'$
where $\ell(w)=\ell(w(s_1,s_2))+\ell(w')$ as required.
\qed\vspace{3mm}
\noindent
{\it Proof of Proposition \ref{sizig2}}.
Induction on the length of $w$ and Lemma \ref{sizig3}
show that it is sufficient to prove the following:
if $w=w_1w_2$ with $\ell(w)=\ell(w_1)+\ell(w_2)$ and one
has $s_1w_1=w_1s_2$, $s_2w_2=w_2s_3$ where $\ell(s_1w)=\ell(w)+1$,
then the commutativity of the diagram (\ref{sizig})
for the triples $(w_1,s_1,s_2)$ and $(w_2,s_2,s_3)$
implies its commutativity for $(w,s_1,s_3)$.
This can be easily checked using the fact that we have a
$B^+$-action.
\qed\vspace{3mm}
\subsection{Grothendieck groups}
As before we can define a notion of $W$-gluing data ${\cal D}\Phi$ for
the derived categories $({\cal D}^b({\cal C}_w))$, which induce the
$W$-gluing data $\Phi$ for $({\cal C}_w)$.
Let us denote by $\phi_w:K_0({\cal C}_w')\rightarrow
K_0({\cal C}_{ww'})$ the corresponding homomorphisms of
Grothendieck groups. If in addition the categories ${\cal C}_w$
are artinian and noetherian, then according to Theorem \ref{K0}
the image of the homomorphism $K_0({\cal C}(\Phi))\rightarrow\oplus_w
K_0({\cal C}_w)$ coincides with the subgroup
$$K(\Phi)=\{ (c_w)\in\oplus_{w\in W} K_0({\cal C}_w) \ | \
\phi_w c_{w'}-c_{ww'}\in K_{ww',w'},\ w,w'\in W \}.$$
The property that $\nu_{w,w'}$ is an isomorphism
when $\ell(ww')=\ell(w)+\ell(w')$ allows us to give an alternative
definition for the category ${\cal C}(\Phi)$. Namely,
the morphisms $\a_{w',w}:\Phi_{w'}A_w\rightarrow A_{w'w}$
for all $w'\in W$
can be recovered uniquely from the morphisms
$\a_{s,w}:\Phi_sA_w\rightarrow A_{sw}$ provided the latter morphisms
respect the relations in $W$ in the obvious sense.
On the level of Grothendieck groups this is reflected in the
following result.
\begin{prop}\label{simple}
Let ${\cal D}\Phi$ be a $W$-gluing data for
$({\cal D}^b({\cal C}_w))$. Then
\begin{equation}\label{newK0}
K(\Phi)=\{ (c_w)\in\oplus_{w\in W} K_0({\cal C}_w) \ | \
\phi_s c_w-c_{sw}\in K_{sw,w}, \ w\in W, s\in S \}.
\end{equation}
\end{prop}
\noindent {\it Proof} . It is clear that the left-hand side of (\ref{newK0})
is contained in the right-hand side, so we have to check the
inverse inclusion.
If $\ell(sw)=\ell(w)+1$, then $\phi_{sw}=\phi_s\phi_w$. Hence,
$$\phi_{sw} c_{w'}-c_{sww'}=(\phi_s c_{ww'}- c_{sww'})+
\phi_s (\phi_w c_{w'}-c_{ww'}).$$
Thus, it is sufficient to prove that
$$K_{sww',ww'}+\phi_s(K_{ww',w'})\subset K_{sww',w'}$$
provided that $\ell(sw)=\ell(w)+1$.
By definition ${\cal C}_{sww',w'}$ consists of objects
$X\in{\cal C}_{sww'}$ for which the morphism
\begin{equation}\label{mor}
\Phi_s\Phi_{w}\Phi_{w^{-1}}\Phi_s X\rightarrow X
\end{equation}
is zero. Since this morphism factors through $\Phi_s\Phi_s
X\rightarrow X$ we have an obvious inclusion ${\cal C}_{sww',ww'}\subset
{\cal C}_{sww',w'}$. By Lemma \ref{higherder} for $Y\in{\cal C}_{ww'}$
we have $[H^n{\cal D}\Phi_s Y]\in K_{sww',ww'}$ for $n\le -1$. Thus,
it is sufficient to prove the inclusion
$\Phi_s({\cal C}_{ww',w'})\subset{\cal C}_{sww',w'}$. But for
$X=\Phi_s Y$ the morphism (\ref{mor}) factorizes as follows:
$$\Phi_s\Phi_w\Phi_{w^{-1}}\Phi_s\Phi_sY\rightarrow
\Phi_s\Phi_w\Phi_{w^{-1}}Y\rightarrow\Phi_s Y.$$
If $Y\in{\cal C}_{ww',w'}$, then the latter arrow is zero, hence,
$\Phi_s Y\in{\cal C}_{sww',w'}$.
\qed\vspace{3mm}
\section{Symplectic Fourier transform}
\subsection{Functors and distinguished triangles}\label{foursq}
Let $k$ be a field of characteristic $p>0$ that is
either finite or algebraically closed, and $S$ a
scheme of finite type over $k$.
Let $\pi:V\rightarrow S$ be a symplectic vector bundle of rank $2n$ over
$S$, and $\langle , \rangle: V\times_S V\rightarrow{\Bbb G}_a$ the corresponding
symplectic pairing. Let us fix a non-trivial additive character
$\psi:{\Bbb F}_p\rightarrow\overline{{\Bbb Q}}_l^*$.
The Fourier---Deligne transform
${\cal F}={\cal F}_\psi$ is the involution of ${\cal D}^b_c(V,\overline{{\Bbb Q}}_l)$
defined by
$${\cal F}(K)=p_{2!}({\cal L}\otimes p_1^*(K))[2n](n).$$
where $p_i$ are the projections of the product $V\times_S V$ on its
factors, and ${\cal L}={\cal L}_{\psi}(\langle , \rangle)$ is a smooth rank-1
$\overline{{\Bbb Q}}_l$-sheaf on $V\times_S V$ which
is the pullback of the Artin---Schreier sheaf ${\cal L}_{\psi}$ on ${\Bbb G}_a$ under
the morphism $\langle , \rangle$.
Let $s:S\rightarrow V$ be the zero section, $j:U\rightarrow V$ the
complementary open subset to $s(S)$, and $p=\pi\circ j:U\rightarrow S$
the projection of $U$ to $S$. Let us denote
$${\cal F}_!=j^*{\cal F} j_!:{\cal D}^b_c(U,\overline{{\Bbb Q}}_l)\rightarrow {\cal D}^b_c(U,\overline{{\Bbb Q}}_l).$$
\begin{prop}\label{Fouriersq}
For every $K\in {\cal D}^b_c(U,\overline{{\Bbb Q}_l})$
there is a canonical distinguished triangle in
${\cal D}^b_c(U,\overline{{\Bbb Q}}_l)$:
\begin{equation}\label{triangle}
{\cal F}_!^2(K)\rightarrow K\rightarrow p^*p_! K[4n](2n)\rightarrow\ldots
\end{equation}
\end{prop}
\noindent {\it Proof} . Note that ${\cal F}_!(K)\simeq p_{2!}({\cal L}'\otimes p_1^*K)[2n](n)$
where ${\cal L}'={\cal L}|_{U^2}$;
hence,
$${\cal F}_!^2(K)\simeq p_{2!}(({\cal L}'\circ{\cal L}')\otimes p_1^*K)[4n](2n)$$
where ${\cal L}'\circ{\cal L}'=p_{13!}(p_{12}^*{\cal L}'\otimes
p_{23}^*{\cal L}')\in{\cal D}^b_c(U^2,\overline{{\Bbb Q}}_l)$.
Let $k=\operatorname{id}\times j\times\operatorname{id}: V\times_S U\times_S V\rightarrow V^3$ be
the open embedding. Then
$${\cal L}'\circ{\cal L}'\simeq (j\times j)^*p_{13!}
k_!k^*(p_{12}^*{\cal L}\otimes p_{23}^*{\cal L}).$$
Note that we have
a canonical isomorphism
$${\cal L}_{\psi}(\langle x_1, x_2\rangle)\otimes{\cal L}_{\psi}(\langle
x_2, x_3\rangle)\simeq {\cal L}_{\psi}(\langle x_2, x_3-x_1\rangle)$$
on $V^3$, and the latter rank-1 local system is trivial on
the complement to $k$; hence we have the following exact
triangle:
$${\cal L}'\circ{\cal L}'\rightarrow (j\times j)^*({\cal L}\circ{\cal L})\rightarrow
\overline{{\Bbb Q}}_{l,U^2}\rightarrow\ldots,$$
where ${\cal L}\circ{\cal L}=p_{13!}(p_{12}^*{\cal L}\otimes p_{23}^*{\cal L})\simeq
\Delta_*\overline{{\Bbb Q}}_{l,V}[-4n](-2n)$, which induces the exact triangle
(\ref{triangle}).
\qed\vspace{3mm}
Let us denote by
$\sideset{^p}{_!}{{\cal F}}=\sideset{^p}{^0}{H}{\cal F}_!:\operatorname{Perv}(U)\rightarrow\operatorname{Perv}(U)$
the right-exact functor induced by ${\cal F}_!$. Then we have a
canonical morphism of functors
$\nu:(\sideset{^p}{_!}{{\cal F}})^2\rightarrow\operatorname{Id}$.
\begin{cor}\label{zer}
The full subcategory of objects $K\in\operatorname{Perv}(U)$ such
that $\nu_K:(\sideset{^p}{_!}{{\cal F}})^2K\rightarrow K$ is zero coincides with
the subcategory $p^*[2n](\operatorname{Perv}(S))\subset\operatorname{Perv}(U)$.
\end{cor}
\noindent {\it Proof} . For $K\in\operatorname{Perv}(U)$ the exact
triangle (\ref{triangle}) induces the long exact sequence
$$\ldots\rightarrow(\sideset{^p}{_!}{F})^2 K\rightarrow K\rightarrow
p^*[2n](\sideset{^p}{^{2n}}{H}p_!K)(2n)\rightarrow 0.$$
Since $\sideset{^p}{^{4n}}{H}p_!p^*(2n)\simeq\operatorname{Id}_{\operatorname{Perv}(S)}$,
the assertion follows.
\qed\vspace{3mm}
\subsection{Associativity}\label{assofourier}
To check the associativity condition (\ref{asso_simple})
for the above morphism ${\cal F}_{!}^2\rightarrow\operatorname{Id}$,
it is sufficient to check this condition for the
morphism ${\cal F}^2\rightarrow\operatorname{Id}$, which is done in the following
proposition.
\begin{prop} Let ${\cal F}$ be the Fourier transform associated
with a symplectic vector bundle $V$ of rank $2n$. Then the
canonical morphism of functors $c:{\cal F}^2\rightarrow\operatorname{Id}$ satisfies the
associativity condition ${\cal F}\circ c=c\circ{\cal F}:{\cal F}^3\rightarrow{\cal F}$.
\end{prop}
\noindent {\it Proof} . Recall that ${\cal F}$ is given by the kernel ${\cal L}[2n](n)$ on
$V^2$ and $c$ is induced by the canonical morphisms of kernels
\begin{align*}
&p_{13!}(p_{12}^*{\cal L}\otimes p_{23}^*{\cal L}[4n](2n))\rightarrow
p_{13!}(\Delta_{13*}\Delta_{13}^*(p_{12}^*{\cal L}\otimes
p_{23}^*{\cal L}[4n](2n)))\simeq\\
&\simeq p_{13!}\Delta_{13*}\overline{{\Bbb Q}}_{l,V^2}[4n](2n)
\stackrel{\operatorname{Tr}}{\rightarrow}\Delta_*\overline{{\Bbb Q}}_{l,V},
\end{align*}
where $p_{ij}$ are the projections of $V^3$ on the double
products, $\Delta:V\rightarrow V^2$ and $\Delta_{13}:V^2\rightarrow V^3$ are the diagonals,
$\operatorname{Tr}$ is induced by the trace morphism (see \cite{SGA4})
(the triviality of $\Delta_{13}^*(p_{12}^*{\cal L}\otimes p_{23}^*{\cal L})$
follows from the skew-symmetry of ${\cal L}$).
Now ${\cal F}^3$ is given by the kernel
$p_{14!}(L[6n](3n))$ on
$V^2$ where $L=p_{12}^*{\cal L}\otimes p_{23}^*{\cal L}\otimes p_{34}^*{\cal L}$
$p_{ij}$ are the projections of $V^4$ on the
double products, and the associativity boils down to the
commutativity of the diagram
\begin{equation}
\begin{array}{ccccc}
& &p_{14!}(L[6n](3n))& &\\
&\ldlar{} & &\ldrar{}&\\
p_{14!}(\Delta_{13*}\Delta_{13}^*L[6n](3n))& &
& & p_{14!}(\Delta_{24*}\Delta_{24}^*L[6n](3n))\\
&\ldrar{\operatorname{Tr}} & &\ldlar{\operatorname{Tr}}& \\
& &{\cal L}[2n](n)& &
\end{array}
\end{equation}
where $\Delta_{ij}:V^3\rightarrow V^4$ are the diagonals,
and the lower diagonal arrows are induced by the isomorphisms
$\Delta_{13}^*L\simeq\Delta_{24}^*L\simeq p_{13}^*{\cal L}$ on $V^3$
and by the trace morphisms.
Changing the coordinates on $V^4$ by $(x,y,z,t)\mapsto
(x,y-t,z-x,t)$ one can see that this reduces to the
commutativity of the diagram
\begin{equation}\label{comm_tr}
\begin{array}{ccccc}
p_!({\cal L}[4n](2n)) &\lrar{} p_!(i_{1*}i_1^*{\cal L}[4n](2n))\simeq &
p_!(\overline{{\Bbb Q}}_{l,V}[4n](2n))\\
\ldar{} & &\ldar{\operatorname{Tr}}\\
p_!(i_{2*}i_2^*{\cal L}[4n](2n))&\simeq p_!(\overline{{\Bbb Q}}_{l,V}[4n](2n))
\lrar{\operatorname{Tr}} & \overline{{\Bbb Q}}_l
\end{array}
\end{equation}
where $i_1:V\times 0\rightarrow V^2$, $i_2:0\times V\rightarrow V^2$ are the
natural closed embeddings, and $p$ denotes the projection to
$\operatorname{Spec}(k)$.
Here is the argument due to M.~Rapoport verifying the
commutativity of (\ref{comm_tr}). Obviously, we may assume
that the base is a point and $n=1$, so that $V={\Bbb A}^2$.
Let $i:I\hookrightarrow \P^1\times {\Bbb A}^2$ be the
tautological line bundle over the projective line (the
incidence correspondence), and $p_2^*{\cal L}$ be the pullback of
${\cal L}$ under the projection $p_2:\P^1\times{\Bbb A}^2\rightarrow{\Bbb A}^2$.
Then $i$ is an embedding of a lagrangian subbundle in the
trivial rank-2 symplectic bundle over $\P^1$; hence,
we have a sequence of canonical morphisms
$$q^*p_!({\cal L}[4](2))\widetilde{\rightarrow}
\pi_!(p_2^*{\cal L}[4](2))\rightarrow
\pi_!(i_*i^*p_2^*{\cal L}[4](2))\widetilde{\rightarrow}
\pi_!(\overline{{\Bbb Q}}_{l,I}[4](2))\widetilde{\rightarrow} q^*\overline{{\Bbb Q}}_l$$
where $\pi$ denotes the projection to $\P^1$, and $q$ is the
projection of $\P^1$ to $\operatorname{Spec}(k)$. Let
$\phi:q^*p_!({\cal L}[4](2))\rightarrow q^*\overline{{\Bbb Q}}_l$ be the composed
morphism. Then $\phi=q^*\widetilde{\phi}$ for some morphism
$\widetilde{\phi}:p_!({\cal L}[4](2))\rightarrow\overline{{\Bbb Q}}_l$ (since in fact
$p_!({\cal L}[4](2))\simeq\overline{{\Bbb Q}}_l$). On the other hand, the two morphisms
$p_!({\cal L}[4](2))\rightarrow\overline{{\Bbb Q}}_l$ in (\ref{comm_tr})
are the restrictions of $\phi$ to the points $x,y\in\P^1$
corresponding to the coordinate lines in ${\Bbb A}^2$; hence, they
are both equal to $\widetilde{\phi}$.
\qed\vspace{3mm}
\section{Gluing on the basic affine space}\label{glu}
\subsection{Setup}
Let $k$ be a field of characteristic $p>2$, which is either
finite or algebraically closed, let
$G$ be a connected, simply-connected, semisimple algebraic group over
$k$. Assume that $G$ is split over $k$
and fix a split maximal torus $T\subset G$ and a Borel subgroup
$B$ containing $T$. Also we denote by $W=N(T)/T$ the Weil group of $G$,
and by $S\subset W$ the set of simple reflections.
Let $X=G/U$ be the corresponding
basic affine space, where
$U$ is the unipotent radical of $B$.
Following \cite{KL} we are going to construct a $W$-gluing
data such that ${\cal C}_w$ is
the category $\operatorname{Perv}(X)$ of perverse sheaves on $X$ for every
$w$.
To define the gluing functors we need some additional data.
First, we fix a nontrivial additive character
$\psi:{\Bbb F}_p\rightarrow\overline{{\Bbb Q}_l}^*$. We denote by ${\cal L}_{\psi}$ the
corresponding Artin---Schreier sheaf on ${\Bbb G}_{a,{\Bbb F}_p}$.
Second, for every simple root $\a_s$ we fix an isomorphism
of the corresponding 1-parameter subgroup $U_s\subset U$ with
the additive group ${\Bbb G}_{a,k}$. This defines uniquely a
homomorphism $\rho_s:\operatorname{SL}_{2,k}\rightarrow G$ such that
$\varphi_s$ induces the given isomorphism of ${\Bbb G}_{a,k}$
(embedded in $\operatorname{SL}_{2,k}$ as upper-triangular matrices)
with $U_s$ (see \cite{KL}). Let
$$n_s=\rho_s\left(\matrix 0 & 1 \\ -1 & 0 \endmatrix\right).$$
For every $w\in W$ with a reduced decomposition
$w=s_1\ldots s_l$ we set $n_w=n_{s_1}\ldots n_{s_l}\in G$.
Then $n_w$ does not depend on a choice of a reduced
decomposition, so we get a canonical system of representatives
for $W$ in $N(T)$.
\subsection{Gluing functors} For every element $w\in W$ we consider the
subtorus
$$T_w=\prod_{\a\in R(w)}{\a^{\vee}({\Bbb G}_m)}\subset T$$
where $R(w)\subset R^+$ is the set of positive roots $\a\in R^+$
such that $w(\a)\in -R^+$, $\a^{\vee}$ is the corresponding
coroot. By definition we have a surjective homomorphism
$$\prod_{\a\in R(w)}{\a^{\vee}}:{\Bbb G}_{m,k}^{R(w)}\rightarrow T_w.$$
It is easy to see that
$$T_w=\prod_{s\in S_w} T_s$$
where $S_w\subset S$ is the set of simple reflections $s$ such
that $s\leq w$ with respect to the Bruhat order (see
\cite{KL}, 2.2.1).
Now we define $X(w)\subset X\times_k X$ as the subvariety
of pairs $(gU,g'U)\subset X\times_k X$ such that
$g^{-1}g'\in Un_wT_wT$. There is a canonical projection
$\operatorname{pr}_w:X(w)\rightarrow T_w$ sending $(gU,g'U)$ to the unique
$t_w\in T_w$ such that $g^{-1}g'\in Un_wt_wU$.
The morphism $\operatorname{pr}_w$ is smooth of relative dimension
$\dim X +\ell(w)$, surjective, with connected geometric fibers.
The last ingredient in the definition of the gluing functors is
the morphism
$$\sigma_w:{\Bbb G}_{m,k}^{R(w)}\rightarrow{\Bbb G}_{a,k}:
(z_{\a})_{\a\in R(w)}\mapsto -\sum_{\a\in R(w)} z_{\a}.$$
Now we set
$$K(w)=K_{\psi}(w)=
\operatorname{pr}_w^*(\prod_{\a\in R(w)}\a^{\vee})_!\sigma_w^*{\cal L}_{\psi}
[2\ell(w)](\ell(w)).$$
As shown in (\cite{KL}, 2.2.8) this is, up to shift by $\dim X$,
an irreducible
perverse sheaf on $X(w)$. Finally, one defines
$\overline{K(w)}$ to be the Goresky---MacPherson extension of
$K(w)$ to the closure $\overline{X(w)}$ of $X(w)$ in $X\times X$.
The gluing functors
$F_{w,!}:{\cal D}^b_c(X,\overline{{\Bbb Q}}_l)\rightarrow{\cal D}^b_c(X,\overline{{\Bbb Q}}_l)$
are defined by
$$F_{w,!}(A)=p_{2,!}(p_1^*(A)\otimes \overline{K(w)}).$$
In the case when $w=s$ is a simple reflection
the morphism $\operatorname{pr}_s:X(s)\rightarrow T_s\simeq{\Bbb G}_{m,k}$ extends to
$\overline{\operatorname{pr}}_s:\overline{X(s)}\rightarrow{\Bbb G}_{a,k}$ and we have
$$\overline{K(s)}\simeq (-\overline{\operatorname{pr}}_s)^*{\cal L}_{\psi}.$$
This leads to the alternative construction of the functor
$F_{s,!}$ using the embedding of
$G/U$ in a rank-2 vector bundle and the corresponding
partial Fourier transform. Namely,
let us denote $M_s=\rho_s(\operatorname{SL}_{2,k})\subset G$ and
consider the projection $p_s:X=G/U\rightarrow G/Q_s$, where
$Q_s=M_sU\subset G$ (note that $U$ normalizes $M_s$).
Now $p_s$ is the complementary open subset
to the zero section in a $G$-equivariant rank-2 symplectic vector bundle
$\pi_s:V_s\rightarrow G/Q_s$ (see \cite{KL}). Furthermore, we have
$\overline{X(s)}\simeq X\times_{G/Q_s} X$ and the morphism
$-\overline{\operatorname{pr}}_s$ from $\overline{X(s)}$ to ${\Bbb G}_{a,k}$ coincides with
the restriction of the symplectic pairing on $V_s$. It follows
that $F_{s,!}=j^*{\cal F} j_!$ where $j:X\hookrightarrow V_s$ is the embedding,
${\cal F}$ is the (symplectic) Fourier transform for $V_s$ (see
the previous section), so ${\cal F}^2\simeq\operatorname{Id}$. Since $j_!$ is right
$t$-exact with respect to perverse $t$-structures, so is
$F_s$. As is shown in \cite{KL} (see also \ref{recKL} below)
for every reduced decomposition $w=s_1\ldots s_l$ one has a
canonical isomorphism of functors
$$F_{w,!}\simeq F_{s_1,!}\circ\ldots\circ F_{s_l,!}.$$
It follows that all the functors $F_{w,!}$ are right
$t$-exact.
\begin{thm} The functors $F_{w,!}$ define a quasi-action of
$W$ on ${\cal D}^b_c(X,\overline{{\Bbb Q}}_l)$.
\end{thm}
It follows that the functors $\sideset{^p}{^0}{H} F_{w,!}$ can be used
to define a $W$-gluing for $|W|$ copies of $\operatorname{Perv}(X)$.
The resulting glued category is denoted by $\AA$.
This theorem was stated as Theorem 2.6.1 in \cite{KL}.
However, the proof presented in loc. cit. is insufficient.
Namely, it is proved in \cite{KL} that the functors
$F_{w,!}$ generate the action of the positive braid monoid
on ${\cal D}^b_c(X,\overline{{\Bbb Q}}_l)$, and the morphisms $F_{s,!}^2\rightarrow\operatorname{Id}$ are
constructed. To finish the proof one has to show
that the other conditions of Theorem \ref{quasiact}
are satisfied. The condition (1) was checked in
\ref{assofourier}, and the condition (2) will be checked
below in \ref{assocomp}.
\subsection{Isomorphism}\label{recKL}
Let us recall from \cite{KL} the construction of
the canonical isomorphism
$$F_{w,!}\simeq F_{w_1,!}\circ F_{w_2,!}$$
associated with a decomposition $w=w_1w_2$ such that
$\ell(w)=\ell(w_1)+\ell(w_2)$.
One starts with a commutative diagram
\begin{equation}\label{Xw_1w_2cart}
\begin{array}{ccccc}
X(w_1)\times_X X(w_2) &\lrar{} &T_{w_1}\times T_{w_2}\\
\ldar{} & & \ldar{m_{w_1,w_2}}\\
X(w) &\lrar{} &T_w
\end{array}
\end{equation}
where the map
$m_{w_1,w_2}:T_{w_1}\times T_{w_2}\rightarrow T_{w_1w_2}$ is given
by $(t_1,t_2)\mapsto w_2^{-1}(t_1)t_2$.
It is easy to check that this diagram is cartesian. Moreover,
since $R(w_1w_2)$ is the disjoint union of $w_2^{-1}(R(w_1))$
and $R(w_2)$, by the base change we get an isomorphism
\begin{equation}\label{w_1w_2}
K(w)\simeq p_{13,!}(p_{12}^*K(w_1)\otimes p_{23}^*K(w_2)),
\end{equation}
where $p_{ij}$ are the projections from
$X(w_1)\times_X X(w_2)\subset X^3$.
To derive from this the isomorphism
\begin{equation}\label{cw_1w_2}
\overline{K(w)}\simeq \overline{K(w_1)}\circ \overline{K(w_2)}
\end{equation}
it remains to check that the right-hand side is an irreducible
perverse sheaf up to shift by $\dim X$.
This is done in \cite{KL}, (2.4)---(2.5).
\subsection{Geometric information}\label{Xw_1w_2}
We need some more information about varieties
$X(w)$.
For every $w_1, w_2\in W$ such that $l(w_1w_2)=l(w_1)+l(w_2)$
consider the natural maps
\begin{align*}
&i_{w_1,w_2}: X(w_1)\times_X X(w_2)\rightarrow X(w)\times
T_{w_1}:
(g_1U, g_2U, g_3U)\mapsto ((g_1U, g_3U), \operatorname{pr}_{w_1}(g_1U,g_2U),\\
&i'_{w_1,w_2}: X(w_1)\times_X X(w_2)\rightarrow X(w)\times
T_{w_2}:
(g_1U, g_2U, g_3U)\mapsto ((g_1U, g_3U), \operatorname{pr}_{w_2}(g_2U,g_3U),
\end{align*}
where $w=w_1w_2$.
\begin{lem}\label{iw_1w_2}
The morphism $i_{w_1,w_2}$ (resp. $i'_{w_1,w_2}$) is an isomorphism onto
the locally closed subvariety of pairs
$(x,t_1)\in X(w)\times
T_{w_1}$ such that $\operatorname{pr}_{w}(x)t_1^{-1}\in T_{w_2}$
(resp. $(x,t_2)\in X(w)\times
T_{w_2}$ such that $\operatorname{pr}_{w}(x)t_2^{-1}\in
w_2^{-1}(T_{w_1})$).
\end{lem}
\noindent {\it Proof} . The cartesian diagram (\ref{Xw_1w_2cart}) allows one to
identify $X(w_1)\times_X X(w_2)$ with the subvariety
in $X(w)\times T_{w_1}\times T_{w_2}$ consisting
of $(x,t_1,t_2)$ such that $\operatorname{pr}_w(x)=w_2^{-1}(t_1)t_2$.
Together with the fact that $w_2(t)t^{-1}\in T_{w_2}$
for any $t\in T$, this implies the assertion.
\qed\vspace{3mm}
Let ${\cal B}=G/B$ be the flag variety of $G$,
and $O(w)\subset{\cal B}\times{\cal B}$ the $G$-orbit corresponding to
$w\in W$. The canonical projection $X(w)\rightarrow O(w)$ can
be considered as a $T\times T_w$-torsor. Namely, we have the
natural action of $T\times T$ on $X\times X$ such that
$(t,t')(gU,g'U)=(gtU,g't'U)$. Now the subgroup
$\{(t,t')\ |\ t^{-1}t'\in T_w \}$ (which is naturally
isomorphic to $T\times T_w$) preserves $X(w)$ and induces
the above torsor structure. Let us denote by $\overline{X(w)}\subset
X\times X$ and $\overline{O(w)}\subset{\cal B}\times{\cal B}$ the Zariski
closures.
\begin{lem}\label{tors} For every $s\in S$ the morphism
$\overline{X(s)}\rightarrow \overline{O(s)}$ is a $T\times T_s$-torsor.
\end{lem}
This follows essentially from the rank-1 case when
$\overline{X(s)}=({\Bbb A}^2-0)^2$, $\overline{O(s)}=\P^1\times\P^1$.
\subsection{Associativity}\label{assocomp}
Now we will check the second condition of Theorem \ref{quasiact}
for the functors $F_{w,!}$. Clearly, it is sufficient to
check the commutativity of the corresponding diagram of
morphisms between kernels on $X\times X$.
Let us recall the construction of the morphism
$$c_{s,s}:\overline{K(s)}\circ\overline{K(s)}\rightarrow\Delta_*\overline{{\Bbb Q}}_{X,l}.$$
Consider the natural embedding
\begin{equation}\label{De_s}
\Delta_s:\overline{X(s)}\hookrightarrow\overline{X(s)}\times_X\overline{X(s)}:
(g_1U,g_2U)\mapsto (g_2U,g_1U,g_2U).
\end{equation}
We have the morphism
$$\overline{\operatorname{pr}}_{s,s}:\overline{X(s)}\times_X\overline{X(s)}\rightarrow {\Bbb G}_{a,k}:
(x,x')\mapsto\overline{\operatorname{pr}}_s(x)+\overline{\operatorname{pr}}_s(x')$$
such that $p_{12}^*\overline{K(s)}\otimes p_{23}^*\overline{K(s)}=
(-\overline{\operatorname{pr}}_{s,s})^*{\cal L}_{\psi}$. The composition
$\overline{\operatorname{pr}}_{s,s}\Delta_s$ is the constant map to
$\{0\}\in{\Bbb G}_{a,k}$, hence we get the canonical isomorphism
$$\Delta_s^*(p_{12}^*\overline{K(s)}\otimes p_{23}^*\overline{K(s)})
\simeq\overline{{\Bbb Q}}_{\overline{X(s)},l}[2](1).$$
Now $c_{s,s}$ corresponds by adjunction to the morphism
$$\Delta^*(\overline{K(s)}\circ\overline{K(s)})\simeq
p_{1,!}(\Delta_s^*(p_{12}^*\overline{K(s)}\otimes p_{23}^*\overline{K(s)})\simeq
p_{1,!}(\overline{{\Bbb Q}}_{\overline{X(s)},l}[2](1))\stackrel{\operatorname{tr}}{\rightarrow}
\overline{{\Bbb Q}}_{X,l}$$
where $\operatorname{tr}$ is the relative trace morphism for $p_1$.
\begin{thm}\label{sizker}
Let $sw=ws'$, where
$w\in W$, $s,s'\in S$ and $l(sw)=l(w)+1$.
Then the following diagram in ${\cal D}^b_c(X\times X,\overline{{\Bbb Q}}_l)$ is
commutative:
\begin{equation}\label{sizigker}
\begin{array}{ccccc}
\overline{K(s)}\circ\overline{K(w)}\circ\overline{K(s')} &\lrar{} &
\overline{K(w)}\circ\overline{K(s')}\circ\overline{K(s')}\\
\ldar{} & & \ldar{c_{s',s'}}\\
\overline{K(s)}\circ\overline{K(s)}\circ\overline{K(w)} &\lrar{c_{s,s}} &
\overline{K(w)}
\end{array}
\end{equation}
where unmarked arrows are induced by the isomorphisms
(\ref{cw_1w_2}).
\end{thm}
The main step in the proof is the following lemma.
\begin{lem}\label{mainsiz} Under the conditions of Theorem \ref{sizker}
there is a canonical isomorphism
$$\a:\overline{X(s)}\times_X X(w)\rightarrow X(w)\times_X\overline{X(s')}$$
of schemes over $X\times X$ and a canonical isomorphism
$$\overline{p}_s^*\overline{K(s)}\otimes p_w^*K(w)\simeq
\a^*(p_w^*K(w)\otimes \overline{p}_{s'}^*\overline{K(s)})$$
which is compatible with the isomorphisms (\ref{w_1w_2}),
where e.g. $\overline{p}_s:\overline{X(s)}\times_X X(w)\rightarrow \overline{X(s)}$
is the natural projection, etc.
\end{lem}
\noindent {\it Proof} . Consider the morphism
$$\overline{i'}_{s,w}:\overline{X(s)}\times_X X(w)\rightarrow
X\times X\times T_w:
(g_1U, g_2U, g_3U)\mapsto (g_1U, g_3U, \operatorname{pr}_w(g_2U,g_3U))$$
extending the morphism $i'_{s,w}$ defined in \ref{Xw_1w_2}.
It is easy to see that $\overline{i'}_{s,w}$ is an embedding.
Let us consider the locally closed subvariety
$Y(s,w)=O(sw)\cup O(w)\subset{\cal B}\times{\cal B}$.
We have the following isomorphism of
${\cal B}\times{\cal B}$-schemes:
$$\overline{O(s)}\times_{{\cal B}} O(w)\simeq Y(s,w).$$
Using Lemma \ref{tors} we see that the natural projection
$$\overline{p}_{s,w}:\overline{X(s)}\times_X X(w)\rightarrow Y(s,w)$$
is a $T\times T_s\times T_w$-torsor, where
$T\times T_s\times T_w$ is identified with the subgroup
$$\{(t_1,t_2,t_3)\in T\times T\times T|\ t_1^{-1}t_2\in T_s,
t_2^{-1}t_3\in T_w\}$$
acting naturally on $\overline{X(s)}\times_X X(w)$.
Let $p:X\times X\times T_w\rightarrow{\cal B}\times{\cal B}$ be the projection.
Then $p$ is a $T\times T\times T_w$-torsor, and
via $\overline{i'}_{s,w}$ we can identify the morphism
$$p^{-1}(Y(s,w))\rightarrow Y(s,w)$$
with the $T\times T\times T_w$-torsor over $Y(s,w)$ induced from
$\overline{p}_{s,w}$ by the embedding $T\times T_s\times T_w\hookrightarrow
T\times T\times T_w$. In particular, the image of
$\overline{i'}_{s,w}$ is a closed subvariety in $p^{-1}(Y(s,w))$.
Thus, $\operatorname{im}(\overline{i'}_{s,w})$ is the closure of $\operatorname{im}(i'_{s,w})$
in $p^{-1}(Y(s,w))$.
Similarly, we have an embedding
$$\overline{i}_{w,s'}:X(w)\times_X \overline{X(s)}\rightarrow
X\times X\times T_w$$
extending $i_{w,s'}$, such that $\operatorname{im}(\overline{i}_{w,s'})$
is the closure of $\operatorname{im}(i_{w,s'})$ in $p^{-1}(Y(w,s'))$, where
$Y(ws')=O(ws')\cup O(w)$. Now since $sw=ws'$ we have
$Y(s,w)=Y(w,s')$ and we claim that
$\operatorname{im}(i'_{s,w})=\operatorname{im}(i_{w,s'})$, which implies immediately
that $\operatorname{im}(\overline{i'}_{s,w})=\operatorname{im}(\overline{i}_{w,s'})$. Indeed,
according to Lemma \ref{iw_1w_2} the image of $i'_{s,w}$
consists of pairs $(x,t)\in X(sw)\times T_w$
such that $\operatorname{pr}_{sw}(x)t^{-1}\in w^{-1}(T_s)$, while
the image of $i_{w,s'}$ consists of
$(x,t)$ such that $\operatorname{pr}_{sw}(x)t^{-1}\in T_s'$. Now
the equality $w^{-1}sw=s'$ implies that $w^{-1}(T_s)=T_{s'}$
which proves our claim.
Let
$$\a:\overline{X(s)}\times_X X(w)\rightarrow X(w)\times_X\overline{X(s')}$$
be the unique isomorphism compatible with embeddings
$i'_{s,w}$ and $i_{w,s'}$.
It remains to check that the sheaves
$\overline{p}_s^*\overline{K(s)}\otimes p_w^*K(w)$ and
$p_w^*K(w)\otimes\overline{p}_{s'}^*\overline{K(s')}$ correspond to each
other under $\a$. Since $K(w)$ is the inverse image of a sheaf
on $T_w$ it is sufficient to check that the sheaves
$\overline{p}_s^*\overline{K(s)}$ and $\overline{p}_{s'}^*\overline{K(s')}$ correspond
to each other under $\a$. Since both these sheaves are local
systems (up to shift) it is sufficient to check that
$p_s^*K(s)$ and $p_{s'}^*K(s')$ correspond to each other
under the restriction of $\a$ to the open subset $X(s)\times_X X(w)$.
It is easy to check that the following diagram is commutative
\begin{equation}
\begin{array}{ccccc}
X(s)\times_X X(w) &\lrar{\a} &
X(w)\times_X X(s')\\
\ldar{\operatorname{pr}_s} & & \ldar{\operatorname{pr}_s'}\\
T_s &\lrar{w^{-1}} & T_{s'}
\end{array}
\end{equation}
Moreover, since $l(sw)=l(w)+1$ we have
$w^{-1}(\a_s^{\vee})=\a_{s'}^{\vee}$, hence
the bottom arrow becomes the identity under the
identification of $T_s$ (resp. $T_{s'}$) with ${\Bbb G}_m$
via $\a_s^{\vee}$ (resp. $\a_{s'}^{\vee}$),
and our assertrion follows immediately.
\qed\vspace{3mm}
\begin{lem}\label{cor}
Let $Y$ be a scheme, $A$ a correspondence
over $Y\times Y$, $C$ and $C'$ symmetric
correspondences over $Y\times Y$ such that an isomorphism
of $Y\times Y$-schemes is given by
$$\a:C\times_Y A\widetilde{\rightarrow} A\times_Y C'.$$
Let
\begin{align*}
&\Delta_C:C\hookrightarrow C\times_Y C:(y_1,y_2)\mapsto (y_2,y_1,y_2),\\
&\Delta_{C'}\sigma:C'\hookrightarrow C'\times_Y C':(y_1,y_2)\mapsto (y_1,y_2,y_1)
\end{align*}
be the natural embeddings (where $\sigma:C'\rightarrow C'$ is the
permutation of factors in $Y\times Y$). Then the following diagram
is commutative:
\begin{equation}
\begin{array}{ccccc}
C\times_Y A &\lrar{\a'} &
A\times_Y C'\\
\ldar{\Delta_C} & & \ldar{\Delta_{C'}\sigma}\\
C\times_Y C\times_Y A &\lrar{} & A\times_Y C'\times_Y C'
\end{array}
\end{equation}
where the bottom arrow is induced by $\a$,
$\a'$ is an isomorphism given by
$$\a'(y_1,y_2,y_3)=(y_2,y_3,y_2'),$$
for $(y_1,y_2,y_3)\in C\times_Y A$ where
$\a(y_1,y_2,y_3)=(y_1,y_2',y_3)$.
\end{lem}
The proof is straightforward. Note that $\a'$ commutes with
projections to $A$, and $\a^{-1}\circ\a'$ is an involution of
$C\times_Y A$.
\vspace{2mm}
\noindent
{\it Proof of Theorem \ref{sizker}}.
First we note that
$\overline{K(s)}\circ\overline{K(w)}\circ\overline{K(s')}\in
\sideset{^p}{^{\leq\dim X}}{{\cal D}}(X\times X)$.
Indeed, the functor $K\mapsto \overline{K(s)}\circ K$
from ${\cal D}^b_c(X\times X,\overline{{\Bbb Q}}_l)$ to itself coincides
with the functor ${\cal F}_!$ of (\ref{foursq}) for the
rank-2 symplectic bundle $V_s\times X\rightarrow G/Q_s\times X$,
and hence it is right $t$-exact with respect to the perverse
$t$-structure. Similarly, the
functor $K\mapsto K\circ\overline{K(s')}$ is right $t$-exact,
hence our claim follows from the fact that
$\overline{K(w)}[\dim X]$ is a
perverse sheaf on $X\times X$. It follows that
we can replace all objects in the diagram (\ref{sizigker})
by their $\sideset{^p}{^{\dim X}}H$. Since
$\overline{K(w)}$ is the Goresky---MacPherson extension from
$X(w)\subset X\times_k X$, it is sufficient to check the
commutativity of the restriction of (\ref{sizigker}) to
$X(w)$. Applying Lemma \ref{cor} to the correspondences
$A=X(w)$, $C=\overline{X(s)}$ and $C'=\overline{X(s')}$ we obtain
the commutative diagram
\begin{equation}
\begin{array}{ccccc}
\overline{X(s)}\times_X X(w) &\lrar{\a'} &
X(w)\times_X \overline{X(s')}\\
\ldar{\Delta_s} & & \ldar{\Delta_{s'}\sigma}\\
\overline{X(s)}\times_X \overline{X(s)}\times_X X(w) &\lrar{} &
X(w)\times_X \overline{X(s')}\times_X \overline{X(s')}
\end{array}
\end{equation}
From the construction of the morphisms $c_{s,s}$ and
$c_{s',s'}$ and Lemma \ref{mainsiz}, we see that
it is sufficient to prove that the trivializations (up to
shift and twist)
of $\Delta_s^*(\overline{p}_{s,1}^*\overline{K(s)}\otimes
\overline{p}_{s,2}^*\otimes p_w^*K(w))$
and of
$\Delta_{s'}^*(\overline{p}_{s',1}^*\overline{K(s')}\otimes
\overline{p}_{s',2}\otimes p_w^*K(w))$
are compatible via the above commutative diagram with
$\a'$ and the isomorphism of Lemma \ref{mainsiz}
(here $\overline{p}_{s,1}$ and $\overline{p}_{s,2}$ are the projections onto the
first and the second factors $\overline{X(s)}$).
To this end we can replace $\overline{X(s)}$, $\overline{X(s')}$, $\overline{K(s)}$
and $\overline{K(s')}$ by $X(s)$, $X(s')$, $K(s)$ and $K(s')$,
respectively.
Now this follows immediately from the fact that the isomorphism
$\a:X(s)\times_X X(w)\simeq X(w)\times_X X(s')$ is compatible
with the projections to $T_s\times T_w\simeq T_w\times T_{s'}$
(see the proof of Lemma \ref{mainsiz}).
\qed\vspace{3mm}
\subsection{Grothendieck group of the glued category}
It is easy to check that every functor $F_{w,!}$ has the right
adjoint $F_{w,*}$. Indeed, $F_{w,!}$ is the composition
of the functors $F_{s,!}$ corresponding to simple reflections.
Now $F_{s,!}=j^*{\cal F} j_!$ where ${\cal F}$ is the Fourier transform,
and $j$ is an open embedding; hence it is left adjoint
to $j^*{\cal F} j_*$. In fact, it is shown in \cite{KL} that
$F_{w,*}(A)=p_{2,*}(p_1^*A\otimes\overline{K(w)})$.
The Proposition \ref{simple} combined with Theorem \ref{main}
gives a simple description of the Grothendieck group of the
abelian category $\AA$ resulting from gluing
on $G/U$.
As we have seen in Corollary \ref{zer} the subgroup
$K_{sw,w}\subset K_0(\operatorname{Perv}(G/U))$ coincides with the image of
the natural (injective) homomorphism
$p_s^*:K_0(\operatorname{Perv}(G/Q_s))\rightarrow K_0(\operatorname{Perv}(G/U))$. Hence,
we get the following description of $K_0(\AA)$.
\begin{thm}\label{K_0A} The group $K_0(\AA)$ is isomorphic
to the group
$${\cal K}=\{ (c_w)\in\oplus_{w\in W} K_0(\operatorname{Perv}(G/U)) \ | \
\phi_s c_w-c_{sw}\in p_s^*(K_0(\operatorname{Perv}(G/Q_s))), \ w\in W,
s\in S \},$$
where $\phi_s$ is an operator on $K_0(\operatorname{Perv}(G/U))$
induced by the partial Fourier transform $F_{s,!}$.
\end{thm}
\section{Cubic Hecke algebra}
\subsection{A property of the Fourier transform}
Let $\pi:V\rightarrow S$ be a symplectic rank-2 bundle, let
${\cal F}:{\cal D}^b_c(V,\overline{{\Bbb Q}}_l)\rightarrow{\cal D}_c^b(V,\overline{{\Bbb Q}}_l)$ be
the corresponding Fourier
transform, $j:U\rightarrow V$ the complement to the zero section, and
${\cal F}_!=j^*{\cal F} j_!:{\cal D}_c^b(U,\overline{{\Bbb Q}}_l)\rightarrow{\cal D}_c^b(U,\overline{{\Bbb Q}}_l)$.
\begin{lem}\label{mainl}
There is a canonical isomorphism of functors
\begin{equation}\label{shift}
{\cal F}_!\circ p^*\simeq p^*[1](1)
\end{equation}
where $p=\pi\circ j:U\rightarrow S$.
\end{lem}
\noindent {\it Proof} . We start with canonical isomorphisms
\begin{align*}
&{\cal F}\circ s_*\simeq \pi^*[2](1),\\
&{\cal F}\circ\pi^*\simeq s_*[-2](-1)
\end{align*}
where $s:S\rightarrow V$ is the zero section (see \cite{Laumon}).
Now applying ${\cal F}$ to the exact triangle
$$j_!p^*F\rightarrow\pi^*F\rightarrow s_*F\rightarrow\ldots$$
and using the above isomorphisms we obtain the
exact triangle on $V$:
$${\cal F} j_! p^*F\rightarrow s_*F[-2](-1)\rightarrow\pi^*F[2](1)\ldots$$
Restricting to $U\subset V$ we get the required isomorphism.
\qed\vspace{3mm}
\subsection{Cubic relation}
For every scheme $Y$ we denote by $K_0(Y)=K_0({\cal D}_c^b(Y,\overline{{\Bbb Q}}_l))$
the Grothendieck group of the category
${\cal D}_c^b(Y,\overline{{\Bbb Q}}_l)$. We define the action of the algebra
${\Bbb Z}[u,u^{-1}]$ (where $u$ is an indeterminate) on $K_0(Y)$
by setting $u\cdot [F]=[F(-1)]$, where $F\mapsto F(1)$ is the Tate twist.
\begin{prop}
Let $\phi:K_0(U)\rightarrow K_0(U)$ be the operator induced by ${\cal F}_!$.
Then $\phi$ satisfies the
equation
\begin{equation}\label{cubic}
(\phi+u^{-1})(\phi^2-1)=0.
\end{equation}
\end{prop}
\noindent {\it Proof} . From the previous lemma we have
$(\phi+u^{-1})|_{\operatorname{im}(p^*)}=0$.
On the other hand, from Proposition \ref{Fouriersq} we have
$\operatorname{im}(\phi^2-1)\subset\operatorname{im}(p^*)$, hence the assertion.
\qed\vspace{3mm}
\begin{cor}\label{s2-1} Let $R={\Bbb Z}[u,u^{-1},(u^2-1)^{-1}]$. The submodule
$K_0(G/Q_s)\otimes_{{\Bbb Z}[u,u^{-1}]} R\subset K_0(G/U)
\otimes_{{\Bbb Z}[u,u^{-1}]} R$ coincides with the image of the operator
$\phi^2-1$ on the latter $R$-module.
\end{cor}
Let $r:U\rightarrow\P(V)$ be the projection to the projectivization
of $V$, $q:\P(V)\rightarrow S$ the projection to the base, so that
$p=q\circ r$.
\begin{prop}\label{Fourproj} The following relation between operators
$K_0(\P(V))\rightarrow K_0(U)$ holds:
\begin{equation}
\phi r^*= r^* - u^{-1}p^*q_!= r^* (\operatorname{id} - u^{-1}q^*q_!).
\end{equation}
\end{prop}
\noindent {\it Proof} . By definition we have
$${\cal F}_!(r^*G)=p_{2!}(p_1^*r^*G\otimes{\cal L}|_{U^2})[2](1)\simeq
p'_{2!}(p_1^{\prime *}G\otimes (r\times\operatorname{id}_U)_!({\cal L}|_{U^2}))[2](1)$$
where
$p_i$ (resp. $p'_i$) are the projections of $U^2$
(resp. $\P(V)\times U$) on its factors.
Note that $(r\times\operatorname{id}_U)_!({\cal L}|_{U^2})\simeq
((r\times\operatorname{id}_V)_!({\cal L}|_{U\times V}))|_{\P(V)\times U}$.
To compute the latter sheaf we decompose $r$ as follows:
$r=l\circ k$ where $k:U\hookrightarrow I$ is the open embedding into
the incidence correspondence $I\subset\P(V)\times V$
($k:v\mapsto (\langle v\rangle,v)$), $l:I\rightarrow\P(V)$ is the
projection. Let $s:\P(V)\rightarrow \P(V)\times 0\subset I$ be the zero
section. Then since the images of $k$ and $s$ are
complementary to each other we have an exact triangle
\begin{equation}\label{trian}
(k\times\operatorname{id}_V)_!({\cal L}|_{U\times V})\rightarrow\widetilde{{\cal L}}\rightarrow
(s\times\operatorname{id}_V)_*(s\times\operatorname{id}_V)^*\widetilde{{\cal L}}\rightarrow\ldots
\end{equation}
where $\widetilde{{\cal L}}$ is the pullback of ${\cal L}$ by the morphism
$I\times V\rightarrow V\times V$ induced by the projection $I\rightarrow V$.
Now we have $(l\times\operatorname{id}_V)_!(s_*s^*\widetilde{{\cal L}})\simeq
\overline{{\Bbb Q}}_{l,\P(V)\times V}$ and
$(l\times\operatorname{id}_V)_!(\widetilde{{\cal L}})\simeq \overline{{\Bbb Q}}_{l,I}[-2](-1)$.
Indeed, the latter isomorphism follows from the fact that
$(l\times\operatorname{id}_V)_!(\widetilde{{\cal L}})$ is supported on
$I\subset\P(V)\times V$ and the trivialiaty of
$\widetilde{{\cal L}}|_{(l\times\operatorname{id}_V)^{-1}(I)}$.
Applying the functor $(l\times\operatorname{id}_V)_!$ to the triangle
(\ref{trian}) and using these isomorphisms we obtain
the exact triangle
$$(r\times\operatorname{id}_V)_!({\cal L}|_{U\times V})\rightarrow
\overline{{\Bbb Q}}_{l,I}[-2](-1)\rightarrow \overline{{\Bbb Q}}_{l,\P(V)\times V}\rightarrow\ldots$$
Note that the restriction of the projection $I\rightarrow V$ to
$I\cap (\P(V)\times U)$ is an isomorphism. Hence, passing to
Grothendieck groups we obtain
$$[{\cal F}_!r^*G]=[r^*G]-[p^*q_!G(1)]$$
as required.
\qed\vspace{3mm}
\subsection{Action of the cubic Hecke algebra}
Now we return to the situation of section \ref{glu}.
Let ${\cal H}$ be the Hecke algebra defined as the quotient of
the group algebra with coefficients in ${\Bbb Z}[u,u^{-1}]$
where $u$ is indeterminate,
of the generalized braid group corresponding to $(W,S)$ by
the relations $(s+1)(s-u)=0$, $s\in S$.
Recall that there is an action of ${\cal H}$ on $K_0(G/B)$ such
that ${\Bbb Z}[u,u^{-1}]$ acts in the standard way (using Tate twist) and
the action of $s$ is given by the
correspondence $O(s)\subset (G/B)^2$:
$O(s)=\{ (gB,g'B)\ |\ g^{-1}g'\in BsB\}$. In terms of the
projective bundle $q_s:G/B\rightarrow G/M_sB$ associated with $s$
we have $T_s=q_s^*q_{s!}-\operatorname{id}$.
\begin{prop} The functors $F_{s,!}$, $s\in S$ extend to the action on
$K_0(G/U)$ of the cubic Hecke algebra ${\cal H}^c$ which is obtained
as the quotient of the group algebra ${\Bbb Z}[u,u^{-1}][B]$ of the braid group
$B$ corresponding to $(W,S)$ by the relations
$(s+u)(s^2-1)=0$, $s\in S$.
This action preserves $K_0(G/B)\subset K_0(G/U)$
and restricts to the standard action of the quadratic Hecke algebra
${\cal H}$ on $K_0(G/B)$ via the ${\Bbb Z}[u,u^{-1}]$-linear homomorphism
$${\cal H}^c\rightarrow{\cal H}:s\mapsto -s^{-1}.$$
\end{prop}
The proof reduces to a simple computation for the Fourier transform
on a symplectic bundle of rank 2.
\section{Adjoint functors and canonical complexes for $W$-gluing}
\label{complex}
\subsection{Adjoint functors for parabolic subgroups}
Let $(W,S)$ be a finite Coxeter group, $J\subset S$ a subset,
and $W_J\subset W$ the subgroup generated
by simple reflections in $J$.
We will frequently use the following fact
(see \cite{B}, IV, Exercise 1.3): every left
(or right) $W_J$-coset
contains a unique element of minimal length.
Furthermore, an element $w\in W$ is the shortest element in
$W_Jw$ if and only if $l(sw)=l(w)+1$ for every $s\in J$,
and in this case we have $l(w'w)=l(w')+l(w)$ for every
$w'\in W_J$.
In particular, for any coset $W_Jx$ and
any $w\in W$ there exists the unique element $p_{W_Jx}(w)\in W_Jx$
such that $n_{W_Jx}(w):=wp_{W_Jx}(w)^{-1}$ has minimal possible length.
Namely, $n_{W_Jx}(w)$ is the shortest element in the coset
$wx^{-1}W_J$.
Let $({\cal C}_w,w\in W)$ be a collection of abelian
categories, $\Phi_w:{\cal C}_{w'}\rightarrow{\cal C}_{ww'}$,
a $W$-gluing data, and ${\cal C}(\Phi)$ the corresponding
glued category.
For every subset $P\subset W$ let us denote by $\Phi_P$ the
gluing data on categories ${\cal C}_w$, $w\in P$, induced by $\Phi$.
We have natural restriction functors
$j_P^*:{\cal C}(\Phi)\rightarrow{\cal C}(\Phi_P)$.
We claim that for every coset $W_Jx\subset W$
the functor $j^*_{W_Jx}$ has the left adjoint
functor $$j_{W_Jx,!}:{\cal C}(\Phi_{W_Jx})\rightarrow{\cal C}(\Phi).$$
Indeed, let $A=(A_w,w\in W_Jx;\a_{w,w'})$
be an object of ${\cal C}(\Phi_{W_Jx})$.
Set $j_{W_Jx,!}=(A_w,w\in W;\a_{w,w'})$
where $$A_w=\Phi_{n(w)}A_{p(w)},$$
$n(w):=n_{W_Jx}(w)$, $p(w):=p_{W_Jx}(w)$, and
the morphisms $\a_{w,w'}:\Phi_w(A_{w'})\rightarrow A_{ww'}$
are defined as follows. Let us write
$p(ww')=w_1p(w')$ with $w_1\in W_J$. Then we have
$$wn(w')=n(ww')w_1$$
and $l(n(ww')w_1)=l(n(ww'))+l(w_1)$.
Hence, we can define $\a_{w,w'}$ as the composition
$$\Phi_w(A_{w'})=\Phi_w\Phi_{n(w')}(A_{p(w')})\rightarrow
\Phi_{wn(w')}(A_{p(w')})
\simeq\Phi_{n(ww')}\Phi_{w_1}(A_{p(w')})
\setlength{\unitlength}{0.50mm}
\lrar{\Phi_{n(ww')}(\a_{w_1,p(w')})}
\Phi_{n(ww')}(A_{w_1p(w')})=A_{ww'}.
$$
One can easily check that $j_{W_Jx,!}$ is indeed an
object of ${\cal C}(\Phi)$.
\begin{prop}\label{jWJx!}
The functor $j_{W_Jx,!}$ is left adjoint to
$j^*_{W_Jx}$.
\end{prop}
\noindent {\it Proof}. Let $A=(A_w,w\in W_Jx;\a_{w,w'})$ be an object of
${\cal C}(\Phi_{W_Jx})$, and $B=(B_w,w\in W;\b_{w,w'})$ an object
of ${\cal C}(\Phi)$. We have an obvious map
\begin{equation}\label{adjmap}
\operatorname{Hom}_{{\cal C}(\Phi)}(j_{W_Jx,!}A,B)\rightarrow
\operatorname{Hom}_{{\cal C}(\Phi_{W_Jx})}(A,j^*_{W_Jx}B).
\end{equation}
The inverse map is constructed as follows.
Assume that we are given a morphism
$f=(f_w,w\in W_Jx):A\rightarrow j^*_{W_Jx}B$.
Then for every $w\in W$ we define the morphism
$\widetilde{f}_w:\Phi_{n(w)}A_{p(w)}\rightarrow B$
as the composition
$$
\setlength{\unitlength}{0.35mm}
\Phi_{n(w)}A_{p(w)}\lrar{\Phi_{n(w)}(f_{p(w)})}
\Phi_{n(w)}B_{p(w)}\lrar{\b_{n(w),p(w)}} B_w.$$
It is easy to check that $\widetilde{f}=(\widetilde{f}_w,w\in W)$
is the morphism in ${\cal C}(\Phi)$ between $j_{W_Jx,!}A$ and
$B$ and that the obtained map
$$\operatorname{Hom}_{{\cal C}(\Phi_{W_Jx})}(A,j^*_{W_Jx}B)\rightarrow
\operatorname{Hom}_{{\cal C}(\Phi)}(j_{W_Jx,!}A,B):f\mapsto\widetilde{f}$$
is inverse to (\ref{adjmap}).
\qed\vspace{3mm}
If we have an inclusion $W_Jx\subset P\subset W$, then the restriction
functor
$j_{W_Jx,P}^*:{\cal C}(\Phi_P)\rightarrow{\cal C}(\Phi_{W_Jx})$ has the left adjoint
$$j_{W_Jx,P;!}:=j^*_Pj_{W_Jx,!}:{\cal C}(\Phi_{W_Jx})\rightarrow
{\cal C}(\Phi_P)$$
(this follows from the proof of the above proposition).
Moreover, by construction the composition $j_{W_Jx,P}^*j_{W_Jx,P;!}$
is the identity functor on ${\cal C}(\Phi_{W_Jx})$. Hence,
we can apply Theorem \ref{braverman} to conclude that for any subset
$P\subset W$ which is a union of subsets of the form $W_Jx$, the category
${\cal C}(\Phi_P)$ is obtained by gluing from the categories
${\cal C}(\Phi_{W_Jx})$ for $W_Jx\subset P$.
Let $J\subset K\subset S$. Then one has canonical isomorphisms
$$j^*_{W_Jx}\simeq j^*_{W_Jx,W_Kx}\circ j^*_{W_Kx},$$
$$j_{W_Jx,!}\simeq j_{W_Kx,!}\circ j_{W_Jx,W_Kx;!}.$$
Also one has the canonical morphism of functors
\begin{equation}\label{WJK}
j_{W_Jx,!}j^*_{W_Jx}\rightarrow j_{W_Kx,!}j^*_{W_Kx}.
\end{equation}
Assume that every functor $\Phi_w$ has the left derived
$L\Phi_w:{\cal D}^-({\cal C}_w')\rightarrow{\cal D}^-({\cal C}_{ww'})$ and that these functors
satisfy $L\Phi_{w_1}\circ L\Phi_{w_2}\simeq L\Phi_{w_1w_2}$
when $\ell(w_1w_2)=\ell(w_1)+\ell(w_2)$.
The following proposition gives a sufficient condition for the existence of
the derived functor for $j_{W_Jx,!}$.
\begin{prop}\label{leftderived} Assume that for every $w\in W$
there is a family of objects
${\cal R}_w\subset \operatorname{Ob}{\cal C}_w$ that are $\Phi_{w'}$-acyclic for every
$w'\in W$ and such that every object of ${\cal C}_w$
can be covered by an object in ${\cal R}_w$. Then the
functor $j_{W_Jx,!}$ has the left derived
$Lj_{W_Jx,!}:{\cal D}^-({\cal C}(\Phi_{W_Jx}))\rightarrow{\cal D}^-({\cal C}(\Phi))$ which
is left adjoint to the restriction functor
$j^*_{W_Jx}:{\cal D}^-({\cal C}(\Phi))\rightarrow{\cal D}^-({\cal C}(\Phi_{W_Jx}))$.
Furthermore, one has an isomorphism of functors
$$j^*_w\circ Lj_{W_Jx,!}\simeq L\Phi_{n(w)}j^*_{p(w)}.$$
\end{prop}
\noindent {\it Proof} . Let ${\cal R}\subset\operatorname{Ob}{\cal C}(\Phi_{W_Jx})$ be the family
of objects that are direct sums of objects of the form
$j_{y,W_Jx;!}R_y$ where $y\in W_Jx$, $R_y\in{\cal R}_y$.
Clearly, every object of ${\cal C}(\Phi_{W_Jx})$ can be covered by
an object in ${\cal R}$.
We claim that ${\cal R}$ is an adapted class of objects for the functor
$j_{W_Jx,!}$. Indeed, it suffices to prove that for every $w\in W$ the
object $\Phi_{p(w)y^{-1}}R_y$ is $\Phi_{n(w)}$-acyclic, where
$w=n(w)p(w)$ is the decomposition used in the definition of $j_{W_Jx,!}$.
Now we use the fact that
$$\ell(wy^{-1})=\ell(n(w)p(w)y^{-1})=\ell(n(w))+\ell(p(w)y^{-1}).$$
Therefore, by our assumption
$$\Phi_{wy^{-1}}R_y=L\Phi_{wy^{-1}}R_y\simeq L\Phi_{n(w)}\circ
L\Phi_{p(w)y^{-1}}R_y=L\Phi_{n(w)}(\Phi_{p(w)y^{-1}}R_y)$$
and our claim follows. Thus, the left derived
functor for $j_{W_Jx,!}$ exists and can be computed
using resolutions in ${\cal R}$. The remaining assertions
can be easily checked using such resolutions.
\qed\vspace{3mm}
Note that Theorem \ref{adapted} implies that the conditions
of the previous proposition are satisfied for gluing on the
basic affine space.
\subsection{Canonical complex}
Let us fix a complete order on $S$: $S=\{s_1,\ldots,s_n\}$.
Given an object $A\in{\cal C}(\Phi)$ we construct a homological
coefficient system on the $(n-1)$-simplex $\Delta_{n-1}$ with values in
${\cal C}(\Phi)$. Namely, to a subset
$J=\{i_1<\ldots<i_k\}\subset [1,n]$ we assign the
object
$$A(J)=\oplus j_{W_{S-J}x,!}j^*_{W_{S-J}x}A$$
where the sum is taken over all right $W_{S-J}$-cosets.
For every inclusion $J\subset J'$ we have the canonical morphism
$A(J')\rightarrow A(J)$ with components (\ref{WJK}).
Thus, we can consider the corresponding chain complex
\begin{equation}
C_{\cdot}(A):C_{n-1}=A([1,n])\rightarrow\ldots\rightarrow
C_1=\oplus_{|J|=2}A(J)\rightarrow C_0=\oplus_{|J|=1}A(J)
\end{equation}
The sum of adjunction morphisms
$$C_0(A)=\oplus_{|J|=n-1,W_Jx\subset W}j_{W_Jx,!}j^*_{W_Jx}A\rightarrow
A$$
induces the morphism
\begin{equation}\label{H0}
H_0(C_{\cdot}(A))\rightarrow A.
\end{equation}
It turns out that $C_{\cdot}(A)$ is almost a resolution of
$A$. To describe its homology we need to introduce the
functor $\iota:{\cal C}(\Phi)\rightarrow{\cal C}(\Phi)$. For $A=(A_w;\a_{w,w'})$
we set $\iota A=(\Phi_{w_0}A_{w_0w},\widetilde{\a}_{w,w'})$
where $w_0$ is the longest element in $W$, the morphism
$\widetilde{\a}_{w,w'}:\Phi_w\Phi_{w_0}A_{w_0w'}\rightarrow
\Phi_{w_0}A_{w_0ww'}$ is equal to the composition
$$\Phi_w\Phi_{w_0}A_{w_0w'}\simeq
\Phi_{w_0}\Phi_{w_0ww_0}A_{w_0w'}
\setlength{\unitlength}{0.50mm}
\lrar{\Phi_{w_0}(\a_{w_0ww_0,w_0w'})}\Phi_{w_0}A_{w_0ww'}.$$
Here we used the following identity in the braid group:
$$\tau(w)\tau(w_0)=\tau(w)\tau(w^{-1}w_0)\tau(w_0ww_0)=
\tau(w_0)\tau(w_0ww_0).$$
Note that for every $y\in W$ we have the natural morphism
$$\a_y:\iota A\rightarrow j_{y,!}j^*_yA$$
with components
$$\Phi_{wy^{-1}}(\a_{yw^{-1}w_0,w_0w}):\Phi_{w_0}A_{w_0w}\rightarrow
\Phi_{wy^{-1}}A_y.$$
It is easy to see that one has the canonical morphism
\begin{equation}\label{Hn-1}
\iota A\rightarrow H_{n-1}(C_{\cdot}(A)),
\end{equation}
induced by the morphism
$$\iota A\lrar{((-1)^{l(y)}\a_y)} C_{n-1}(A)=\oplus_{y\in W}
j_{y,!}j^*_yA.$$
\begin{thm}\label{homology}
One has $H_i(C_{\cdot}(A))=0$ for $i\neq 0,n-1$,
$H_0(C_{\cdot}(A))\simeq A$, and $H_{n-1}(C_{\cdot}(A))\simeq\iota A$.
\end{thm}
\noindent {\it Proof}. Let us consider the complex
$\widetilde{C}_{\cdot}(A)$
obtained from $C_{\cdot}$ by adding the terms
$\widetilde{C}_n=\iota A$ and $\widetilde{C}_{-1}=A$ with additional differentials
induced by (\ref{H0}) and (\ref{Hn-1}).
By definition for every $w\in W$ the complex
$j_w^*\widetilde{C}_{\cdot}(A)$ looks as follows:
\begin{align*}
&\Phi_{w_0}A_{w_0w}\rightarrow\oplus_{x\in W}\Phi_{wx^{-1}}A_x\rightarrow
\oplus_{|J|=1,x\in W_J\backslash W}
\Phi_{n_{W_Jx}(w)}A_{p_{W_Jx}(w)}\rightarrow\ldots\\
&\rightarrow\oplus_{|J|=n-1,x\in W_J\backslash W}
\Phi_{n_{W_Jx}(w)}A_{p_{W_Jx}(w)}\rightarrow A_w.
\end{align*}
Recall that as $x$ runs through the set of cosets
$W_J\backslash W$,
the element $n_{W_Jx}(w)$ runs through all the elements
$y\in W$ such that $l(ys)=l(y)+1$ for all $s\in J$.
Let us denote by $P_j\subset W$ the set of all $y\in W$
such that $l(ys_j)=l(y)+1$.
For $J\subset [1,n]$ we denote $P_J=\cap_{j\in J} P_j$.
Then we can rewrite the complex $j^*_w\widetilde{C}_{\cdot}(A)$
as
\begin{align*}
&\Phi_{w_0}A_{w_0w}\rightarrow\oplus_{y\in W}\Phi_{y}A_{y^{-1}w}\rightarrow
\oplus_{j,y\in P_j}\Phi_yA_{y^{-1}w}\rightarrow
\ldots\\
&\rightarrow\oplus_{|J|=n-1,y\in P_J}\Phi_{y}A_{y^{-1}w}\rightarrow A_w.
\end{align*}
Consider the increasing filtration $F_0\subset F_1\subset\ldots$
on $j^*_w\widetilde{C}_{\cdot}(A)$ such that
$F_n$ contains only summands $\Phi_y A_{y^{-1}w}$ with
$l(y)\le n$. Then the differentials in
$\operatorname{gr}_{F}j^*_w\widetilde{C}_{\cdot}(A)$ are only $\pm\operatorname{id}$ or zeroes.
Note that $P_S=\{1\}$ while
$\cup_{j=1}^n P_j=W-\{w_0\}$.
Hence the complex $\operatorname{gr}_F j_w^*\widetilde{C}_{\cdot}(A)$ is acyclic,
and so is $\widetilde{C}_{\cdot}(A)$.
\qed\vspace{3mm}
Note that we can attach the end of the complex
$C_{\cdot}(\iota A)$ to the beginning of the complex
$C_{\cdot}(A)$ via the map
$$C_0(\iota A)\rightarrow \iota A\rightarrow C_{n-1}(A)$$
to get the complex
$\widetilde{C}=\widetilde{C}_{\cdot}(A)$ with homologies
$H_0(\widetilde{C})=A$ and $H_{2n-1}(\widetilde{C})=\iota^2(A)$ (all other homologies
vahish).
The functor $\iota^2$ sends the object
$A=(A_w,\a_{w,w'})\in{\cal C}(\Phi)$ to the object
$(\Phi_{\pi} A_w, \Phi_{\pi}(\a_{w,w'}))$
where $\pi=\tau(w_0)^2\in B$ is the canonical central element.
The importance of the above construction is that the members
of the complex $C_{\cdot}$ are direct sums of objects of the
form $j_{W_Jx,!}(\cdot)$ where $J\subset S$ is a proper subset.
Hence, it can be used for the induction process.
For example, suppose we know that for all
proper subsets $J\subset S$ the categories ${\cal C}(\Phi_{W_Jx})$
have finite cohomological dimension (this is true for
the gluing on the basic affine space if the rank of $G$ is equal to $2$).
The derived category
version of the above construction gives a canonical
morphism of functors $\operatorname{Id}\rightarrow\Phi_{\pi}[2n]$. Now
an object $A\in{\cal C}(\Phi)$ has finite projective dimension
if and only if some power of the morphism
$A\rightarrow\Phi_{\pi}^k(A)[2nk]$ vanishes.
\subsection{Canonical complex for a ``half" of $W$}
Let us fix an element $s_i\in S$. Recall that we denote
by $P_i$ the subset of $w\in W$ such that $l(ws_i)>l(w)$.
One has a decomposition of $W$ into the disjoint union
of $P_i$ and $P_is_i$. Let us consider gluing data
$\Phi_{P_i}$ and $\Phi_{P_is_i}$
corresponding to these two pieces.
We are going to construct an analogue of the complex
$C_{\cdot}$ for these partial gluing data.
As before, for every $A\in{\cal C}(\Phi_{P_i})$
we can define a homological
coefficient system on $\Delta_{n-1}$ with values in
${\cal C}(\Phi_{P_i})$ as follows.
For every $j\in [1,n]$ let us denote
$W^{(j)}=W_{[1,n]-j}$.
Now to a nonempty subset
$J=\{i_1<\ldots<i_k\}\subset [1,n]$ our coefficient system assigns the
object
$$A(J)=\oplus j_{W_{S-J}x,P_i;!}
j^*_{W_{S-J}x,P_i}A$$
where the sum is taken over all
$x\in W_{S-J}\backslash W$ such that
$W^{(j)}x\subset P_i$ for every $j\in J$.
For every inclusion $J\subset J'$ we have the canonical morphism
$A(J')\rightarrow A(J)$. Let us consider the corresponding chain complex
\begin{equation}\label{chaincomplex}
C_{\cdot}(P_i,A):C_{n-1}=A([1,n])\rightarrow\ldots\rightarrow
C_1=\oplus_{|J|=2}A(J)\rightarrow C_0=\oplus_{|J|=1}A(J).
\end{equation}
\begin{thm}\label{complexPi} One has $H_0(C_{\cdot}(P_i,A))\simeq A$
and $H_j(C_{\cdot}(P_i,A))=0$ for $j\neq 0$.
\end{thm}
The proof of this theorem will be given in \ref{proofPi}.
\subsection{Homological lemma}
Let $T$ be a finite set, and $(T^{1}_i)$ and $(T^2_i)$
two families of subsets of $T$ indexed by $i\in [1,n]$.
Assume that we have a family $(B_t)$ of objects of some
additive category indexed by $t\in T$. Then we can construct
a homology coefficient system on $\Delta_{n-1}$ by setting
$B(J)=\oplus_{t\in T(J)}B_t$ where
$$T(J)=\cap_{j\in J}T^1_j\cap\cap_{j\in\overline{J}}T^2_j,$$
$J\subset [1,n]$ is a subset, $\overline{J}$ is the complementary
subset. For $J\subset K$ we have the natural map
$B(K)\rightarrow B(J)$ which is the following composition of the projection
and the embedding:
$$\oplus_{t\in T(K)}B_t\rightarrow\oplus_{t\in T(J)\cap T(K)}B_t\rightarrow
\oplus_{t\in T(J)}B_t.$$
Let $D_{\cdot}=C_{\cdot}(\Delta_{n-1},B(\cdot))$ be the corresponding
chain complex.
\begin{lem}\label{homlem} Assume that for every $t\in T$ the sets
$I^1(t)=\{i\ |\ t\in T^1_i\}$ and $I^2(t)=\{i\ |\ t\not\in T^2_i\}$
are different. Then $H_i(D_{\cdot})=0$ for $i>0$,
$H_0(D_{\cdot})\simeq B(\emptyset)=\oplus_{t\in T(\emptyset)}B_t$
where $T(\emptyset)=\cap_{j=1}^nT^2_j$.
\end{lem}
\noindent {\it Proof} . Let $\widetilde{D}_{\cdot}$ be the complex
$$B([1,n])\rightarrow\oplus_{|J|=n-1}B(J)\rightarrow\ldots\rightarrow
\oplus_{|J|=1}B(J)\rightarrow B(\emptyset)$$
obtained from $D_{\cdot}$ by attaching one more term
$B(\emptyset)$. Then $\widetilde{{\cal D}}_{\cdot}$ is the direct sum
over $t\in T$ of the complexes $\widetilde{D}(t)$ where
$\widetilde{D}_i(t)=\oplus_{J:t\in T(J),|J|=i}B_t$.
Note that $t\in T(J)$ if and only $I^2(t)\subset J\subset I^1(t)$.
Hence, the condition $I^1(t)\neq I^2(t)$ implies
that the complex $\widetilde{D}_i(t)$ is exact.
\qed\vspace{3mm}
\subsection{Convexity}
\begin{lem}\label{W'Pi}
Let $W'=W_J\subset W$ for some $J\subset S$,
and let $y\in P_i$ be an element. Then $W'y\subset P_i$ if and only if
$ys_iy^{-1}\not\in W'$.
\end{lem}
\noindent {\it Proof}. The second condition
is equivalent to the requirement that the cosets $W'y$ and
$W'ys_i$ are different. This is in turn equivalent
to the condition that the double coset
$W'y\langle s_i\rangle$, where $\langle s_i\rangle=\{1,s_i\}$,
has $2|W'|$ elements. Let $y_0$ be the shortest
element in this double coset (see \cite{B}, IV, Exercise 1.3).
Then every element in $W'y\langle s_i\rangle$ can be written
uniquely in the form $w_1y_0w_2$ where $w_1\in W'$, $w_2\in \langle
s_i\rangle$ and $l(w_1y_0w_2)=l(w_1)+l(y_0)+l(w_2)$. Since there
are $2|W'|$ such expressions this gives a bijection between
$W'\times\langle s_i\rangle$ and $W'y\langle s_i\rangle$.
The condition $y\in P_i$ implies that
$y=w'y_0$ for some $w'\in W'$, i.~e. $W'y=W'y_0$. But for
every $w_1\in W'$ one has $l(w_1y_0s_i)=l(w_1y_0)+1$, hence
$W'y_0\subset P_i$.
\qed\vspace{3mm}
\begin{lem}\label{appear}
For every $y\in P_i$ the set of $j\in [1,n]$ such that
$W^{(j)}y\subset P_i$ coincides with the set of $j$ such that
$s_j$ appears in a reduced decomposition of $ys_iy^{-1}$.
\end{lem}
\noindent {\it Proof} . Applying Lemma \ref{W'Pi}
to $W'=W^{(j)}$ we obtain that $W^{(j)}y\subset P_i$ if and
only if $ys_iy^{-1}\not\in W^{(j)}$. The latter condition
means that $s_j$ appears in every (or some) reduced decomposition
of $ys_iy^{-1}$.
\qed\vspace{3mm}
Consider the graph with vertices $W$ and
edges between $w$ and $sw$ for every $w\in W$, $s\in S$.
To give a path from $w_1$ to $w_2$ in this graph is
the same as giving a decomposition of $w_2w_1^{-1}$
into a product of simple reflections. The shortest paths,
{\it geodesics}, correspond to reduced decompositions.
Let us call a subset $P\subset W$ convex if every geodesic
between vertices in $P$ lies entirely in $P$.
\begin{prop}\label{convex}
The subset $P_i$ is convex.
\end{prop}
\noindent {\it Proof} . First let us show that for every pair of elements
$y,w\in P_i$ there exists a geodesic from $y$ to $w$ which
lies entirely in $P_i$. Arguing by induction in $l(wy^{-1})$,
we see that it is sufficient to check that there exists
$s_k\in S$ such that either $s_ky\in P_i$ and
$l(wy^{-1}s_k)<l(wy^{-1})$, or $ws_k\in P_i$ and
$l(s_kwy^{-1})<l(wy^{-1})$. Assume that there is no such
$s_k$. Note that by Lemma \ref{W'Pi} the condition $s_ky\in P_i$
is equivalent to $ys_iy^{-1}\neq s_k$. In particular,
this condition fails to be true for at most one $k$.
Thus, for $w\neq y$ our assumption implies that
$ys_iy^{-1}=s_k$ for some $k$, $ws_iw^{-1}=s_l$ for some $l$
and $l(wy^{-1}s_j)>l(wy^{-1})$ for all $j\neq k$,
$l(s_jwy^{-1})>l(wy^{-1})$ for all $j\neq l$. Let us set
$x=wy^{-1}\in W$. Then $xs_kx^{-1}=s_l$ and $x$ is the
shortest element in $W^{(l)}xW^{(k)}$. Using Lemma \ref{sizig3}
one can easily see that this is impossible.
Note that every two geodesics with common ends are connected
by a sequence of transformations of the following kind:
replace the segment
\begin{equation}\label{seg1}
y\rightarrow s_j y\rightarrow s_k s_j y\rightarrow\ldots\rightarrow w_0(s_j,s_k)y
\end{equation}
by the segment
\begin{equation}\label{seg2}
y\rightarrow s_k y\rightarrow s_j s_k y\rightarrow\ldots\rightarrow w_0(s_j,s_k)y
\end{equation}
where $w_0(s_i,s_j)$ is the longest element in the subgroup
generated by $s_i$ and $s_j$. Thus, it is sufficient to check
that if segment (\ref{seg1}) lies in $P_i$, then the corresponding
segment (\ref{seg2}) does as well. Applying Lemma \ref{W'Pi} we see
that for $y\in P_i$ the segment (\ref{seg1}) lies in $P_i$
if and only if
\begin{equation}\label{ysi}
ys_iy^{-1}\not\in\{s_j, s_ks_js_k, s_js_ks_js_ks_j,
\ldots\}.
\end{equation}
Now if $l(w_0(s_i,s_j))$ is odd, then the latter set
remains the same if we switch $s_j$ and $s_k$. Otherwise
(if $l(w_0(s_i,s_j))$ is even)
the path inverse to segment (\ref{seg2}) has the form
$$w_0(s_j,s_k)\rightarrow s_j w_0(s_j,s_k)\rightarrow\ldots\sigma_k y\rightarrow y.$$
Hence, the conditions $w_0(s_j,s_k)\in P_i$ and (\ref{ysi})
imply that this segment lies in $P_i$.
\qed\vspace{3mm}
\subsection{Proof of Theorem \ref{complexPi}}
\label{proofPi}
Let us denote by $Q_{i,j}$ the set of $y\in P_i$ such
that $W^{(j)}y\subset P_i$. Then for any $w\in P_i$ the complex
$j^*_wC_{\cdot}(P_i,A)$ can be written as
\begin{equation}
\oplus_{x\in Q(w,[1,n])}\Phi_{wx^{-1}}A_x\rightarrow\ldots
\rightarrow\oplus_{|J|=2, x\in Q(w,J)}
\Phi_{wx^{-1}}A_x
\rightarrow\oplus_{|J|=1, x\in Q(w,J)}
\Phi_{wx^{-1}}A_x
\end{equation}
where
$Q(w,J)=\cap_{j\in J} Q_{i,j}\cap\cap_{k\in \overline{J}}P_k^{-1}w$,
$A_x=j_x^*A$.
We can filter this complex by the length of $wx^{-1}$ as in the
Proof of Theorem \ref{homology}. Now the associated graded factor
$\operatorname{gr}_Fj^*_wC_{\cdot}(P_i,A)$ is the complex
of Lemma \ref{homlem} with $T=P_i$,
$T^1_j=Q_{i,j}$ and $T^2_j=P_i\cap P_j^{-1}w$.
Since $\cap_{j=1}^n P_k^{-1}w=\{ w\}$ it remains to show
that the conditions of Lemma \ref{homlem} are satisfied in
our situation. Indeed, assume that the set of $j$
such that $W^{(j)}y\subset P_i$ coincides with the set of $j$
such that $wy^{-1}\not\in P_j$. Let us denote this set by
$S_1\subset [1,n]$. Then for every $j\in S_1$ we have
$l(wy^{-1}s_j)<l(wy^{-1})$. Therefore,
$l(w_0wy^{-1}s_j)>l(w_0wy^{-1})$ for $j\in S_1$, where
$w_0\in W$ is the longest element. This means that
$w_0wy^{-1}$ is the shortest element in the coset
$w_0wy^{-1}W_{S_1}$. Hence, $l(w_0wy^{-1}w_1)=
l(w_0wy^{-1})+l(w_1)$ for every $w_1\in W_{S_1}$.
This can be rewritten as $l(wy^{-1}w_1)=l(wy^{-1})-l(w_1)$
for every $w_1\in W_{S_1}$. In other words, for every
$w_1\in W_{S_1}$ there exists a geodisic from $y$ to $w$
passing through $w_1y$. According to Lemma \ref{convex}
this implies that $w_1y\in P_i$ for every $w_1\in W_{S_1}$.
Now by Lemma \ref{appear}
we have $ys_iy^{-1}\in W_{S_1}$. Thus, taking
$w_1=ys_iy^{-1}$ we obtain that $ys_i\in P_i$ ---
contradiction.
\qed\vspace{3mm}
\subsection{Comparison with Beilinson---Drinfeld gluing}
For every nonempty subset $J\subset [1,n]$
let us denote
${\cal C}_J=\oplus_{\cal C}(\Phi_{W_{S-J}x})$
where the sum is taken over
$x\in W_{S-J}x\backslash W$ such that
$W^{(j)}x\subset P_i$ for every $j\in J$. Then
$({\cal C}_J)$ is a family of abelian categories cofibered
over the category of nonempty subsets $J\subset [1,n]$
and their embeddings. Namely, if $J\subset K\subset [1,n]$,
then we have obvious restriction functors
$j^*_{J,K}:{\cal C}_J\rightarrow{\cal C}_K$. Following Beilinson and Drinfeld \cite{Beil}
we denote by ${\cal C}_{\operatorname{tot}}$ the category of
cocartesian sections of $({\cal C}_J)$. An object of
${\cal C}_{\operatorname{tot}}$ is a collection of objects $A_J\in{\cal C}_J$
and isomorphisms $\a_{J,K}:A_K\widetilde{\rightarrow}j^*_{J,K}A_J$ for
$J\subset K$, such that
for $J\subset K\subset L$ one has
\begin{equation}\label{aJKL}
\a_{J,L}=j^*_{K,L}\a_{J,K}\circ \a_{K,L}.
\end{equation}
For every $J\subset [1,n]$ we have an obvious restriction functor
$j^*_J:{\cal C}(\Phi_{P_i})\rightarrow{\cal C}_J$ and these functors fit
together into the functor $(j^*_{\cdot}):{\cal C}(\Phi_{P_i})\rightarrow{\cal C}_{\operatorname{tot}}$.
\begin{thm} The functor
$(j^*_{\cdot}):{\cal C}(\Phi_{P_i})\rightarrow{\cal C}_{\operatorname{tot}}$ is an
equivalence of categories.
\end{thm}
\noindent {\it Proof} . For every $J\subset [1,n]$ we have the left adjoint
functor $j_{J,!}:{\cal C}_J\rightarrow{\cal C}(\Phi_{P_i})$ to
$j^*_J$. Namely, $j_{J,!}$
is equal to $j_{W_Jx,P_i;!}$ on
${\cal C}(\Phi_{W_{S-J}x})\subset {\cal C}_J$.
Now for every $(A_J,\a_{J,K})\in {\cal C}_{\operatorname{tot}}$
the objects $j_{J,!}A_J$ form a homological coefficient system
on $\Delta_{n-1}$, so we can consider the corresponding
chain complex ${\cal C}_{\cdot}(A_{\cdot})$. We claim that the
functor
$$(A_J)\mapsto H_0({\cal C}_{\cdot}(A_{\cdot}))$$
is quasi-inverse to $(j^*_{\cdot})$.
Notice that for $A\in{\cal C}(\Phi_{P_i})$ the complex
${\cal C}_{\cdot}(j^*_{\cdot}A)$ coincides with the complex
(\ref{chaincomplex}), hence by
Theorem \ref{complexPi} it is a resolution of $A$.
It remains to show that for $(A_{\cdot})\in{\cal C}_{\operatorname{tot}}$ and
for every $J\subset [1,n]$ there is a system of compatible
isomorphisms
$$H_0(j^*_J{\cal C}_{\cdot}(A_{\cdot}))\simeq A_J.$$
It is sufficient to construct
canonical isomorphisms
$\a_k:H_0(j^*_{\{k\}}({\cal C}_{\cdot}(A_{\cdot})))\widetilde{\rightarrow} A_{\{k\}}$
for every $k\in [1,n]$.
These morphisms are induced by the canonical
projections ${\cal C}_0(A_{\cdot})\rightarrow A_{\{k\}}$. To check that
$\a_k$ are isomorphisms it is sufficient to restrict everything
by some functor $j^*_w$ and to apply arguments from the proof of Theorem
\ref{complexPi}.
\qed\vspace{3mm}
\subsection{Cohomological dimension}
Following \cite{Beil} let us consider the
category $\operatorname{sec}_{-}=\operatorname{sec}_{-}({\cal C}_{\cdot})$ whose objects are
collections $A_J\in{\cal C}_J$, $J\subset [1,n]$,
$\a_{J,K}:A_K\rightarrow j_{J,K}^*A_J$ for $J\subset K$
satisfying (\ref{aJKL}). We consider ${\cal C}_{\operatorname{tot}}$ as the
subcategory in $\operatorname{sec}_{-}$ and denote by ${\cal D}^b_{\operatorname{tot}}$ the
full subcategory in the derived category
${\cal D}^b(\operatorname{sec}_{-})$ consisting of complexes $C^{\cdot}$
with $H^i(C^{\cdot})\in{\cal C}_{\operatorname{tot}}$.
The standard $t$-structure on ${\cal D}^b(\operatorname{sec}_{-})$ induces
a $t$-structure on ${\cal D}^b_{\operatorname{tot}}$ with core ${\cal C}_{\operatorname{tot}}$.
As shown in \cite{Beil} for every $M,N\in{\cal C}_{\operatorname{tot}}$
there is a spectral sequence converging to
$\operatorname{Ext}^{p+q}_{{\cal D}_{\operatorname{tot}}}(M,N)$ with
$E_1^{p,q}=\oplus_{|J|=p+1}\operatorname{Ext}^q_{{\cal C}_J}(M_J,N_J)$.
\begin{thm}\label{anBe}
Assume that every object of ${\cal C}_w$ can be
covered by an object which is acyclic with respect to
all the functors $\Phi_{w'}$.
Then for every $M,N\in{\cal C}_{\operatorname{tot}}$ the natural
map $\operatorname{Ext}^i_{{\cal C}_{\operatorname{tot}}}(M,N)\rightarrow\operatorname{Ext}^i_{{\cal D}_{\operatorname{tot}}}(M,N)$
is an isomorphism.
\end{thm}
\noindent {\it Proof} . It is sufficient to prove that for every element
$e\in\operatorname{Ext}^i_{{\cal D}_{\operatorname{tot}}}(M,N)$ there exists a surjection
$M'\rightarrow M$ in ${\cal C}_{\operatorname{tot}}$ such that
the corresponding homomorphism $\operatorname{Ext}^i_{{\cal D}_{\operatorname{tot}}}(M,N)\rightarrow
\operatorname{Ext}^i_{{\cal D}_{\operatorname{tot}}}(M',N)$ sends $e$ to zero.
Note that for every $w\in P_i$ the functor
$$j_{w,P_i;!}:{\cal C}_w\rightarrow{\cal C}(\Phi_{P_i})\simeq{\cal C}_{\operatorname{tot}}
\rightarrow\operatorname{sec}_{-}$$
is left adjoint to the restriction functor
$j_w^*:\operatorname{sec}_{-}\rightarrow{\cal C}_w$. Furthermore, if an object
$P\in{\cal C}_w$ is acyclic with respect to all the functors
$\Phi_{w'}$, then it is also $j_{w,P_i;!}$-acyclic.
Thus, for such an object $P$ we have
$\operatorname{Ext}^i_{{\cal D}_{\operatorname{tot}}}(j_{w,P_i;!}P,B)\simeq
\operatorname{Ext}^i_{{\cal C}_w}(P,j^*_wB)$. Thus, we can start by
choosing surjections $P_w\rightarrow j^*_wA$ such that
$P_w$ is acyclic with respect to all $\Phi_{w'}$
Then for every $w$ we can choose a surjection
$P'_w\rightarrow P_w$ killing the image of $e$
in the space $\operatorname{Ext}^i_{{\cal C}_w}(P_w,j^*_wB)$,
and take
$M'=\oplus_w j_{w,P_i;!}P'_w$ with the natural morphism
$M'\rightarrow \oplus_w j_{w,P_i;!}j^*_wM\rightarrow M$.
\qed\vspace{3mm}
\begin{cor} Assume that the conditions of Theorem \ref{anBe}
are satisfied and in addition all categories ${\cal C}(\Phi_{W_Jx})$,
where $J\subset [1,n]$ is a proper subset, have
finite cohomological dimension (this is true for the gluing
on the basic affine space if the rank of $G$ is equal to $2$).
Then the category ${\cal C}(\Phi_{P_i})$ also
has finite cohomological dimension.
\end{cor}
\subsection{Gluing from two halves}
Theorem \ref{complexPi} also implies that
the restriction functor
$j_{P_i}^*:{\cal C}(\Phi)\rightarrow{\cal C}(\Phi_{P_i})$ has the left adjoint.
Namely,
for every $A\in{\cal C}(\Phi_{P_i})$ we have
$$A=\operatorname{coker}
(\oplus_{|J|=2}A(J)\rightarrow \oplus_j A(j)).$$
Hence, for any $B\in{\cal C}(\Phi)$ we have
$$\operatorname{Hom}(A,j^*_{P_i}B)\simeq
\ker(\oplus_j\operatorname{Hom}(A(j),j^*_{P_i}B)\rightarrow\oplus_{|J|=2}
\operatorname{Hom}(A(J),j^*_{P_i}B)).$$
Now by adjunction we have
$$\operatorname{Hom}(A(J),j^*_{P_i}B)\simeq\oplus
\operatorname{Hom}(j^*_{W_{S-J}x,P_i}A,j^*_{W_{S-J}x}B)
\simeq\oplus
\operatorname{Hom}(j_{W_{S-J}x,!}j^*_{W_{S-J}x,P_i}A,B)
$$
where the sum is taken over $x\in W_{S-J}\backslash W$
such that $W^{(j)}x\subset P_i$ for every $j\in J$.
It follows that we have an isomorphism
\begin{align*}
\operatorname{Hom}(A,j^*_{P_i}B)\simeq
&\ker(\operatorname{Hom}(\oplus_{j,x} j_{W^{(j)}x,!}j^*_{W^{(j)}x,P_i}A,B)\rightarrow
\operatorname{Hom}(\oplus_{|J|=2,x}
j_{W_{S-J}x,!}j^*_{W_{S-J}x,P_i}A,B))\\
&\simeq \operatorname{Hom}(j_{P_i,!}A,B)
\end{align*}
where
$$j_{P_i,!}A=\operatorname{coker}(
j_{W_{S-J}x,!}j^*_{W_{S-J}x,P_i}A\rightarrow
\oplus_{j,x} j_{W^{(j)}x,!}j^*_{W^{(j)}x,P_i}A).$$
Theorem \ref{complexPi} and the above construction work
almost literally for the subset $P_is_i\subset W$ instead
of $P_i$. Now we can consider the functors
$j^*_{P_is_i}j_{P_i,!}$ and $j^*_{P_i}j_{P_is_i,!}$
as gluing data on the pair of categories
${\cal C}(\Phi_{P_i})$ and ${\cal C}(\Phi_{P_is_i})$. Theorem \ref{braverman}
implies that the corresponding glued category is equivalent
to ${\cal C}(\Phi)$.
\subsection{Supports of simple objects}
Now we are going to apply the explicit construction of the adjoint functors
corresponding to cosets of parabolic subgroups in $W$ to the study of
simple objects in the glued category.
\begin{prop}\label{supp1}
Let $P\subset W$ be a subset,
$S$ a simple object of ${\cal C}(\Phi_P)$,
and $x\in P$ an element such that $s_ix\in P$ for some simple
reflection $s_i$. Assume that $S_x=0$ and $S_{s_ix}\neq0$.
Then $S_w=0$ for every $w\in P\cap P_ix$.
\end{prop}
\noindent {\it Proof} . Let $W_i\subset W$ be the subgroup generated by $s_i$. Our
assumptions imply that $j^*_{W_ix,P}S\neq 0$, hence the
adjunction morphism
$$A=j_{W_ix,P;!}j^*_{W_ix,P}S\rightarrow S$$
is surjective. Now since $S_x=0$, the explicit construction of
$j_{W_ix,P,!}$ tells us that the $A_w=0$ if $w$ is closer to $x$
than to $s_ix$. The latter condition is equivalent to $wx^{-1}\in P_i$.
By surjectivity for such $w$ we also have $S_w=0$.
\qed\vspace{3mm}
For every object $A=(A_w,w\in W)$ of ${\cal C}(\Phi)$ we
denote by $\operatorname{Supp}(A)$ the set of $w\in W$ such that $A_w\neq 0$.
The above proposition gives serious restrictions on a subset $\operatorname{Supp}(S)$
for a simple object $S\in{\cal C}(\Phi)$.
\begin{thm}\label{supp2} Let $S$ be a simple object of ${\cal C}(\Phi)$.
Then either $\operatorname{Supp}(S)=W$ or $\operatorname{Supp}(S)$ is an intersection of
subsets of the form $P_ix$. In particular, if $\operatorname{Supp}(S)\neq W$,
then $\operatorname{Supp}(S)$ is convex.
\end{thm}
\noindent {\it Proof} . Note that an intersection of
subsets of the form $P_ix$ is convex by Lemma \ref{convex}.
Thus, it suffices to prove that $W-\operatorname{Supp}(S)$ is a union of
subsets of the form $P_ix$. Let $w\not\in\operatorname{Supp}(S)$. Choose a
geodesic path
$$w\rightarrow s_{i_1}w\rightarrow\ldots s_{i_k}\ldots s_{i_1}w=x$$
of minimal length from $w$ to an element $x$ of $\operatorname{Supp}(S)$. Then we can
apply Proposition \ref{supp1} to $x'=s_{i_k}x$ to conclude that
$P_{i_k}x'\subset W-\operatorname{Supp}(S)$. On the other hand,
clearly $w\in P_{i_k}x'$.
\qed\vspace{3mm}
\section{Extensions in the glued category}
\subsection{Adapted objects and finiteness of dimensions}
The proof of the following theorem is due to L.~Positselski.
\begin{thm}\label{adapted}
Let $k:V\hookrightarrow X$ be an affine open subset
such that the projection $G\rightarrow X$ splits over $V$.
Then the functors $F_{w,!}k_!$ and $F_{w,*}k_*$ are $t$-exact.
\end{thm}
\noindent {\it Proof} . Since $k_!$ is $t$-exact and $F_{w,!}$ is $t$-exact
from the right, it is sufficient to prove that
$F_{w,!}k_!$ is $t$-exact from the left. Now by definition
the functor $F_{w,!}$ is given by the kernel
$\overline{K(w)}$ on $X\times X$, hence the functor
$F_{w,!}k_!$ is given by the kernel $\overline{K(w)}|_{V\times X}$
on $V\times X$. Since the projection $p_2:V\times X\rightarrow X$
is affine, the functor $p_{2!}$ is left $t$-exact; hence
it is sufficient to prove that for any $A\in\operatorname{Perv}(V)$
the object $p_1^*A\otimes\overline{K(w)}|_{V\times X}$ is
a perverse sheaf on $V\times X$. Let $s:V\rightarrow G$ be
a splitting of the projection $G\rightarrow X$ over $V$.
Consider the isomorphism
$$\nu:V\times X\rightarrow V\times X:(v,x)\mapsto (v,s(v)x).$$
Then it is sufficient to check that
$\nu^*(p_1^*A\otimes\overline{K(w)})\simeq p_1^*A\otimes
\nu^*\overline{K(w)}$ is perverse. The sheaf $\nu^*\overline{K(w)}$
is the Goresky---MacPherson extension of
$\nu^*K(w)$ on $\nu^{-1}(X(w))$. By definition
$$\nu^{-1}(X(w))=\{(v,x)\in V\times X|\ x\in X_w\}$$
where $X_w=Un_wT_wU/U\subset X$ is a locally closed subvariety
of $X$. Note that
the projection $\operatorname{pr}_w:X(w)\rightarrow T_w$ factors as the composition
of projections $p_w:X(w)\rightarrow X_w$ and $q_w:X_w\rightarrow T_w$
where $p_w$ is smooth of relative dimension $\dim X$,
$q_w$ is smooth of relative dimension $l(w)$.
Also we have $K(w)=\operatorname{pr}_w^*L_w[l(w)]$ where $L_w$ is a perverse
sheaf on $T_w$. Hence, $K(w)=p_w^*(q_w^*L_w[l(w)])$ where
$q_w^*L_w[l(w)]$ is a perverse sheaf on $X_w$.
Now we have
$p_w\circ\nu|_{\nu^{-1}(X(w)}=p_2|_{\nu^{-1}(X(w))}$
where $p_2:V\times X\rightarrow X$ is the projection, and hence
$$\nu^*K(w)\simeq p_2^*(q_w^*L_w[l(w)]).$$
It follows that $\nu^*\overline{K(w)}\simeq p_2^*M_w$
where $M_w$ is the Goresky---MacPherson extension of
$q_w^*L_w[l(w)]$ to $X$. Thus,
$$p_1^*A\otimes\nu^*\overline{K(w)}\simeq A\boxtimes M_w,$$
and the latter sheaf is perverse by \cite{BBD}, 4.2.8.
The exactness of functor $F_{w,*}k_*$ follows
from the isomorphism
$$F_{w,\psi,*}=D\circ F_{w,\psi^{-1},!}\circ D$$
where $D$ is the Verdier duality (see \cite{KL}, 2.6.2(i)).
\qed\vspace{3mm}
\begin{thm}\label{findim} For any $A,B\in\AA$ the spaces
$\operatorname{Ext}^i_{\AA}(A,B)$ are finite-dimensional.
\end{thm}
\noindent {\it Proof} . It is sufficient to prove that for every
$A\in\AA$ there exists a surjection $A'\rightarrow A$
in $\AA$ such that $\operatorname{Ext}^i_{\AA}(A,B)$
is finite-dimensional for every $B\in\AA$.
We start with the surjection
$$\oplus_w j_{w,!}A_w\rightarrow A$$
where $A_w=j_w^*A$.
Now we choose a finite covering $(U_k)$ of $X$
by affine open subsets such that the projection
$G\rightarrow X$ splits over every $U_k$. Let
$j_k:U_k\hookrightarrow X$ be the corresponding open embeddings.
Now we replace every $A_w$ by $j_{k,!}j_k^*A_w$
to get the surjection
$$A'=\oplus_{i,w} j_{w,!}j_{k,!}j_k^*A_w\rightarrow A.$$
It remains to prove that
$\operatorname{Ext}^i_{\AA}(j_{w,!}j_{k,!}C,B)$ is finite-dimensional
for every $C\in\operatorname{Perv}(U_k)$. According to Theorem \ref{adapted}
the functor $j_{w,!}j_{k,!}:\operatorname{Perv}(U_k)\rightarrow\AA$ is exact.
Since it is left adjoint to $j_k^*j_w^*:\AA\rightarrow\operatorname{Perv}(U_k)$
we get the isomorphism
$$\operatorname{Ext}^i_{\AA}(j_{w,!}j_{k,!}C,B)\simeq
\operatorname{Ext}^i_{\operatorname{Perv}(U_k)}(C,j_k^*j_w^*B),$$
where the latter space is finite-dimensional, as follows from
Beilinson's theorem \cite{Be}.
\qed\vspace{3mm}
\subsection{Vanishing of $Ext^1$}
Let $(W,S)$ be a finite Coxeter system,
$\Phi$ be a $W$-gluing data.
\begin{thm}\label{Ext1van}
Assume that for every $w\in W$ and $s\in S$ the
following condition holds: for every object $A\in{\cal C}_w$ such
that the canonical morphism $\Phi_s^2(A)\rightarrow A$ is zero, one
has $\Phi_s(A)=0$. Let $S$ and $S'$ be simple objects in ${\cal C}(\Phi)$
such that $\operatorname{Supp}(S)\cap\operatorname{Supp}(S')=\emptyset$.
Then $\operatorname{Ext}^1_{{\cal C}(\Phi)}(S,S')=0$.
\end{thm}
\noindent {\it Proof} . By assumption we have a partition of $W$ into two subsets
$I$ and $I'$ such that $S_w=0$ for $w\in I'$ and $S'_{w}=0$
for $w\in I$. Then we have $E_w=S_w$ for $w\in I$, and $E_w=S'_w$
for $w\in I'$. Now we claim that the structural morphisms
$\Phi_sE_w\rightarrow E_{sw}$ vanish unless $w$ and $sw$ belong to the
same subset of this partition. By the definition of $W$-gluing
this would imply that $E\simeq S\oplus S'$. So assume for
example that $w\in I$ and $sw\in I'$. Then the morphism
$\Phi_s^2 S_w\rightarrow S_w$ is zero, and hence $\Phi_s E_w=\Phi_s S_w=0$.
Similarly, if $w\in I'$ and $sw\in I$, then $\Phi_s E_w=\Phi_s S'_w=0$.
\qed\vspace{3mm}
\begin{cor}\label{Ext1inj}
Let $\Phi$ be a $W$-gluing data of finite type.
Under the assumptions of Theorem \ref{Ext1van},
for every pair of simple objects
$S=(S_w)$ and $S'=(S'_w)$ of ${\cal C}(\Phi)$
the natural map
$$\operatorname{Ext}^1_{{\cal C}(\Phi)}(S,S')\rightarrow
\oplus_{w\in W}\operatorname{Ext}^1_{{\cal C}_w}(S_w,S'_w)$$
is injective.
\end{cor}
\noindent {\it Proof} . If the supports of $S$ and $S'$ do not intersect then we are done
by Theorem \ref{Ext1van}. So we can assume that
there exists $w\in W$ such that $S_w$ and $S'_w$ are both
non-zero. Then according
to Lemmas \ref{ressim} and \ref{gormac}
we have $S\simeq j_{w,!*}S_w$ and
$S'\simeq j_{w,!*}S'_w$. Now assume that we have an extension
\begin{equation}\label{extSS'}
0\rightarrow S'\rightarrow E\rightarrow S\rightarrow 0
\end{equation}
in ${\cal C}(\Phi)$ that induces a trivial extension
\begin{equation}\label{extw}
0\rightarrow S'_w\rightarrow E_w\rightarrow S_w\rightarrow 0.
\end{equation}
We claim that the adjunction morphism
$j_{w,!}E_w\rightarrow E$ is surjective. Indeed, this
follows immediately from the
commutative diagram with exact rows:
\begin{equation}
\begin{array}{cccccc}
j_{w,!}S'_w & \lrar{} & j_{w,!}E_w & \lrar{} &
j_{w,!}S_w &\lrar{} 0\\
\ldar{}& &\ldar{}& &\ldar{}\\
S'& \lrar{} & E & \lrar{} &
S &\lrar{} 0\\
\end{array}
\end{equation}
Similarly, one proves that the adjunction
morphism $E\rightarrow j_{w,*}E_w$ is injective. Thus,
$E\simeq j_{w,!*}E_w$, and a splitting of the
extension (\ref{extw})
induces a splitting of (\ref{extSS'}).
\qed\vspace{3mm}
\begin{rems}
\noindent
1. The conditions of Theorem \ref{Ext1van} are satisfied for
the gluing on the basic affine space as one can see combining
Corollary \ref{zer} and Lemma \ref{mainl}.
\noindent 2. There is an analogue of Theorem \ref{Ext1van}
for arbitrary gluing data of finite type. In this case one
should impose the condition that $\Phi_{ij}({\cal C}_{ij})=0$
(in the notation of section \ref{groth}). For example,
this condition is satisfied for the usual geometric gluing
data associated with an open covering (see \cite{KL}).
\end{rems}
Theorem \ref{Ext1van} can be strengthened using the adjoint
functors.
\begin{thm} Assume that we have a gluing data $\Phi$ as in Theorem
\ref{Ext1van}.
Let $P\subset W$ be a subset such that there exists left and right adjoint
functors $j_{P,!},j_{P,*}:{\cal C}(\Phi_P)\rightarrow{\cal C}(\Phi)$ to the natural
restriction functor
$j_P^*:{\cal C}(\Phi)\rightarrow{\cal C}(\Phi_P)$.
Let $S$ and $S'$ be simple objects in ${\cal C}(\Phi)$ such that
$\operatorname{Supp}(S)\cap P\neq \emptyset$, $\operatorname{Supp}(S')\cap P\neq \emptyset$ but
$\operatorname{Supp}(S)\cap\operatorname{Supp}(S')\cap P=\emptyset$. Then
$\operatorname{Ext}^1_{{\cal C}(\Phi)}(S,S')=0$.
\end{thm}
\noindent {\it Proof} . The proof of Theorem \ref{Ext1van} shows that there are no
non-trivial extensions between $j_P^*S$ and $j_P^*S'$.
Now given an extension between $S$ and $S'$ the same argument as in
Corollary \ref{Ext1inj} shows that it splits.
\qed\vspace{3mm}
For example, let $W=S_3$, $\Phi$ be a $W$-gluing data of finite type.
Then the conditions of this theorem are satisfied for any proper connected
subgraph $P\subset S_3$. Together with Theorem \ref{supp2} this implies
that in this case $Ext^1_{{\cal C}(\Phi)}(S,S')=0$ unless either
$\operatorname{Supp}(S)\subset\operatorname{Supp}(S')$ or $\operatorname{Supp}(S')\subset\operatorname{Supp}(S)$.
\section{Mixed glued category}
\subsection{Definition}
We are going to define a mixed analogue of the gluing on the
basic affine space. Namely, we consider the situation when
the field $k$ is the algebraic closure of ${\Bbb F}_p$ ($p>2$),
and we fix the finite subfield ${\Bbb F}_q\subset k$ such that
$G$, $T$, and $B$ are defined over ${\Bbb F}_q$.
Following \cite{BBD} we denote by subscript
$0$ objects defined over this finite field.
Thus $X=G/U$ is obtained by extension of scalars from
$X_0$, the corresponding scheme over ${\Bbb F}_q$.
Let $\operatorname{Fr}:X\rightarrow X$ be the corresponding geometric Frobenius.
Since the Fourier transform commutes with $\operatorname{Fr}$, there is a
well-defined functor $\operatorname{Fr}^*$ on the glued category
$\AA=\AA_{\psi}$.
Let $\AA^{\operatorname{Fr}}$ be the category of objects $A\in\AA$ together
with isomomorphisms $\a:A\widetilde{\rightarrow}\operatorname{Fr}^*A$.
Let $\AA_m$ be the subcategory in $\AA^{\operatorname{Fr}}$ consisting of
$A=(A_w)\in\AA$ such that all $A_w$ are mixed
perverse sheaves on $X_0$, with the canonical morphism $\a$.
Then $\AA_m$ is obtained by gluing from $|W|$ copies
of the category $\operatorname{Perv}_m(X_0)$ of mixed perverse sheaves on
$X_0$ (since the relevant functors between perverse
sheaves preserve mixedness). We will call an object of $\AA_m$
{\it pure} if all its components are pure perverse sheaves of the
same weight.
\subsection{Action of Frobenius on extensions}
Let ${\cal C}$ be a $\overline{{\Bbb Q}_l}$-linear abelian category, and
$\operatorname{Fr}^*:{\cal C}\rightarrow{\cal C}$ a $\overline{{\Bbb Q}_l}$-linear exact functor. Let ${\cal C}^{\operatorname{Fr}}$
be the category of pairs $(A,\a)$ where $A\in{\cal C}$,
$\a:A\widetilde{\rightarrow}\operatorname{Fr}^*A$ is an isomorphism. Assume that
${\cal C}_0$ is a full subcategory of ${\cal C}^{\operatorname{Fr}}$.
Let us denote by letters with subscript $0$
objects in ${\cal C}_0$ and omit the subscript when considering
the corresponding objects of ${\cal C}$. For every pair of objects
$A_0$ and $B_0$ of ${\cal C}_0$ there is a natural automorphism
$\operatorname{Fr}$ on $\operatorname{Ext}^*_{{\cal C}}(A,B)$ induced by $\operatorname{Fr}^*$. Let us assume that
for every $i$ the $\overline{{\Bbb Q}_l}$-space $\operatorname{Ext}^i_{{\cal C}}(A,B)$
is finite-dimensional and the eigenvalues of the above endomorphism
are $l$-adic units. Finally, notice that if $V$ is a
finite-dimensional $\overline{{\Bbb Q}_l}$-vector space with an
automorphism $\phi$, then we can define a functor of twist with
$(V,\phi)$ on ${\cal C}^{\operatorname{Fr}}$ by sending $(A,\a)$ to the object
$A'=A\otimes_{\overline{{\Bbb Q}_l}} V$ with the isomorphism
$\a'=\a\otimes\phi:A'\widetilde{\rightarrow}\operatorname{Fr}^*A'$. Our last assumption
is that the subcategory ${\cal C}_0\subset{\cal C}^{\operatorname{Fr}}$ is stable under
twists by finite-dimensional $\overline{{\Bbb Q}_l}$-vector spaces with
continuous $\hat{{\Bbb Z}}$-action.
\begin{thm}\label{Frinv}
With the above assumptions the following two
conditions are equivalent:
\begin{enumerate}
\item canonical maps
$\operatorname{Ext}^i_{{\cal C}_0}(A_0,B_0)\rightarrow\operatorname{Ext}^i_{{\cal C}}(A,B)^{\operatorname{Fr}}$
are surjective for all $A_0,B_0\in{\cal C}_0$, $i\ge 0$,
\item for every $A_0,B_0\in{\cal C}_0$ and every element
$e\in\operatorname{Ext}^i_{{\cal C}}(A,B)$ there exists a morphism
$f:B_0\rightarrow B'_0$ such that the image of $e$ in
$\operatorname{Ext}^i_{{\cal C}}(A,B')$ is zero.
\end{enumerate}
\end{thm}
\noindent {\it Proof}. (1)$\implies$(2). Obviously it is sufficient to check the
required condition for elements $e$ belonging to one of the
generalized eigenspaces of $\operatorname{Fr}$. Twisting $A_0$ or $B_0$ with a
one-dimensional $\hat{{\Bbb Z}}$-representation we can assume that
$(\operatorname{Fr}-1)^n\cdot e=0$ for some $n\ge 1$.
Consider the element $e_1=(\operatorname{Fr}-1)^{n-1}\cdot e$. Then
$e_1$ is invariant under $\operatorname{Fr}$; hence by assumption it is induced by some
extension in ${\cal C}_0$ between $A_0$ and $B_0$. In particular,
there exists a morphism $f:B_0\rightarrow B'_0$ such that the
image of $e_1$ in $\operatorname{Ext}^i(A,B')$ is zero. Let $e'$ be the
image of $e$ in $\operatorname{Ext}^i(A,B')$. Then $(\operatorname{Fr}-1)^{n-1}e'=0$, so
we can apply induction to finish the proof.
\noindent
(2)$\implies$(1). We use induction in $i$.
For $i=0$ the required surjectivity follows from the
assumption that ${\cal C}_0$ is a full subcategory of ${\cal C}^{\operatorname{Fr}}$.
Let $i\ge 1$, $e\in\operatorname{Ext}^i(A,B)$ be an element
invariant under $\operatorname{Fr}$. Then by assumption there exists
an embedding $B_0\hookrightarrow B'_0$ killing $e$. In other words, $e$
is the image of some element $e'\in\operatorname{Ext}^{i-1}(A,B'/B)$
under the boundary map
$\delta:\operatorname{Ext}^{i-1}(A,B'/B)\rightarrow\operatorname{Ext}^i(A,B)$. Notice that the element
$\operatorname{Fr}(e')-e'$ lies in the kernel of $\delta$; hence it comes
from some element $d\in\operatorname{Ext}^{i-1}(A,B')$. Applying our
assumption to this element we find an embedding
$B'_0\hookrightarrow B''_0$ such that $d$ is killed in
$\operatorname{Ext}^{i-1}(A,B'')$. Let $e''$ be the image of $e'$ in
$\operatorname{Ext}^{i-1}(A,B''/B)$. Then $e''$ is invariant under $\operatorname{Fr}$
and maps to $e$ under the boundary map
$\delta':\operatorname{Ext}^{i-1}(A,B''/B)\rightarrow\operatorname{Ext}^i(A,B)$. Now the proof is
finished by applying the induction hypothesis to $e''$.
\qed\vspace{3mm}
\subsection{Weights of Ext-groups}
In this section we will temporarily use the notation
$\Phi_w=\sideset{^p}{^0}{H}F_{w,!}$ for Kazhdan---Laumon
gluing functors (reserving symbols $F_{\cdot}$ for
the filtration on mixed objects).
\begin{prop}\label{pure}
Every simple object of $\AA_m$ is pure.
Every object $A$ of $\AA_m$ has a canonical increasing
filtration such that $\operatorname{gr}_n(A)$ is pure of weight $n$.
Every morphism in $\AA_m$ is strictly compatible with
this filtration.
\end{prop}
\noindent {\it Proof} . Since $\AA_m$ is obtained by gluing,
from Lemmas \ref{ressim} and \ref{gormac} we conclude that
every simple object of $\AA_m$ has form
$j_{w,!*}(A_w)$ for some simple mixed perverse sheaf $A_w$ on
$X_0$. Recall that by \cite{BBD}, 5.3.4 $A_w$ is pure
of some weight $n$.
Now since the symplectic Fourier transform preserves weights
we obtain that $j_{w,!}(A_w)$ has weights $\le n$, while
$j_{w,*}(A_w)$ has weights $\ge n$. Hence, $j_{w,!*}$ is pure
of weight $n$.
Let $A=(A_w)$ be an object of $\AA_m$, and let $F_{\cdot}(A_w)$ be
the canonical filtration on $A_w$ such that
$F_n(A_w)/F_{n-1}(A_w)$ is pure of weight $n$. Then
the weights of $\Phi_s(F_n(A_w))$ are $\le n$; hence
the structural morphisms $\Phi_s(A_w)\rightarrow A_{sw}$ induce
the morphisms $\Phi_s(F_n(A_w))\rightarrow F_n(A_{sw})$. Thus,
$(F_n(A_w))$ is a subobject of $A$ for every $n$, and
the $(F_{\cdot}(A_w))$ is the filtration with required
properties.
\qed\vspace{3mm}
\begin{rem} Another proof of existence of canonical
filtrations in $\AA_m$ can be obtained using Corollary
\ref{Ext1inj} and \cite{BBD}, 5.1.15 and 5.3.6.
\end{rem}
\begin{thm}\label{bound}
Let $A_0$ and $B_0$ be pure objects in $\AA_m$
of weights $a$ and $b$, respectively. Then
weights of Frobenius in $\operatorname{Ext}^i_{\AA}(A,B)$ are
$\ge i+b-a$.
\end{thm}
\noindent {\it Proof} . Without loss of generality we can assume that $A_0$ and
$B_0$ are simple.
For $i=0$ the assertion is clear, since
$\operatorname{Hom}_{\AA}(A,B)\hookrightarrow\oplus_{w\in W}\operatorname{Hom}(j_w^*A,j_w^*B)$
and the weights of Frobenius on the latter space are
$\ge b-a$.
Similarly, for $i=1$ the statement follows from Corollary \ref{Ext1inj}
and \cite{BBD}, 5.1.15.
So let us assume that $i>1$ and
that the assertion is true for $i-1$.
Using twist one can easily see that
it is sufficient to prove that for $i>a-b$ one
has $\operatorname{Ext}^i_{\AA}(A,B)^{\operatorname{Fr}}=0$.
We claim that equivalent conditions
of Theorem \ref{Frinv} are satisfied for the pair
of categories $\AA_m\subset\AA$. Indeed,
the action of Frobenius on $\operatorname{Ext}^i$-spaces extends
to continuous $\hat{{\Bbb Z}}$-action as follows from
the proof of Theorem \ref{findim}. Next,
the equivalent conditions of \ref{Frinv} are satisfied for the pair
$\operatorname{Perv}_m(X_0)\subset\operatorname{Perv}(X)$ since by Beilinson's theorem
the $\operatorname{Ext}$-groups in these categories can be computed
in derived categories of construcible sheaves, and
then the condition (1) of \ref{Frinv}
follows from ``local to global" spectral sequence
(see e.g., \cite{BBD} 5.1). Now we can check the condition (2) of
\ref{Frinv} for $\AA_m\subset\AA$ as follows. Let
$e\in\operatorname{Ext}^i_{\AA}(A,B)$ be an element. For every $w\in W$
consider the embedding $j^*_wB_0\hookrightarrow C_{w,0}$ such that
$C_{w,0}$ is $j_{w,*}$-acyclic (such an embedding exists
by Theorem \ref{adapted}). Then
$\operatorname{Ext}^i(A,j_{w,*}C_w)\simeq\operatorname{Ext}^i(j^*_wA,C_w)$; hence we
can find an embedding $C_{w,0}\rightarrow C'_{w,0}$ such that
the image of $e$ is killed in $\operatorname{Ext}^i(j^*_wA,C'_w)$. Then
$$B_0\hookrightarrow B'_0=\oplus_w j_{w,*}C'_{w,0}$$
is the required embedding. Thus the morphism
$\operatorname{Ext}^i_{\AA_m}(A,B)\rightarrow\operatorname{Ext}^i_{\AA}(A,B)^{\operatorname{Fr}}$
is surjective.
Now given an Ioneda extension class
in $\AA_m$
\begin{equation}\label{Ion}
0\rightarrow B_0\rightarrow E^1_0\rightarrow E^2_0\rightarrow\ldots E^i_0\rightarrow A_0\rightarrow 0
\end{equation}
we can replace $E^1_0$ and $E^2_0$ by their quotients
$E^1_0/F_{b-1}(E^1_0)$ and $E^2_0/F_{b-1}(E^1_0)$ (where
$F_{\cdot}(\cdot)$ is the weight filtration) to get an equivalent
extension class (\ref{Ion}) such that weights of $E_0^1$
are $\ge b$. Let $C_0=E^1_0/B_0$. Then
the class $e$ of (\ref{Ion}) is an image under the boundary
map
$$\operatorname{Ext}^{i-1}(A_0,C_0)\rightarrow\operatorname{Ext}^i(A_0,B_0)$$
of some class $e_1\in\operatorname{Ext}^{i-1}(A_0,C_0)$.
Since weights of $C_0$ are $\ge b$ we have an exact sequence
$$\operatorname{Ext}^{i-1}(A,F_b(C))\rightarrow\operatorname{Ext}^{i-1}(A,C)\rightarrow
\operatorname{Ext}^{i-1}(A,C/F_b(C)).$$
For every extension class $e$ in $\AA_m$ let us denote
by $c(e)$ the corresponding extension class in $\AA$.
By the induction hypothesis the image of $c(e_1)$ in
$\operatorname{Ext}^{i-1}(A,C/F_b(C))$ is zero, hence, $c(e_1)$ comes
from some class $e'_1\in\operatorname{Ext}^{i-1}(A,F_b(C))$.
Now the class $c(e)$ is the image of $c(e_1)$
under the boundary map
$$\operatorname{Ext}^{i-1}(A,C)\rightarrow\operatorname{Ext}^i(A,B);$$
hence it lies in the image of the boundary map
$$\operatorname{Ext}^{i-1}(A,F_b(C))\rightarrow \operatorname{Ext}^i(A,B)$$
corresponding to the exact sequence
$$0\rightarrow B_0\rightarrow F_b(E^1_0)\rightarrow F_b(C_0)\rightarrow 0.$$
But the image of this exact sequence in $\AA$ splits,
hence $c(e)=0$.
\qed\vspace{3mm}
\begin{cor}\label{findimwt}
For every $A_0,B_0\in\AA_m$ and every
$n\in{\Bbb Z}$ let $\operatorname{Ext}^*_{\AA}(A,B)_n$ denotes the weight-$n$
component of $\operatorname{Ext}^*_{\AA}(A,B)$. Then all the spaces
$\operatorname{Ext}^*_{\AA}(A,B)_n$ are finite-dimensional and
$\operatorname{Ext}^*_{\AA}(A,B)_n=0$ for $n<<0$.
\end{cor}
\section{Induction for representations of braid groups}
\label{induction}
\subsection{Formulation of the theorem}
Let $(W, S)$ be a finite Coxeter group, $B$ the
corresponding braid group, and $B^+\subset B$ the monoid of
positive braids.
We fix a subset $J\subset S$ of simple reflections. Let
$W_J\subset W$ be the subgroup generated by simple reflections
in $J$. Then $(W_J, J)$ is a Coxeter group and we denote
by $B_J$ the corresponding braid group.
Let $\operatorname{Mod}_J-B$ be the category of representations of $B$
of the form $\oplus_{x\in W/W_J} V_x$ such that the action of
$b\in B$ sends $V_x$ to $V_{\overline{b}x}$ and the
following condition is satisfied: for every $s\in S$ and
every $x\in W/W_J$
such that $sx\neq x$ one has $s^2|_{V_x}=\operatorname{id}_{V_x}$.
Morphisms in
$\operatorname{Mod}_J-B$ are morphisms of $B$-modules preserving
direct sum decompositions. Let $\operatorname{Mod}_J-B^+$ be the similar
category of $B^+$-representations.
Let $x_0\in W/W_J$ be the coset
containing the identity.
\begin{thm}\label{braidmain}
The functor $\oplus_{x\in W/W_J} V_x\mapsto V_{x_0}$
is an equivalence of $\operatorname{Mod}_J-B$ with the category $\operatorname{Mod}-B_J$
of $B_J$-representations. Similarly the category
$\operatorname{Mod}_J-B^+$ is equivalent to $\operatorname{Mod}-B_J^+$.
\end{thm}
\subsection{Arrangements of hyperplanes}
Let us realize $W$ as the group generated by reflections
in a Euclidean vector space $V$ over ${\Bbb R}$.
Let ${\cal H}$ be the corresponding arrangement of hyperplanes in
$V$, $X$ be the complement in $V_{{\Bbb C}}$
to all the complex hyperplanes $H_{{\Bbb C}}$, $H\in {\cal H}$. For
a subset $J\subset S$ of simple roots we denote by $X_J$
the similar space associated with $W_J$.
By a theorem of P.~Deligne \cite{D1} the spaces
$X/W$ and $X_J/W_J$ are $K(\pi,1)$ with fundamental groups
$B$ and $B_J$, respectively. Let $X'_J$ be the complement
in $V_{{\Bbb C}}$ to all the complex hyperplanes $H_{{\Bbb C}}$ such
that the corresponding reflection $r_H$ belongs to $W_J$.
Then $X'_J$ is a $W_J$-invariant open subset
containing $X$. Furthermore, there is a $W_J$-equivariant
retraction $X'_J\rightarrow X_J$ which induces a homotopic equivalence
of $X'_J/W_J$ with $X_J/W_J$. Let us consider the cartesian
power $Y_J=(X'_J/W_J)^{W/W_J}$ with the natural action of
$W$ by permutations and let us denote by
$X\times_W Y_J$ the quotient of $X\times Y_J$ by the diagonal
action of $W$. Let
$f:X\times_W Y_J\rightarrow X/W$ be the natural projection.
Then $f$ is a fibration with fiber $Y_J$. Let $x_0\in X$
be a fixed point. Then $y_0=(x_0,\ldots,x_0)\in Y_J$
is a point fixed by $W$ and hence the map
$$X\rightarrow X\times Y_J:x\mapsto (x,y_0)$$
descends to a section
$\sigma_0:X/W\rightarrow X\times_W Y_J$ of $f$. Let $\overline{x_0}$ be the
image of $x_0$ in $X/W$. Using $\sigma_0$ we obtain a canonical
identification
$$\pi_1(X\times_W Y_J,\sigma_0(\overline{x_0}))\simeq
\pi_1(Y_J,y_0)\rtimes\pi_1(X/W,\overline{x_0}).$$
Note that $\pi_1(X/W,\overline{x_0})\simeq B$ acts
on $\pi_1(Y_J,y_0)\simeq B_J^{W/W_J}$ via the
action of $W$ by permutations. In particular,
we have the canonical surjection
$$\pi_1(X\times_W Y_J,\sigma_0(\overline{x_0}))\rightarrow
B_J^{W/W_J}\rtimes W.$$
Now let us consider a $W$-equivariant map
$$\widetilde{\sigma}:X\rightarrow X\times Y_J: x\mapsto (x,\pi_J(\overline{w^{-1}x})_{w\in W/W_J})$$
where $\pi_J:X'_J\rightarrow X'_J/W_J$ is the natural projection.
Then $\widetilde{\sigma}$ induces a section
$\sigma:X/W\rightarrow X\times_W Y_J$ of $f$, and hence a homomorphism
$$\sigma_*:B\rightarrow\pi_1(X\times_W Y_J,\sigma(\overline{x_0})).$$
To identify the latter group with
$\pi_1(X\times_W Y_J,\sigma_0(\overline{x_0}))$
we have to construct a path from
$\sigma_0(x_0)$ to $\sigma(x)$. In other words for every $wW_J\in
W/W_J$ we have to construct a path from $\pi_J(x_0)$ to
$\pi_J(w^{-1}x_0)$ in $X'_J/W_J$. To do this we
take the canonical representative $w\in W$ of every $W_J$-coset
and consider the corresponding path between $x_0$ and
$w^{-1}x_0$ in $X$ (there is a canonical choice up to
homotopy, corresponding to the section $\tau:W\rightarrow
B=\pi_1(X/W,\overline{x_0})$),
then project it to $X'_J/W_J$.
This gives the required identification of the fundamental
groups, hence we get a homomorphism
$$f_J:B\rightarrow B_J^{W/W_J}\rtimes W.$$
It is easy to check that for every $s\in S$ one has
$$f_J(s)=((b_w)_{w\in W/W_J},\overline{s})$$
where $w$ runs over canonical representatives of $W/W_J$,
$b_w=\tau(w^{-1}sw)$ if $w^{-1}sw\in W_J$, and $b_w=1$
otherwise. Note that since $w$ is a canonical representative
the condition $w^{-1}sw\in W_J$ implies that
$l(sw)=l(w)+l(w^{-1}sw)$; hence $w^{-1}sw$ is in fact a simple
reflection.
\subsection{Proof of Theorem \ref{braidmain}}
The inverse functor
$\operatorname{Mod}-B_J\rightarrow\operatorname{Mod}_J-B$ is constructed as follows.
Let $V_0$ be a representation of $B_J$. Then there is a
natural action of $B_J^{W/W_J}\rtimes W$ on
$V=\oplus_{x\in W/W_J}V_0$ such that $B_J^{W/W_J}$
acts component-wise while $W$ acts by permutations of
components in the direct sum. Using the homomorphism
$f_J$ we obtain the action of $B$ on $V$. We claim
that $V$ belongs to the subcategory
$\operatorname{Mod}_J-B$. Indeed, by the definition of $f_J$,
if $sx\neq x$ for $x\in W/W_J$ and $s\in S$; then
$s$ acts by the permutation of factors on
$V_x\oplus V_{sx}\subset V$.
It remains to observe that for any $V\in\operatorname{Mod}_J-B$ we can
identify all the components $V_x$ with $V_{x_0}$ using
the action of the canonical representative for $x$.
This gives an isomorphism of $V$ with the $B$-representation
associated with $V_{x_0}$ as above.
\subsection{Commutator subgroup of the pure braid group}
When $J$ consists of one element we have $B_J={\Bbb Z}$ and
the homomorphism $f_J$ factors through the
canonical homomorphism
$f:B\rightarrow \oplus_{t\in T}{\Bbb Z}\rtimes W$
where $T$ is the set of
all reflections in $W$ (=the set of elements that are conjugate to
some element of $S$).
By definition $f(s)=(e_{\overline{s}},\overline{s})$,
where $e_t$ is the standard basis of $\oplus_{t\in T}{\Bbb Z}$.
\begin{prop} Assume that $W$ is finite. Then
$f$ is surjective with the kernel
$[P,P]$, the commutator subgroup of the pure braid group
$P\subset B$.
\end{prop}
\noindent {\it Proof} . In the above geometric picture $f|_P$ corresponds to taking
the link indices of a loop in $X$ with complex
hyperplanes $H_{{\Bbb C}}$ for all $H\in{\cal H}$. Thus, the map $f|_P$
can be identified with the natural projection
$\pi_1(X)\rightarrow H_1(X)=\pi_1(X)/[\pi_1(X),\pi_1(X)].$
\qed\vspace{3mm}
\subsection{Induction for actions of positive braid monoids}
There is a version of Theorem \ref{braidmain} concerning
the actions of braid monoids (or groups) on categories.
Namely, assume that we have a $B$-action on an additive category
${\cal C}=\oplus_{x\in W/W_J}{\cal C}_x$ such that
the functor $T(b)$ corresponding to $b\in B$ sends ${\cal C}_x$ to
${\cal C}_{\overline{b}x}$. Assume also that for $s\in S$ and $x\in
W/W_J$ such that $sx\neq x$, the functor
$T(s):{\cal C}_x\rightarrow{\cal C}_{sx}$ is an equivalence. Then we can
reconstruct ${\cal C}$ and the action of $B$ on it from the
action of $B_J$ on ${\cal C}_{x_0}$ exactly as in Theorem
\ref{braidmain}. More precisely, if
$w\in W$ is a canonical representative of $x\in W/W_J$, then
the functor $T(w):{\cal C}_{x_0}\rightarrow {\cal C}_x$ is an equivalence,
and the corresponding equivalence
$${\cal C}\simeq \oplus_{x\in W/W_J}{\cal C}_{x_0}$$
is compatible with the $B$-actions, where the action of $B$
on the right-hand side is the composition of the natural
$B_J^{W/W_J}\rtimes W$-action with $f_J$.
This is reflected in the following result concerning
$W$-gluing.
Let $({\cal C}_w,\Phi)$ be a $W$-gluing data, and $J\subset S$
a subset of simple reflections. Then it induces a gluing
data $\Phi_J$ for the categories $({\cal C}_w), w\in W_J$.
\begin{thm}
Assume that for every
$s\in S$ and every $w\in W$ such that $w^{-1}sw\not\in W_J$,
the morphism $\nu_s:\Phi_s^2|_{{\cal C}_w}\rightarrow\operatorname{Id}_{{\cal C}_w}$
is an isomorphism. Then the glued category ${\cal C}(\Phi)$
is equivalent to ${\cal C}(\Phi_J)$.
\end{thm}
\noindent {\it Proof} . Note that our condition on the gluing functors
implies that for every $w\in W$, which is the shortest
element of $wW_J$ and every $w'\in W_J$,
the functors $\Phi_w|_{{\cal C}_{w'}}$ and
$\Phi_{w^{-1}}|_{{\cal C}_{ww'}}$ are quasi-inverse to each other.
We claim that the functors
$j^*=j^*_{W_J}:{\cal C}(\Phi)\rightarrow{\cal C}(\Phi_J)$
and $j_!=j_{W_J,!}:{\cal C}(\Phi_J)\rightarrow{\cal C}(\Phi)$ are
quasi-inverse to each other. Indeed, we always have
$j^*j_!=\operatorname{Id}$. Now let $A=(A_w; \a_{w,w'})$ be an object of
${\cal C}(\Phi)$. The canonical adjunction morphism
$j_!j^*A\rightarrow A$ has as components the morphisms
$$\a_{n(w),p(w)}:\Phi_{n(w)}A_{p(w)}\rightarrow A_w.$$
Since the functors $\Phi_{n(w)}$ and $\Phi_{n(w)^{-1}}$
between ${\cal C}_{p(w)}$ and ${\cal C}_w$ are quasi-inverse to each
other, the associativity condition on $\a$ implies
that $\a_{n(w),p(w)}$ is an isomorphism.
\qed\vspace{3mm}
\section{Good representations of braid groups}\label{goodsec}
\subsection{Some ideals in the group ring of the pure
braid group}\label{ideals}
Let $(W,S)$ be a Coxeter group, and $B$ and $P$ the
corresponding (generalized) braid group and pure braid groups, respectively.
Recall that $P$ is the kernel of the
natural homomorphism $B\rightarrow W:b\mapsto\overline{b}$.
In other words, $P$ is the normal
closure of the elements $\{ s^2, s\in S\}$ in $B$.
Below we view $S$ as a subset in $B$ and denote by $\overline{s}$
($s\in S$) the corresponding elements in $W$.
\begin{thm} There exists a unique collection of right ideals
$(I_w, w\in W)$ in ${\Bbb Z}[P]$ such that $I_1=0$,
$$I_{\overline{s}}=(s^2-1){\Bbb Z}[P]$$ for every $s\in S$, and
$$I_{ww'}=I_w+\tau(w)I_{w'}\tau(w)^{-1}$$
for every pair $w,w'\in W$ such that $l(ww')=l(w)+l(w')$.
\end{thm}
\noindent {\it Proof} . We use the induction in $l(w)$, so we assume that
all $I(y)$ with $l(y)<l(w)$ are already constructed and
satisfy the property
$$I_{yy'}=I_y+\tau(y)I_{y'}\tau(y)^{-1}$$
for $y,y'\in W$ such that $l(yy')=l(y)+l(y')<l(w)$.
Choose a decomposition $w=\overline{s}w_1$ with $l(w)=l(w_1)+1$. Then
we must have $I_w=I_{\overline{s}}+sI_{w_1}s^{-1}$. It remains to show that the
right-hand side
does not depend on a choice of decomposition.
Let $w=\overline{s'}w'_1$ be another decomposition with $l(w)=l(w'_1)+1$.
Assume first that $s$ and $s'$ commute. Then $w_1=\overline{s'}y$ and
$w'_1=\overline{s}y$ where $l(y)=l(w)-2$, so that by the induction hypothesis
we have
$$I_{w_1}=I_{\overline{s'}}+s'I_{y}(s')^{-1},$$
$$I_{w'_1}=I_{\overline{s}}+sI_{y}s^{-1}.$$
It follows that
$$I_{\overline{s}}+sI_{w_1}s^{-1}=
I_{\overline{s'}}+s'I_{w'_1}(s')^{-1}=I_{s}+ I_{s'}+(ss')I_y(ss')^{-1}.$$
Now if $s$ and $s'$ do not commute there is a defining relation of the form
$$ss's\ldots=s'ss'\ldots$$
where both sides have the same length $m$.
Let us write this relation in the form
$$sr=s'r'$$
where $r=s's\ldots$ and $r'=ss'\ldots$ are the corresponding elements of length
$m-1$.
In this case we have $w_1=\overline{r}y$, $w'_1=\overline{r'}y$ where $l(y)=l(w)-m$,
so by the induction hypothesis we have
$$I_{w_1}=I_{\overline{r}}+rI_{y}r^{-1}.$$
Hence
$$I_{\overline{s}}+sI_{w_1}s^{-1}=
I_{\overline{s}}+sI_{\overline{r}}s^{-1}+(sr)I_y(sr)^{-1}.$$
Comparing this with the similar expression for
$I_{\overline{s'}}+s'I_{w'_1}(s')^{-1}$ we see that it is sufficient to
prove the equality
$$I_{\overline{s}}+sI_{\overline{r}}s^{-1}=
I_{\overline{s'}}+s'I_{\overline{r'}}(s')^{-1}.$$
In other words, we have to check that
\begin{equation}
I_{\overline{s}}+sI_{\overline{s'}}s^{-1}+ss'I_{\overline{s}}(ss')^{-1}+\ldots=
I_{\overline{s'}}+s'I_{\overline{s}}(s')^{-1}+s'sI_{\overline{s'}}(s's)^{-1}+\ldots
\end{equation}
where both sides contain $m$ terms. More precisely, we
claim that the terms of the right-hand side coincide
with those of the left-hand side in inverse order.
This follows immediately from the identity
$rI_{s''}r^{-1}=I_s$ where $s''=s$ if $m$ is even, $s''=s'$
if $m$ is odd.
\qed\vspace{3mm}
Note that $I_w$ is generated by $l(w)$ elements and
$I_w\subset I_y$ if $y=ww'$ and $l(ww')=l(w)+l(w')$.
In the case when
$W$ is finite, this implies that $I_{w_0}$ contains all the ideals $I_w$,
where $w_0$ is the longest element in $W$.
\begin{cor} Assume that $W$ is finite. Then
$I_{w_0}=I$ where $I$ is the augmentation ideal in ${\Bbb Z}[P]$.
\end{cor}
\noindent {\it Proof}. Let $s\in S$. Then we have
$$I_{w_0}=I_s+sI_{sw_0}s^{-1}.$$
Hence,
$$sI_{w_0}s^{-1}=I_s+s^2I_{sw_0}=I_s+I_{sw_0};$$
here the last equality follows from the definition of $I_s$.
In particular, $sI_{w_0}s^{-1}\subset I_{w_0}$ for every $s\in S$.
It follows that
$$s^{-1}I_{w_0}s=s^{-2}(sI_{w_0}s^{-1})\subset s^{-2}I_{w_0}\subset
I_{w_0}+I_s=I_{w_0}.$$
Thus, $I_{w_0}$ is stable under the action of $B$ by conjugation.
Since it also contains $(s^2-1)$ for every $s\in S$, it should be
equal to $I$.
\qed\vspace{3mm}
\subsection{Main definition}
Let $V$ be a representation of the braid group $B$.
For every $w\in W$ let us denote
$$V_w=I_wV\subset V$$
where $I_w\subset{\Bbb Z}[P]$ is the ideal defined above.
Then $V_s=(s^2-1)V$ for $s\in S$, and
$$V_{w_1w_2}=V_{w_1}+\tau(w_1)V_{w_2}$$
if $l(w_1w_2)=l(w_1)+l(w_2)$.
Define
$K_W(V)$ to be the following subspace of $V^W$:
$$K_W(V)=\{(x_w), w\in W: x_w\in V, x_{\overline{s}w}-sx_w\in V_s,
\forall s\in S, w\in W\}.$$
\begin{prop} For every $(x_w)\in K_W(V)$ and every
$w_1,w_2\in W$, one has
$x_{w_1w_2}-\tau(w_1)x_{w_2}\in V_{w_2}$.
For every $y\in W$
there is a natural embedding
$i_y:V\rightarrow K_W(V)$ given by
$$i_y(v)=(\tau(wy^{-1})v, w\in W).$$
\end{prop}
The proof is straightforward.
\begin{ex} Let $R={\Bbb Z}[u,u^{-1},(u^2-1)^{-1}]$ where $u$ is an indeterminate.
It follows from Theorem \ref{K_0A} and Corollary \ref{s2-1}
that the $R$-module $K_0(\AA)\otimes_{{\Bbb Z}[u,u^{-1}]}R$ is naturally
isomorphic to $K_W(K_0(G/U)\otimes_{{\Bbb Z}[u,u^{-1}]} R)$ where
$\AA$ is the Kazhdan---Laumon category. The embeddings
$i_y$ correspond to functors $Lj_{y,!}$ (left adjoint to restrictions).
\end{ex}
\begin{defi} A $B$-representation $V$ is called {\it good}
if $K_W(V)$ is generated by the subspaces $i_y(V)$, $y\in W$.
\end{defi}
In the situation of the above example it would be very desirable
to prove that an appropriate localization of $K_0(G/U)$ is
a good representation of $B$ since this would imply that the objects
of finite projective dimension generate the corresponding localization
of $K_0(\AA)$. Then the following result could be applied to construct a
bilinear pairing on appropriate finite-dimensional quotient of $K_0(\AA)$.
\begin{prop}\label{form}
Let $V$ be a finite-dimensional representation of $B$, and
$V^*$ the dual representation.
Then both $V$ and $V^*$ are good if and only if there exists a
non-degenerate pairing $\chi:K_W(V)\otimes K_W(V^*)\rightarrow{\Bbb C}$
such that $\chi(i_y(v_y),v')=\langle v_y, p_y v'\rangle$
and $\chi(v,i_y(v'_y))=\langle p_y v, v'_y\rangle$ for every
$y\in W$, $v_y\in V$, $v'_y\in V^*$, $v\in K_W(V)$,
$v'\in K_W(V^*)$.
\end{prop}
\noindent {\it Proof} . If such a pairing $\chi$ exists, then
the orthogonal to the subspace in $V$ generated by
elements of the form $i_y(v_y)$, $y\in W$, $v_y\in V$,
is zero, and hence, $V$ is good. Similarly, $V^*$ is good.
Now assume that both $V$ and $V^*$ are good.
Then we have the surjective map
$$
\pi:\oplus_{y\in W} V\rightarrow K_W(V):
(v_y,y\in W)\mapsto \sum_y i_y(v_y)$$
and the similar map $\pi':\oplus_{y\in W} V^*\rightarrow K_W(V^*)$.
It is easy to see that the kernel of $\pi$
consists of collections $(v_y,y\in W)$ such that
$\sum_{y\in W}\tau(wy^{-1})v_y$ for every $w\in W$.
The kernel of $\pi'$ has a similar description.
Now we define a pairing
$$\widetilde{\chi}:(\oplus_{y\in W} V)\otimes(\oplus_{y\in W}V^*)\rightarrow{\Bbb C}$$
by the formula
$$\widetilde{\chi}((v_y,y\in W),(v'_y,y\in W))=
\sum_{y,w\in W}\langle v_w,\tau(wy^{-1})v'_y\rangle=
\sum_{y,w\in W}\langle \tau(yw^{-1})v_w,v'_y\rangle.$$
Since the left and right kernels of $\widetilde{\chi}$ coincide with
$\ker(\pi)$ and $\ker(\pi')$, respectively, it descends
to the non-degenerate pairing $\chi$ between $K_W(V)$ and $K_W(V^*)$.
\qed\vspace{3mm}
Let $Br_n$ be the Artin braid group such that the
corresponding Coxeter group is the symmetric group $S_n$.
\begin{prop}\label{B2} Any representation of $Br_2={\Bbb Z}$ is good.
\end{prop}
\noindent {\it Proof} . A representation of $Br_2$ is a space $V$ with an
operator $\phi:V\rightarrow V$. The corresponding space $K_{S_2}(V)$
consists of pairs $(v_1,v_2)\in V^2$ such that
$v_2-\phi v_1\in (\phi^2-1)V$. The maps $i_1$ and $i_s$
(where $s$ is the generator of $Br_2$) are given by
$i_1(v)=(v,\phi v)$, $i_s(v)=(\phi(v), v)$.
Now for any $(v_1,v_2)\in K_{S_2}(V)$ we have
$$(v_1,v_2)=i_1(v_1+\phi(v))+i_s(v)$$
where $v\in V$ is such that
$v_2-\phi v_1=(\phi^2-1)v$.
\qed\vspace{3mm}
\begin{thm}\label{B3}
Let $V$ be a representation of $Br_3$ over a
field $k$, such that for generators $s_1$ and $s_2$, the
following identity is satisfied in $V$:
$$(s^2-1)(s-\lambda)=0$$
where $\lambda\in k$ is an element such that
$\lambda^6\neq 1$. Then $V$ is good.
\end{thm}
\subsection{Criterion} We need an auxiliary result which allows one to
check that a subspace of $V^2$ is of the form $K_{S_2}(V)$
for some action of $Br_2={\Bbb Z}$ on $V$. It is more natural
to generalize this construction as follows. Let
$V=V_0\oplus V_1$ be a super-vector space, and $\phi:V\rightarrow V$
an odd operator. Then $\phi^2$ is even, and
we have the super-subspace $V_{\phi}=(\phi^2-1)V\subset V$.
Now we define the vector space
$K(\phi)=\{v\in V|\ \phi(v)-v\in V_{\phi}\}$.
Note that this is a non-homogeneous subspace of $V$,
equipped with two sections $i_0$ and $i_1$ of
the projections to $V_0$ and $V_1$, namely,
$i_{\a}(v_{\a})=v_{\a}+\phi(v_{\a})$ for $\a=0,1$. We
want to characterize non-homogeneous subspaces of
$V$ arising in this way.
\begin{prop}\label{gluecheck}
Let $K\subset V$ be a non-homogeneous subspace, and
$i_0:V_0\rightarrow K$ and $i_1:V_1\rightarrow K$ sections of the
projections of $K$ to $V_0$ and $V_1$. Let $K^h$ be
the maximal homogeneous subspace of $K$. Let
$\phi:V\rightarrow V$ be the odd operator with components
$p_1i_0$ and $p_0i_1$ where $p_{\a}:K\rightarrow V_{\a}$
are the natural projections. Then we have inclusions
$V_{\phi}\subset K^h$ and $K(\phi)\subset K$. The equality
$K=K(\phi)$ holds if and only if $V_{\phi}=K^h$.
\end{prop}
\subsection{Proof of Theorem \ref{B3}}. We will prove first that
$K_{S_3}(V)=K_{S_2}(V')$ where $V'$ is some representation of
$Br_2$. Namely, let $V'$ be the space of triples
$(x,y,z)\in V^3$ such that $y-s_1x\in V_{s_1}$ and
$y-s_2z\in V_{s_2}$. We have the natural embedding
$$\kappa:K_{S_3}(V)\hookrightarrow V'\oplus V':
(x_w,w\in S_3)\mapsto ((x_1,x_{s_1},x_{s_2s_1}),
(x_{s_2},x_{s_1s_2},x_{s_2s_1s_2})).$$
To apply Proposition \ref{gluecheck} to the
image of $\kappa$ we have to define maps
$i_0,i_1:V'\rightarrow \kappa K_{S_3}(V)$ such that
$p_{\a}i_{\a}=\operatorname{id}$ for $\a=0,1$. Let us set
\begin{align*}
&i_0(x,y,z)=((x,y,z),(s_2x,s_1s_2x-s_2s_1s_2y+s_2s_1z,s_1z)),\\
&i_1(x,y,z)=((s_2x,s_1s_2x-s_2s_1s_2y+s_2s_1z,s_1z),(x,y,z)).
\end{align*}
Then the operator $\phi=p_1i_0=p_0i_1:V'\rightarrow V'$ has the following
form:
$$\phi(x,y,z)=(s_2x,s_1s_2x-s_1s_2s_1y+s_2s_1z,s_1z).$$
Now according to Proposition \ref{gluecheck} in order to
check that $\kappa K_{S_3}(V)$ is obtained by gluing from
$(V'\oplus V',\phi)$ we have to check that the
image of the operator $\phi^2-\operatorname{id}$ is precisely the
subspace $U=\{(x,y,z)\in V'|\ x\in V_{s_2}, z\in V_{s_1}\}$.
Let $u\in U$ be an arbitrary element. Then
$$u=(\phi^2-\operatorname{id})(\lambda^2-1)^{-1}u+u'$$
with $u'=(0,y,0)\in U$. Then $y\in V_{s_1}\cap V_{s_2}$.
Hence, $\tau(w_0)$ acts on $y$ as multiplication by $\lambda^3$.
It follows that $u'=(\phi^2-\operatorname{id})(\lambda^6-1)^{-1}u'$.
Hence, $u'$ and $u$ are in the image of $\phi^2-\operatorname{id}$
as required.
It follows from Proposition \ref{B2} that
$K_{S_3}(V)=K_{S_2}(V')$ is generated by the images of
$i_0$ and $i_1$. Thus, we are reduced to show that $V'$ is
generated by elements of the form $(x,s_1x,s_2s_1x)$,
$(s_1y,y,s_2y)$ and $(s_1s_2z,s_2z,z)$ which is
straightforward.
\qed\vspace{3mm}
\subsection{Representations of quadratic Hecke algebras are good}
Let $H_q$ be the Hecke algebra of $(W,S)$ over
${\Bbb C}$ with complex parameter $q\in{\Bbb C}$. Recall that
$H_q$ is the quotient of the group algebra ${\Bbb C}[B]$ where
$B$ is the corresponding braid group by the quadratic relations
$(s-q)(s+1)=0$ for every $s\in S$.
\begin{thm}\label{quHecke} Assume that $q$ is not a root of unity.
Then every representation of $H_q$ is good.
\end{thm}
\begin{cor}\label{qubraid} Let $V$ be a representation of $B$ such that
$(s-\lambda)(s-\mu)=0$ for every $s\in S$, where
$\lambda\in{\Bbb C}^*$, $\mu\in{\Bbb C}$, and $\frac{\mu}{\lambda}$ is not a root of
unity. Then $V$ is good.
\end{cor}
Recall that we denote $\pi=\tau(w_0)^2\in B$
where $w_0\in W$ is the longest element in $W$.
\begin{lem}\label{mainHecke} Assume that $q$ is not a root of unity. Let
$V$ be an irreducible representation of $H_q$.
Then either $\pi-1$ acts by a non-zero constant on $V$
or $V$ is the one-dimensional representation such that
$s=-1$ on $V$ for every $s\in S$.
\end{lem}
\noindent {\it Proof} . Let $E$ be an irreducible representation of $W$, $E(u)$ be the
corresponding irreducible representation of the Hecke algebra
$H$ of $(W,S)$ over ${\Bbb Q}[u^{\frac{1}{2}},u^{-\frac{1}{2}}]$
defined by Lusztig. According to \cite{Lus} (5.12.2) one has
$$\pi=u^{2l(w_0)-a_E-A_E}$$
on $E(u)$ where the integers $a_E\ge0$ and $A_E\ge0$ are the lowest
and the highest degrees of $u$ appearing with non-zero
coefficient in the formal dimension $D_E(u)$ of $E$. Recall
that $D_E(u)$ is defined from the equation
$$D_E(u)\cdot\sum_{w\in W} u^{-l(w)}\operatorname{Tr}(\tau(w),E(u))^2=
\dim(E)\cdot\sum_{w\in W}u^{l(w)}.$$
It is known that $a_{E\otimes sign}=l(w_0)-A_E$
where $sign$ is the sign representation of $W$.
In particular, $a_E\le A_E\le l(w_0)$.
Thus, we have $\pi=u^l$ on $E(u)$, where $l>0$ unless
$a_E=A_E=l(w_0)$. In the latter case
$D_E(u)=f_E^{-1}u^{l(w_0)}$ where $f_E>0$ is an integer.
Thus, $D_E(1)=\dim(E)=f_E^{-1}=1$ and $E=sign$.
\qed\vspace{3mm}
\noindent
{\it Proof of Theorem \ref{quHecke}}. Since $H_q$ is
finite-dimensional it is sufficient to prove the assertion
for a finite-dimensional representation $V$ of $H_q$.
Clearly, we can assume that $V$ is irreducible. Assume first
that $V$ is one-dimensional and $s=-1$ on $V$ for every $s\in S$.
Then $V_s=0$ for every $s$ so $K_W(V)\simeq V$ and all the
maps $i_y$ are isomorphisms, hence $V$ is good. Thus,
according to Lemma \ref{mainHecke} we can assume that
$\pi-1\neq 0$ on $V$. Now we are going to use
the $K_0$-analogue of the complex defined in section
\ref{complex}. First of all for every coset $W_Jx\in W$
we have the map
$i_{W_Jx}:K_{W_J}(V)\rightarrow K_W(V)$ which is a section of
the projection $p_{W_Jx}$ onto components $W_Jx\subset W$.
Namely, using notation of section \ref{complex}
the $w$-component of $i_{W_Jx}(v_y,y\in W_Jx)$
is $n_{W_Jx}(w)(v_{p_{W_Jx}(w)})$.
To prove that $V$ is good it is sufficient to show that
$K_W(V)$ is generated by images of all maps $i_{W_Jx}$ where
$J\subset S$ is a proper subset. Now the proof of Theorem
\ref{homology} shows that for every $v\in K_W(V)$, we have the identity
$$\sum_{J\subset S,|J|<n,x\in W_J\backslash W}
(-1)^{|J|}i_{W_Jx}p_{W_Jx}(v)=\iota(v)+(-1)^{n-1}v,$$
where $W_{\emptyset}=\{ 1\}$, $\iota(v_w,w\in W)=(w_0v_{w_0w},w\in W)$.
One has $\iota^2(v_w,w\in W)=(\pi v_w,w\in W)$.
It follows that for every $v\in K_W(V)$ the element
$(\pi-1)v$ is a linear combination of elements of the form
$i_{W_Jx}(v')$ where $J\subset S$ is a proper subset, and hence
$v$ itself is such a linear combination.
\qed\vspace{3mm}
\subsection{Good representations and cubic Hecke algebras}
Let $H^c_q$ be the cubic Hecke algebra with complex
parameter $q$, i.e., the quotient of the group algebra ${\Bbb C}[B]$
of the braid group $B$ of $(W,S)$ by the relations
$(s-q)(s^2-1)=0$, $s\in S$,
where $q\in{\Bbb C}$ is a constant. An optimistic conjecture would be that
if $q$ is not a root of unity, then every representation of $B$
that factors through $H^c_q$ is good.
The particular cases are Proposition \ref{B2} and Theorems
\ref{B3}, \ref{quHecke}. Below we check some other particular
cases of this conjecture. However, it seems that in general one needs
to add some higher polynomial identities on the generators in order
for such a statement to be true (see \cite{Wenzl} for an example of
such identities).
\begin{thm}\label{thmAn} Let $(W,S)$ be of type $A_n$ for $n\le 3$.
Then every representation of $H^c_q$ is good provided that
$q$ is not a root of unity.
\end{thm}
\begin{thm}\label{B_2} Let $(W,S)$ be of type $B_2$.
Then every representation of $H^c_q$ is good
provided that $q^8\neq1$.
\end{thm}
The structure of the proof of these two theorems is the same
as of Theorem \ref{B3}.
We choose a simple reflection $s_i\in S$ ($s_2\in S_3$ in
Theorem \ref{B3}) and consider the
partition of $W$ into two pieces: $P_i$ and $P_is_i$.
This way we get an inclusion
$K_W(V)\hookrightarrow V'_0\oplus V'_1$ where
$V'_0\subset \oplus_{P_i}V$ is the space of collections
$(v_w,w\in P_i)$ such that $v_{sw}-sv_{w}\in V_s$ whenever
$w, sw\in P_i$; $V'_1$ is the similar space for $P_is_i\subset W$
instead of $P_i$. Let $p_0$ and $p_1$ be the corresponding
projections of $K_W(V)$ to $V'_0$ and $V'_1$.
We are going to construct sections $s_0$ and $s_1$ of these
projections.
For every coset $W_Jx\subset P_i$ (resp. $W_Jx\subset P_is_i$)
let us consider the map $i_{W_Jx,P_i}=p_0i_{W_Jx}:K_{W_J}\rightarrow V'_0$
(resp. $i_{W_Jx,P_is_i}=p_1i_{W_Jx}:K_{W_J}\rightarrow V'_1$).
The proof of Theorem
\ref{complexPi} shows that for every $v'\in V'_0$ we have the
identity
$$\sum_{J,x}
(-1)^{|J|-1}i_{W_{S-J}x,P_i}p_{W_{S-J}x}(v')=v',$$
where the sum is taken over non-empty subsets $J\subset S$
and $x\in W_{S-J}\backslash W$ such that
$W^{(j)}x\subset P_i$ for every $j\in J$.
Now we define $s_0$ to be the similar sum
$$s_0(v')=\sum_{J,x}(-1)^{|J|-1}i_{W_{S-J}x}p_{W_{S-J}x}(v').$$
\begin{lem}
For every $y\in P_i$ one has
$i_y=s_0i_{y,P_i}$.
\end{lem}
\noindent {\it Proof} . Note that for a coset $W_Jx\subset P_i$ we have
$p_{W_Jx}i_{y,P_i}=i_{x',W_Jx}$, where $x'$ is the element
of $W_Jx$ closest to $y$. Now the proof of Theorem \ref{complexPi}
shows that we have a formal equality
$$\sum_{J,x}(-1)^{|J|-1}x'(W_{S-J}x)=y,$$
where the sum is taken over non-empty subsets $J\subset S$
and $x\in W_{S-J}\backslash W$ such that
$W^{(j)}x\subset P_i$ for every $j\in J$,
$x'(W_{Kx})$ denotes the element in $W_Kx$ closest to $y$.
The assertion follows immediately, since
for every $x'\in W_Kx$ we have $i_{W_Kx}i_{x',W_Kx}=i_{x'}$.
\qed\vspace{3mm}
Similarly, one constructs the section
$s_1:V'_1\rightarrow K_W(V)$.
Thus, it is sufficient to show that the subspace
$K_W(V)\subset V'_0\oplus V'_1$ coincides with the subspace $K(\phi)$
where $\phi|_{V'_0}=p_1s_0:V'_0\rightarrow V'_1$,
$\phi|_{V'_1}=p_0s_1:V'_1\rightarrow V'_0$. By Proposition \ref{gluecheck}
this amounts to showing that the image of $\phi^2-\operatorname{id}$
surjects onto $\overline{V}_0\oplus \overline{V}_1$, where e.g.,
$\overline{V}_0=K_W(V)\cap V'_0$ consists of elements $(v_y, y\in P_i)$
such that
$v_y\in V_{ys_iy^{-1}}$ whenever $ys_iy^{-1}\in S$.
One way to show it would be to consider some natural
filtration on $\overline{V}_0$ preserved by $\phi^2$ and considering the
induced map on the associated graded factors.
The filtration is defined as follows. The subspace
$F_k(V'_0)\subset \overline{V}_0$ for $k\ge 0$ consists
of $(v_y,y\in P_i)\in \overline{V}_0$ such that
$v_y=0$ whenever $l(ys_iy^{-1})\le 2k-1$. There is a similar
filtration on $\overline{V}_1$ (replace $P_i$ by $P_is_i$ everywhere).
We claim that $\phi$ is compatible with these filtrations.
As we will see below this is a consequence of the definition of $s_0$
and of the following result.
\begin{prop}\label{geod_yw} Let $y,w\in W$ be a pair of elements. Assume that
$l(ys_iy^{-1})>l(ws_iw^{-1})$. Then no geodesics from $y$ to $w$
pass through $ys_i$.
\end{prop}
\noindent {\it Proof} . Let us denote $r=ws_iw^{-1}$, $r'=ys_iy^{-1}$, $w_1=yw^{-1}$.
The existence of a geodesic from $y$ to $w$ passing through $ys_i$
is equivalent to the existence of a geodesics from $w_1$ to $1$
passing through $ys_iw^{-1}=w_1r$. In other words this is equivalent
to the equality $l(w_1)=l(w_1r)+l(w_1rw_1^{-1})$, i.e.,
$l(w_1)=l(w_1r)+l(r')$ (since $w_1rw_1^{-1}=r'$).
But one has $l(w_1)\le l(w_1r)+l(r)<l(w_1r)+l(r')$, which contradicts
the above equality.
\qed\vspace{3mm}
\begin{cor} The map $\phi:V'_0\rightarrow V'_1$ sends $F_k(\overline{V}_0)$ to
$F_k(\overline{V}_1)$.
\end{cor}
\noindent {\it Proof} . As we have seen in the proof of Theorem \ref{complexPi},
in the sum defining $p_ws_0$
all the terms cancel except those that correspond to $x\in P_i$,
such that the set $\{s\in S\ |\ l(wx^{-1}s)<l(wx^{-1})\}$ coincides
with the set of $s_j$ such that $W^{(j)}x\subset P_i$. By Lemma \ref{appear}
the latter set coincides with the set $S(xs_ix^{-1})$
of simple reflections appearing
in any reduced decomposition of $xs_ix^{-1}$. In particular,
this implies that for any $w_1\in W_{S(xs_ix^{-1})}$ there exists
a geodesic from $x$ to $w$ passing through $w_1x$. Taking $w_1=xs_ix^{-1}$
we see that for such $x$ there exists a geodesic from $x$ to $w$
passing though $xs_i$. According to Proposition \ref{geod_yw}
this implies that $l(xs_ix^{-1})\le l(ws_iw^{-1})$, hence the
assertion.
\qed\vspace{3mm}
\begin{lem} Let $w\in W$ be an element, $r\in W$ a reflection, and
$S(r)$ the set of simple reflections in a reduced decomposition
of $r$. Assume that $l(wrw^{-1})=l(r)$ and
the set $\{ s\in S\ |\ l(ws)<l(w)\}$ coincides with $S(r)$.
Assume also that $W$ is either of type $A_n$ or of rank 2.
Then $w$ is the longest element of $W_{S(r)}$.
\end{lem}
\noindent {\it Proof} . The rank 2 case is straightforward, so let us prove the statement
for $W=S_n$. Let $r=(ij)$ where $i<j$. Then the assumptions of the lemma are:
1) for any $k\in [1,n]$ one has $w(k)>w(k+1)$ if and only if
$k\in [i,j-1]$, 2) $j-i$=$w(i)-w(j)$. It follows that $w$ maps the
interval $[i,j]$ to the interval $[w(j),w(i)]$ reversing the order
of elements in this interval. One the other hand, $w$ preserves the
order of elements in $[1,i]$ and in $[j,n]$. Hence $w(i)=j$, $w(j)=i$
and $w$ stabilizes all elements outside $[i,j]$.
\qed\vspace{3mm}
It follows from this lemma
that the map
$$\operatorname{gr}\phi:\operatorname{gr}^F_k(\overline{V}_0)\rightarrow \operatorname{gr}^F_k(\overline{V}_1)$$
sends $(v_y,y\in P_i,l(ys_iy^{-1})=2k+1)$ to
$(v_w,w\in P_is_i,l(ws_iw^{-1})=2k+1)$ where
$$v_w=(-1)^{|S(ws_iw^{-1})|-1}\tau(w_0(w))v_{w_0(w)w};$$
here $S(ws_iw^{-1})$ is
the set of simple reflections in any reduced decomposition of
$ws_iw^{-1}$, and $w_0(w)$ is the longest element in $W_{S(ws_iw^{-1})}$.
It follows that $\operatorname{gr}\phi^2$ sends $(v_y)$ to $(\tau(w_0(y))^2v_y)$.
By induction it is easy to see that it would be enough to prove the
surjectivity of this map for the smallest filtration term
$F_l(\overline{V}_0)$ where $l$ is the maximal length of reflections in $W$.
Assume that $W$ is of classical type.
Let $r_0$ be the reflection corresponding to the maximal positive root.
Then $r_0$ has maximal length in its conjugacy class.
We can choose $s_i\in S$ which is conjugate to $r_0$.
Then the space $F_l(\overline{V}_0)$ consists of collections
$(v_y, y\in P_i, ys_iy^{-1}=r_0)$ such that $v_y\in V_{s}$ whenever
$sr_0s\neq r_0$,
and $v_{sy}-sv_y\in V_{s}$ if $sr_0s=r_0$.
If $W=S_n$ is the symmetric group, then $r_0$ is the transposition $(1,n)$.
The space $F_l(\overline{V}_0)$ consists of collections
$(v_y, y\in P_i, ys_iy^{-1}=r_0)$ such that $v_y\in V_{s_1}\cap V_{s_{n-1}}$
and $v_{s_jy}-s_jv_y\in V_{s_j}$ for $1<j<n-1$. Note that
the set of $y\in P_i$ such that $ys_iy^{-1}=r_0$ constitutes one coset
for the subgroup $W_{[2,n-2]}\subset W$ generated by $s_j$ with $1<j<n-1$.
\vspace{2mm}
\noindent
{\it Proof of Theorem \ref{thmAn}}.
Let $W=S_4$.
Then $s_jr_0\neq r_0s_j$ for $j\neq 2$
and $s_2r_0=r_0s_2$. We have an involution $(v_y)\mapsto (v_{s_2y})$
on $F_l(\overline{V}_0)$. Consider the corresponding decomposition
$F_l(\overline{V}_0)=F_l^+\oplus F_l^{-}$. Thus, $F_l^{+}$ consists
of collections $(v_y, y\in P_i, ys_iy^{-1}=r_0)$ such that
$v_{s_jy}\in V_{s_j}$ for $j\neq 2$,
$v_{s_2y}=v_y$, $(s_2-1)v_y\in V_{s_2}$.
Thus, for any component $v_y$ we have $s_j(v_y)=qv_y$ for $j\neq 2$,
$(s_2-1)(s_2-q)v_y=0$. It is easy to deduce from these conditions
that the subrepresentation of $B$ generated by $v_y$ factors through
the quadratic Hecke algebra $H_q$. It remains to apply Theorem
\ref{quHecke}.
\qed\vspace{3mm}
\subsection{Some linear algebra}
\begin{lem}\label{findi}
Let $V$ be a representation of the cubic Hecke
$H^c_q$ corresponding to $(W,S)$ of type $B_2$, where $q\neq 0$. Assume that
$v\in V$ is such that $s_1v=q v$ and
$(s_2-q)(s_2-1)v=0$. Then the subspace ${\Bbb C}[B]v\subset V$
is finite-dimensional.
\end{lem}
\noindent {\it Proof} . We claim that the subspace $V'$ spanned by elements
$$(x, s_2x, s_1s_2x, s_1^{-1}s_2x, s_2s_1s_2x, s_2s_1^{-1}s_2x,
s_1s_2s_1^{-1}s_2x)$$
is closed under $s_1$ and $s_2$.
To prove this first note that $s_1^{-1}s_2^{-1}s_1^{-1}$
commutes with $s_2$, and therefore,
$$(s_2-q)(s_2-1)s_1^{-1}s_2^{-1}x=0.$$
Since $s_1^{-1}s_2x$ is a linear combination of $x$ and $s_1^{-1}s_2^{-1}x$
it follows that
$$(s_2-q)(s_2-1)s_1^{-1}s_2x=0.$$
Similarly since $s_1s_2s_1$ commutes with $s_2$ we get
$$(s_2-q)(s_2-1)s_1s_2x=0.$$
It follows that the subspace spanned by
$$(x, s_2x, s_1s_2x, s_1^{-1}s_2x, s_2s_1s_2x, s_2s_1^{-1}s_2x)$$
contains $s_2^ns_1^ks_2^l$ for any $n,k,l\in{\Bbb Z}$.
It remains to show that $s_2s_1s_2s_1^{-1}s_2x$ and
$s_1^2s_2s_1^{-1}s_2x$ belong to $V'$. Because of the cubic relation
we have $s_1^2s_2s_1^{-1}s_2x\equiv s_1^{-1}s_2s_1^{-1}s_2x\mod V'$.
Since $s_2s_1^{-1}s_2x$ is a linear combination of $s_1^{-1}s_2x$ and
$s_2^{-1}s_1^{-1}s_2x$, it follows that
$s_1^{-1}s_2s_1^{-1}s_2x\in V'$. Finally,
$s_2s_1s_2s_1^{-1}s_2x=s_1^{-1}s_2s_1s_2^2x$ is a linear combination
of $s_1^{-1}s_2s_1s_2x=q^{-1}s_2s_1s_2x$ and
$s_1^{-1}s_2s_1x$, so we are done.
\qed\vspace{3mm}
\begin{lem}\label{q8mu}
Let $s_1$, $s_2$ be a pair of invertible operators
on a vector space $V$ such that $(s_1s_2)^2=(s_2s_1)^2$ and
$(s_1-q)(s_1^2-1)=0$, and let
$v\in V$ be a vector such that $s_1v= q v$,
$(s_2-q)(s_2-1)v=0$, and $(s_1s_2)^2v=\mu v$, where $q\neq 0$ and $\mu$
are constants.
Assume that $q^8\neq 1$. Then $\mu^2\neq 1$.
\end{lem}
\noindent {\it Proof} .
First, we claim that the subspace of $V$ spanned
by $v$ and $s_2v$ is closed under $s_1$ and $s_2$.
Indeed, the identity
$$\mu v=(s_2s_1)^2v=q s_2s_1s_2 v$$
implies that
$$s_1s_2v=q^{-1}\mu s_2^{-1}v=q^{-2}(q+1)\mu v-q^{-2}\mu s_2v.$$
Thus, we can assume that $V$ is generated by $v$ and $s_2v$.
If $s_2v=\nu v$, then $\nu$ is equal to either $1$ or $q$
and $\mu=q^2\nu^2$, so the assertion follows. Otherwise,
$v$ and $s_2v$ constitute a basis of $V$. The matrix of $s_1$
with respect to this basis is
$\left( \matrix q & \frac{\mu(q+1)}{q^2}\\
0 & -\frac{\mu}{q^2} \endmatrix \right)$. Since $s_1$ is diagonalizable
with eigenvalues among $\{\pm 1,q\}$, it follows that
$-\frac{\mu}{q^2}=\pm 1$, i.e. $\mu=\pm q^2$.
\qed\vspace{3mm}
\noindent
{\it Proof of Theorem \ref{B_2}}.
We have $r_0=s_2s_1s_2$, $s_i=s_1$. Thus,
the space $F_l(\overline{V}_0)$ consists of pairs
$(v_{s_2},v_{s_1s_2})\in (V_{s_2})^{\oplus 2}$ such that
$v_{s_1s_2}-s_1v_{s_2}\in V_{s_1}$. We have the natural involution
on $F_l(\overline{V}_0)$ interchanging the components, so that
$F_l^{\pm}$ consists of $(v,\pm v)$, such that $s_2v=qv$,
$(s_1-q)(s_1 \mp 1)v=0$. According to Lemma \ref{findi}
we can assume that $V$ is finite-dimensional.
Also we can assume that $V$ is irreducible as a representation of $B$.
Then the central element $(s_1s_2)^2$ acts as a constant $\mu$ on $V$,
and we can apply Lemma \ref{q8mu} to finish the proof.
\qed\vspace{3mm}
\subsection{Good representations and parabolic induction}
Let $J\subset S$ be a subset, $W_J\subset W$
the corresponding parabolic subgroup, and $V_0$ a
representation of $B_J$, the braid group corresponding to
$(W_J,J)$. In section \ref{induction} we associated with $V_0$
the representation of $B$ in
$$V=\oplus_{x\in W/W_J}V_x,$$
where $V_x=V_0$ such that
$b\in B$ sends $V_x$ to $V_{\overline{b}x}$.
\begin{prop}\label{goodind}
If $V_0$ is a good representation of $B_J$, then
$V$ is a good representation of $B$.
\end{prop}
\noindent {\it Proof} . We have a direct sum decomposition
$$K_W(V)\simeq \oplus_{x\in W/W_J}K_{W,x}(V)$$
where
$$K_{W,x}(V)=\{(v_w,w\in W)\ |\ v_w\in V_{wx},
sv_w-v_{sw}\in (s^2-1)V_{wx} \}.$$
Furthermore, the map
$i_y:V\rightarrow K_W(V)$ for $y\in W$
decomposes into the direct sum of maps
$i_{y,x}:V_{yx}\rightarrow K_{W,x}(V)$ where $x\in W/W_J$.
Thus, it is sufficient to check that for every
$x\in W/W_J$ the images of $i_{y,x}$, $y\in W$ generate
$K_{W,x}(V)$. Let $\widetilde{x}\in W$ be a representative of $x$.
Then we have the isomorphism
$$K_{W,x}(V)\widetilde{\rightarrow}K_{W,x_0}(V):
(v_w)\mapsto (v_{w\widetilde{x}^{-1}})$$
where $x_0\in W/W_J$ is the class containing $1$.
Under this isomorphism the map $i_{y,x}$ corresponds
to $i_{y\widetilde{x},x_0}$. Thus, we can assume that $x=x_0$.
Now we claim that the canonical projection
$$p_{W_J}:K_{W,x_0}(V)\rightarrow K_{W_J}(V_0),$$
leaving only coordinates corresponding to $w\in W_J$, is an isomorphism.
Indeed, let $v=(v_w,w\in W)\in K_{W,x_0}(V)$ be an element.
For any $x=wx_0\in W/W_J$ and $s\in S$ such that $sx\neq x$,
we have $(s^2-1)V_x=0$, hence $v_{sw}=sv_w$. It follows that
for every $w\in W$ we have $v_w=n_{W_J}(w)v_{p_{W_J}(w)}$.
Conversely, the latter formula produces an element of
$K_{W,x_0}(V)$ from an arbitrary element of $K_{W_J}(V_0)$.
Note that for $y\in W_J$, the map $i_{y,x_0}$ corresponds via
$p_J$ to the map $i_y:V_0\rightarrow K_{W_J}(V_0)$. Since the
$B_J$-representation $V_0$ is good, we conclude that
$K_{W,x_0}$ is generated by images of $i_{y,x_0}$, $y\in W_J$.
\qed\vspace{3mm}
The main example where the above proposition works is the
following. Consider the situation of gluing on the
basic affine space of a group $G$ where $G$ and $B$ are defined over
the finite field ${\Bbb F}_q$. Then we have the trace map
$\operatorname{tr}: K_0(\operatorname{Perv}(G/U)^{\operatorname{Fr}})\rightarrow C( G/U({\Bbb F}_q))$
where $\operatorname{Perv}(G/U)^{\operatorname{Fr}}$ is the category of perverse Weil sheaves on
$G/U$, and $C( G/U({\Bbb F}_q))$ is the space of functions
$G/U({\Bbb F}_q)\rightarrow\overline{{\Bbb Q}}_l$.
The action of the braid group $B$ corresponding to $(W,S)$
on $\operatorname{Perv}(G/U)^{\operatorname{Fr}}$
is compatible with the trace map and induces the action of $B$
on $C( G/U({\Bbb F}_q))$.
\begin{thm} Assume that the center of $G$ is connected.
Then the representation of the braid group $B$ on
$C(G/U({\Bbb F}_q))$ is good.
\end{thm}
\noindent {\it Proof} . We have an action of the finite torus $T({\Bbb F}_q)$ on
$G/U({\Bbb F}_q)$ by right translation. Let
$C_{\theta}\subset C(G/U({\Bbb F}_q))$ be
the subspace on which $T({\Bbb F}_q)$ acts through the character
$\theta:T({\Bbb F}_q)\rightarrow\overline{{\Bbb Q}}_l^*$. Then we have the direct
sum decomposition
$$C(G/U({\Bbb F}_q))=\oplus_{\theta}C_{\theta}$$
and the action of $b\in B$ sends
$C_{\theta}$ to $C_{\overline{b}\theta}$.
We claim that if for some character $\theta$ and a simple
reflection $s$, one has $s\theta\neq\theta$; then
$s^2=1$ on $C_{\theta}$. Indeed, let
$f:G/U({\Bbb F}_q)\rightarrow\overline{{\Bbb Q}}_l$ be a function such that
$f(xt)=\theta(x)f(x)$. Then
$$\sum_{\lambda\in{\Bbb F}_q^*}f(x\a_s(\lambda))=0,$$
hence, Proposition \ref{Fouriersq} implies that $s^2f=f$.
Now let $O$ be an orbit of $W$ on the set of characters of
$T({\Bbb F}_q)$. Clearly it is sufficient to prove that
the representation of $B$ on
$C_O=\oplus_{\theta\in O}C_{\theta}$
is good. Now we can choose a representative $\theta_0\in O$
in such a way that all reflections stabilizing $\theta_0$
belong to $S$. Then the stabilizer of $\theta_0$ in $W$
is the parabolic subgroup $W_J\subset W$ corresponding to
some subset $J\subset S$ (this follows from Theorem 5.13 of \cite{DL}).
As we have shown above the representation of $C_O$
belongs to $\operatorname{Mod}_J-B$; hence it is recovered from the
representation of $B_J$ on $C_{\theta_0}$ via
the construction of section \ref{induction}. According
to Proposition \ref{goodind} it is sufficient to check
that the representation of $B_J$ on $C_{\theta}$ is good.
But on $C_{\theta}$ we have $(s+q^{-1})(s-1)=0$ for every
$s\in J$ (this follows from the analogue of Proposition
\ref{Fourproj} for the finite Fourier transorm);
hence we are done by Corollary \ref{qubraid}.
\qed\vspace{3mm}
\subsection{Final remarks}
Recall that according to Theorem \ref{K_0A} and Corollary
\ref{s2-1} we have
$$K_0(\AA^{\operatorname{Fr}})\otimes_{{\Bbb Z}[u,u^{-1}]}\overline{{\Bbb Q}}_l\simeq
K_W(K_0(\operatorname{Perv}(G/U)^{\operatorname{Fr}})\otimes_{{\Bbb Z}[u,u^{-1}]}\overline{{\Bbb Q}}_l)$$
where $\AA^{\operatorname{Fr}}$ is the
analogue of Weil sheaves in the Kazhdan---Laumon category,
and the homomorphism
${\Bbb Z}[u,u^{-1}]\rightarrow\overline{{\Bbb Q}}_l$ sends $u$ to $q$. Hence, we have
the natural trace map
\begin{equation}\label{tracemap}
\operatorname{tr}:K_0(\AA^{\operatorname{Fr}})\rightarrow K_W(C( G/U({\Bbb F}_q))),
\end{equation}
induced by the trace maps on every component.
If the center of $G$ is connected, then the previous theorem
implies the surjectivity of this map.
Assuming that the cohomological dimension of $\AA$ is finite, Kazhdan
and Laumon defined a bilinear pairing on
$K_0(\AA^{\operatorname{Fr}})$ by taking traces of Frobenius on $\operatorname{Ext}$-spaces.
Without this assumption one can still define this pairing for
classes of objects of finite projective dimension.
It is easy to check that
the map (\ref{tracemap}) is compatible with this pairing
and the non-degenerate form on
$K_W(C( G/U({\Bbb F}_q)))$ defined in Proposition \ref{form}
(where $C( G/U({\Bbb F}_q))$ is equipped with the standard scalar product
in which delta-functions constitute an orthonormal basis).
If in addition we know that
$K_0(\operatorname{Perv}(G/U)^{\operatorname{Fr}})\otimes_{{\Bbb Z}[u,u^{-1}]}\overline{{\Bbb Q}}_l$ is
a good representation of $B$, then we can deduce that the map
(\ref{tracemap}) is the quotient of $K_0(\AA^{\operatorname{Fr}})$ by the kernel
of the pairing with $K_0(\AA^{\operatorname{Fr}})$.
Since the representation of the braid group on $K_0(G/U)$ factors
through the cubic Hecke algebra, we can check the latter implication
in the cases where $(W,S)$ is of type $A_2$, $A_3$, or $B_2$.
On the other hand, using Corollary \ref{findimwt} we can always define
a bilinear pairing on $K_0(\AA_m)$ with values in the field of
Laurent series $\overline{{\Bbb Q}}_l((u))$ by looking at the action of Frobenius
on weight-$n$ components of the $\operatorname{Ext}$-spaces. In the cases when
the representation of the braid group on
$K_0(\operatorname{Perv}_m((G/U)_0))\otimes_{{\Bbb Z}[u,u^{-1}]}{\Bbb Q}(u)$
is good, the Laurent series obtained as
values of this pairing are actually rational.
We conjecture that in fact they are always rational. Furthermore,
a similar pairing can be defined in the case when the Frobenius
automorphism is twisted by the action of some $w\in W$. We conjecture
that it still takes values in rational Laurent series and that
the quotient of
$K_0(\AA_m^{w\operatorname{Fr}})\otimes_{{\Bbb Z}[u,u^{-1}]}\overline{{\Bbb Q}}_l((u))$
by the kernel of this pairing is finite-dimensional.
\section{Appendix. Counterexample to the conjecture of
Kazhdan and Laumon.}
In this appendix we will show that the Kazhdan---Laumon
category $\AA$ has infinite
cohomological dimension in the simplest non-trivial case
$G=\operatorname{SL}_3$.
For every $w\in W$ let us denote by
$O_w$ the simple object in $\AA$, such that the
corresponding gluing data is the constant sheaf
$\overline{{\Bbb Q}}_{l,X}[d]$ at the place $w\in W$,
and zero at all the others, where $d=\dim X$.
Recall that for every $w\in W$ the functor
$j_{w,!}:\operatorname{Perv}(X)\rightarrow\AA$ has the left derived one
$Lj_{w,!}$
(see Proposition \ref{leftderived} and the
remark after it). So for every $w\in W$ we can introduce
the following object in the derived category of $\AA$:
$P_w=Lj_{w,!}(\overline{{\Bbb Q}}_{l,X}[d])$.
Note that by Proposition \ref{leftderived}
the functor $Lj_{w,!}$ is left adjoint to the
corresponding restriction functor, hence, we have
$$V^*_{w,w'}:=\operatorname{Ext}^*(P_w,O_{w'})=\cases 0,\ w\neq w',\\ H^*(X)=H^*(G), w=w'.
\endcases$$
(here cohomology is taken with coefficients in $\overline{{\Bbb Q}}_l$).
We want to compare these spaces with the spaces
$$E^*_{w,w'}:=\operatorname{Ext}^*(O_{w}, O_{w'})$$
(in fact, $E^*_{w,w'}$ depends only on $w'w^{-1}$).
This is done with the help of the following lemma.
\begin{lem}\label{labelrom}
For every $i$ we have a canonical isomorphism in $\AA$
$$H^{-i}(P_w)= \oplus_{w', \ell(w')=i} O_{w'w}(i).$$
\end{lem}
\begin{rem} Most of the results in this paper can be translated into the
parallel setting where algebraic $D$-modules on the complex
algebraic variety
$(G/U)_{{\Bbb C}}$
are used instead of
$l$-adic sheaves over the corresponding variety in characteristic
$p$. The main result of \cite{BBP} asserts that the corresponding
``glued'' category is equivalent to the category of modules over the
ring of global differential operators on $(G/U)_{{\Bbb C}}$.
The functor $Rj_*$ (the Verdier dual to the $D$-module
counterpart of the functor $Lj_!$ considered above; it is somewhat
more natural to work with this Verdier dual functor in the $D$-module
setting) is then identified with the derived functor of global sections
from the category of $D$-modules to the category of modules over the
global differential operators.
The $D$-module counterpart of Lemma \ref{labelrom} is easily seen to be
equivalent to the Borel-Weil-Bott Theorem (which computes cohomology
of an equivariant line bundle on the flag variety $(G/B)_{{\Bbb C}}$).
\end{rem}
\noindent
{\it Proof of Lemma \ref{labelrom}}.
Recall that by Proposition \ref{leftderived}
the composition of functors
$j^*_{w'w}\circ Lj_{w,!}$ coincides with the left derived functor of
$\sideset{^p}{^0}{H} F_{w',!}$. Furthermore,
Theorem \ref{adapted} implies that under the
identification of the derived category of $\operatorname{Perv}(X)$ with
$D^b_c(X,\overline{{\Bbb Q}}_l)$ the
latter derived functor coincides with $F_{w',!}$.
Therefore, we have
$$j^*_{w'w}P_w=j^*_{w'w}Lj_{w,!}(\overline{{\Bbb Q}}_{l,X}[d])\simeq
F_{w',!}(\overline{{\Bbb Q}}_{l,X}[d]).$$
According to Lemma \ref{mainl} for every simple reflection $s$ we have
$$F_{s,!}(\overline{{\Bbb Q}}_{l,X}[d])=(\overline{{\Bbb Q}}_{l,X}[d])[1](1).$$
Therefore, for every $w'\in W$ we have
$$F_{w',!}(\overline{{\Bbb Q}}_{l,X}[d])=(\overline{{\Bbb Q}}_{l,X}[d])[\ell(w')](\ell(w')).$$
Hence, for every $i$ we have
$$j^*_{w'w}H^{-i}(P_w)=\cases 0,\ \ell(w')\neq i,\\ \overline{{\Bbb Q}}_{l,X}[d](i),\
\ell(w')=i.\endcases$$
This immediately implies our statement.
\qed\vspace{3mm}
Thus, we have a spectral sequence with the $E_2$-term
$$\oplus_{p\le 0,q\ge 0} \oplus_{\ell(w_1)=-p} E^q_{w_1w,w'}(p)$$
converging to $V^*_{w,w'}$.
Now let us assume that the spaces $E^*_{w,w'}$ are finite-dimensional
and lead this to contradiction.
Note that all our spaces carry a canonical (mixed) action of Frobenius
which is respected by this spectral sequence. We can encode some information
about Frobenius action on such a space
by considering Laurent polynomials in $u$, where
the coefficient with $u^n$ is the super-dimension of the weight-$n$
component. Let $e_{w,w'}$ (resp.
$v_{w,w'}$) be such a Laurent polynomial in $u$ corresponding to
$E^*_{w,w'}$ (resp. $V^*_{w,w'}$).
Then the above spectral sequence implies that
\begin{equation}\label{inconsist}
v_{w,w'}=\sum_{w_1\in W} e_{w_1w,w'}(-u)^{\ell(w_1)}
\end{equation}
Now recall that $v_{w,w'}=0$ for $w\neq w'$ while
$v_{w,w}$ is equal to the Poincare polynomial of $G$
$p_G(u)=\prod(1-u^{e_i})$, with $e_i$ running over the set
of exponents of $G$.
Therefore, we can rewrite (\ref{inconsist}) in the matrix form
as follows.
Let us consider the matrix $E=(e_{w,w'})$
with rows and columns numbered by $W$
and with entries in ${\Bbb Z}[u,u^{-1}]$. Let us also define the matrix $M$
of similar type by setting
$$M_{w,w'}=(-u)^{\ell(w'w^{-1})}.$$
Then (\ref{inconsist}) is equivalent to the following equality of matrices:
\begin{equation}\label{matrixeq}
ME= p_G\cdot I
\end{equation}
where $I$ is the identity matrix.
Now we are going to show that $M$ does not divide $p_G\cdot I$ for
$G=\operatorname{SL}(3)$
in $Mat_6[u,u^{-1}]$ (as $\det (M)$ vanishes at a primitive
$6$-th root of $1$, while $p_G$ does not).
Recall that for any finite group $G$ one can form a matrix
$M^G \in Mat_n ({\Bbb Z}[x_1, .., x_n])$, $n=|G|$; here rows/columns of $M^G$
and variables $x$ of the polynomial ring are indexed by elements of $G$, and
we set $M^G_{g_1,g_2}=x_{g_1g_2}$.
It is well known that
$\det(M^G)$ is the product of factors indexed by irreducible representations
of $G$, and the degree of a factor, as well as the power in which
it enters the decomposition equals the dimension of the representation.
Moreover, the factor corresponding to a ($1$-dimensional) character
of $G$ is $\sum \chi(g) x_g$.
Now our matrix $M$ is obtained from $M^G$ for $G=W$ by
$x_w\mapsto (-u)^{\ell (w)}$
and multiplication by a matrix of the permutation $w\mapsto w^{-1}$.
So $\det(M)$ is divisible by $(\sum_{w\in W} u^{\ell(w)}) \cdot
(\sum_{w\in W} (-u)^{\ell(w)})$.
For $G=\operatorname{SL}(3)$ we get that $\det(M)$ is divisible by
$1+2u + 2 u^2 + u^3$ and $1 - 2u +2 u^2 -u^3$.
The latter polynomial vanishes at a primitive $6$-th root of
$1$. However, $p_{\operatorname{SL}_3}=(1-u^2)(1-u^3)$,
hence the equality (\ref{matrixeq})
is impossible.
Note that this counterexample does not contradict to the results
of section \ref{goodsec} since in that section we considered the
localization of $K_0(\AA)$ on which $u$ does not act as a root of unity.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,752
|
WBSV may refer to:
WBSV-LP, a low-power radio station (93.3 FM) licensed to serve Berrien Springs, Michigan, United States
WFTT-TV, a television station (channel 25, virtual 62) licensed to serve Venice, Florida, United States, which held the call sign WBSV-TV from 1987 to 2000
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,101
|
The Kaye List
by Veronica Jane on January 14, 2023
The Basics of the DNX Vaccine
There is much talk these days about the new DNX vaccine. It's a breakthrough in medical science that promises to help protect us from all kinds of diseases, and it's quickly gaining traction among health care providers and patients alike. But what exactly is the DNX vaccine? And how does it work? Let's take a look at this revolutionary new vaccine so you can make an informed decision on whether or not it's right for you.
How Does the DNX Vaccine Work?
The DNX vaccine works by using mRNA technology, which stands for "messenger RNA". Essentially, this technology helps create proteins within cells that act as antigens – substances that stimulate an immune response when they enter the body. When these proteins are injected into the body via a vaccination, they trigger an immune response that helps protect against certain diseases. This means that instead of introducing foreign bodies such as viruses or bacteria into your system, the vaccines use your own body's defense mechanisms to fight off disease. This makes them safer and more effective than traditional vaccines.
What Does The DNX Vaccine Protect Against?
The DNX vaccine is currently approved to protect against three different types of diseases – measles, mumps, and rubella. It has been proven to be highly effective in protecting against these illnesses, with studies showing it reduces the risk of contracting any one of them by up to 90%. In addition to this, there have been promising results in clinical trials for other diseases such as influenza, hepatitis B and C, rotavirus and human papillomavirus (HPV). With further research and development, there may be even more illnesses that can be prevented with this revolutionary new type of vaccination in future years.
The DNX vaccine is a groundbreaking new type of vaccination that uses mRNA technology to help protect users from various illnesses such as measles, mumps and rubella. Early tests have shown promising results for other potential diseases such as influenza and HPV. If you're looking for a safe and effective way to protect yourself from illness without introducing foreign bodies into your system, then consider exploring the possibilities offered by the DNX vaccine today!
Finding Strength in Partners Against Pain
by Veronica Jane
The Story Behind "Fas de Angelina"
Abuse is Rise. Why Doctors Fail
WHAT IS DIFFERENT ABOUT KOREAN SKIN CARE?
3 Benefits of Having an Armored Vehicle
10 Ways a Cloud Phone System Can …
Faith Freedom Fear Americas Covid Vaccine
Hundreds Suicidal Emergency Rooms. Every
2023© The Kaye List
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,963
|
package assignment10;
import java.io.File;
import java.util.ArrayList;
import java.util.Random;
/**
* Times our program. This class will not run properly without the
* GenerateAlternateSpellingsForTiming class and the SpellingCorrectionTime class
* This to correctly test the generateAltSpellOnlyWithFileWriting() method you
* must comment out any code in the GenerateAlternateSpellingsForTiming that has
* to do with the dictionary Hash Map and the validItems Hash Map.
*
* @author Daryl Bennett & Leland Stenquist
*/
public class TimeSpelling
{
public static void main(String[] args) {
// comment the ones you don't want to run
generateAltSpellOnly();
System.out.println("test1 done");
generateAltSpellOnlyWithFileWriting();
System.out.println("test2 done");
generateAltSpellOnlyWithQueue();
System.out.println("test3 done");
generateAltSpellOnlyWithQueueWithFiles();
System.out.println("test4 done");
runSpellCor();
System.out.println("test5 done");
runSpellCorWithFile();
System.out.println("test6 done");
}
/**
* Times the run time of our creation of alternative spellings and
* only alternative spellings. There is no file creation.
*/
public static void generateAltSpellOnly() {
for (int size = 100; size <= 500; size = size + 10) {
// SETUP TASKS
// fill an array with random strings
ArrayList<String> stringsArrayList = fillArray(size);
// Make sure the size of the collection matches the size we set
// it to.
if (stringsArrayList.size() != size) {
System.out.println("Size does not match");
System.exit(0);
}
// Timing code starting point
long startTime, midpointTime, stopTime;
// First, spin computing stuff until one second has gone by.
// This allows this thread to stabilize.
startTime = System.nanoTime();
while (System.nanoTime() - startTime < 1000000000) {
} // empty block
// Now, run the test.
long timesToLoop = 100;
startTime = System.nanoTime();
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < stringsArrayList.size(); j++) {
createAllAlternateSpellingsOnly(stringsArrayList.get(j));
}
}
midpointTime = System.nanoTime();
// Run an empty loop to capture the cost of running the loop.
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < size; j++) {
}
}
stopTime = System.nanoTime();
// Compute the time, subtract the cost of running the loop
// from the cost of running the loop and computing square roots.
// Average it over the number of runs.
double averageTime = ((midpointTime - startTime) - (stopTime - midpointTime))
/ timesToLoop;
System.out.println(averageTime + "");
}
}
/**
* This method times how long it takes to create all the alternate
* spellings and write them to a file
*/
public static void generateAltSpellOnlyWithFileWriting() {
//to run this would be impossible without commenting out the appropriate code
//from the GenerateAlternateSpellingsForTiming Class.
for (int size = 100; size <= 500; size = size + 10) {
// SETUP TASKS
// fill an array with random strings
ArrayList<String> stringsArrayList = fillArray(size);
// Make sure the size of the collection matches the size we set
// it to.
if (stringsArrayList.size() != size) {
System.out.println("Size does not match");
System.exit(0);
}
// Timing code starting point
long startTime, midpointTime, stopTime;
// First, spin computing stuff until one second has gone by.
// This allows this thread to stabilize.
startTime = System.nanoTime();
while (System.nanoTime() - startTime < 1000000000) {
} // empty block
// Now, run the test.
long timesToLoop = 100;
startTime = System.nanoTime();
//remember the code is only commented out in the GenerateAlternateSpellingsForTiming class to stop it from
//creating a priority queue
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < stringsArrayList.size(); j++) {
GenerateAlternateSpellingsForTiming gA = new GenerateAlternateSpellingsForTiming(stringsArrayList.get(j),true);
gA.deletion();
gA.transposition();
gA.substitution();
gA.insertion();
}
}
midpointTime = System.nanoTime();
// Run an empty loop to capture the cost of running the loop.
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < size; j++) {
}
}
stopTime = System.nanoTime();
// Compute the time, subtract the cost of running the loop
// from the cost of running the loop and computing square roots.
// Average it over the number of runs.
double averageTime = ((midpointTime - startTime) - (stopTime - midpointTime))
/ timesToLoop;
System.out.println(averageTime + "");
}
}
/**
* This code times how long it takes to generate all the alternate spellings and put
* any valid alternatives into a HashMap to check their frequency. No file is created
*/
public static void generateAltSpellOnlyWithQueue() {
for (int size = 100; size <= 500; size = size + 10) {
// SETUP TASKS
// populate the dictionary
SpellingCorrectionTime sc =new SpellingCorrectionTime();
sc.populate(new File("WordStats.txt"));
// fill an array with random strings
ArrayList<String> stringsArrayList = fillArray(size);
// Make sure the size of the collection matches the size we set
// it to.
if (stringsArrayList.size() != size) {
System.out.println("Size does not match");
System.exit(0);
}
// Timing code starting point
long startTime, midpointTime, stopTime;
// First, spin computing stuff until one second has gone by.
// This allows this thread to stabilize.
startTime = System.nanoTime();
while (System.nanoTime() - startTime < 1000000000) {
} // empty block
// Now, run the test.
long timesToLoop = 50;
startTime = System.nanoTime();
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < stringsArrayList.size(); j++) {
sc.process(stringsArrayList.get(j));
}
}
midpointTime = System.nanoTime();
// Run an empty loop to capture the cost of running the loop.
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < size; j++) {
}
}
stopTime = System.nanoTime();
// Compute the time, subtract the cost of running the loop
// from the cost of running the loop and computing square roots.
// Average it over the number of runs.
double averageTime = ((midpointTime - startTime) - (stopTime - midpointTime))
/ timesToLoop;
System.out.println(averageTime + "");
}
}
/**
* This code times how long it takes to generate all the alternate spellings and put
* any valid alternatives into a HashMap to check their frequency. The file is created
*/
public static void generateAltSpellOnlyWithQueueWithFiles() {
for (int size = 100; size <= 500; size = size + 10) {
// SETUP TASKS
SpellingCorrectionTime sc =new SpellingCorrectionTime();
//set boolean to true so the file is created
sc.printCommand =true;
// populate the dictionary
sc.populate(new File("WordStats.txt"));
// fill an array with random strings
ArrayList<String> stringsArrayList = fillArray(size);
// Make sure the size of the collection matches the size we set
// it to.
if (stringsArrayList.size() != size) {
System.out.println("Size does not match");
System.exit(0);
}
// Timing code starting point
long startTime, midpointTime, stopTime;
// First, spin computing stuff until one second has gone by.
// This allows this thread to stabilize.
startTime = System.nanoTime();
while (System.nanoTime() - startTime < 1000000000) {
} // empty block
// Now, run the test.
long timesToLoop = 50;
startTime = System.nanoTime();
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < stringsArrayList.size(); j++) {
sc.process(stringsArrayList.get(j));
}
}
midpointTime = System.nanoTime();
// Run an empty loop to capture the cost of running the loop.
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < size; j++) {
}
}
stopTime = System.nanoTime();
// Compute the time, subtract the cost of running the loop
// from the cost of running the loop and computing square roots.
// Average it over the number of runs.
double averageTime = ((midpointTime - startTime) - (stopTime - midpointTime))
/ timesToLoop;
System.out.println(averageTime + "");
}
}
/**
* Test how long it takes to create the dictionary, create the alternate
* spellings and if valid put them into a Hash Map to return the
* one with the highest frequency. No file is created
*/
public static void runSpellCor() {
for (int size = 100; size <= 500; size = size + 10) {
// SETUP TASKS
SpellingCorrectionTime sc =new SpellingCorrectionTime();
// don't create a file
sc.printCommand =false;
// fill an array with random strings
ArrayList<String> stringsArrayList = fillArray(size);
// Make sure the size of the collection matches the size we set
// it to.
if (stringsArrayList.size() != size) {
System.out.println("Size does not match");
System.exit(0);
}
// Timing code starting point
long startTime, midpointTime, stopTime;
// First, spin computing stuff until one second has gone by.
// This allows this thread to stabilize.
startTime = System.nanoTime();
while (System.nanoTime() - startTime < 1000000000) {
} // empty block
// Now, run the test.
long timesToLoop = 50;
startTime = System.nanoTime();
sc.populate(new File("WordStats.txt"));
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < stringsArrayList.size(); j++) {
sc.process(stringsArrayList.get(j));
}
}
midpointTime = System.nanoTime();
// Run an empty loop to capture the cost of running the loop.
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < size; j++) {
}
}
stopTime = System.nanoTime();
// Compute the time, subtract the cost of running the loop
// from the cost of running the loop and computing square roots.
// Average it over the number of runs.
double averageTime = ((midpointTime - startTime) - (stopTime - midpointTime))
/ timesToLoop;
System.out.println(averageTime + "");
}
}
/**
* Test how long it takes to create the dictionary, create the alternate
* spellings and if valid put them into a Hash Map to return the
* one with the highest frequency. The file is created
*/
public static void runSpellCorWithFile() {
for (int size = 100; size <= 500; size = size + 10) {
// SETUP TASKS
SpellingCorrectionTime sc =new SpellingCorrectionTime();
// create file
sc.printCommand =true;
//fill an array with random strings
ArrayList<String> stringsArrayList = fillArray(size);
// Make sure the size of the collection matches the size we set
// it to.
if (stringsArrayList.size() != size) {
System.out.println("Size does not match");
System.exit(0);
}
// Timing code starting point
long startTime, midpointTime, stopTime;
// First, spin computing stuff until one second has gone by.
// This allows this thread to stabilize.
startTime = System.nanoTime();
while (System.nanoTime() - startTime < 1000000000) {
} // empty block
// Now, run the test.
long timesToLoop = 50;
startTime = System.nanoTime();
sc.populate(new File("WordStats.txt"));
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < stringsArrayList.size(); j++) {
sc.process(stringsArrayList.get(j));
}
}
midpointTime = System.nanoTime();
// Run an empty loop to capture the cost of running the loop.
for (long i = 0; i < timesToLoop; i++) {
for (int j = 0; j < size; j++) {
}
}
stopTime = System.nanoTime();
// Compute the time, subtract the cost of running the loop
// from the cost of running the loop and computing square roots.
// Average it over the number of runs.
double averageTime = ((midpointTime - startTime) - (stopTime - midpointTime))
/ timesToLoop;
System.out.println(averageTime + "");
}
}
/**
* this is the method that generateAltSpellOnly() calls. It only
* generates the alternate spellings
*
* @param s
* - input string
*/
public static void createAllAlternateSpellingsOnly(String s) {
//create a char array of the alphabet
char[] alphabet = { 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i',
'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u',
'v', 'w', 'x', 'y', 'z' };
// create a char array inputChar to contain new word
char[] inputChar;
// create inputString
String inputString = s;
// initialize field variables
inputString = s;
inputChar = new char[inputString.length()];
inputString.getChars(0, inputString.length(), inputChar, 0);
// deletion----------------------------------------------------------
// create new array
String temp = inputString;
for (int i = 0; i < inputString.length(); i++) {
// create new String builder Object with inputString
StringBuilder sb = new StringBuilder(temp);
sb.deleteCharAt(i);
}
// transposition--------------------------------------------------------------------
char[] tempArray = inputString.toCharArray();
// char charTemp = temp[0];
for (int i = 0; i < tempArray.length - 1; i++) {
// doesn't swap elements if they are the same
// ---saves us a useless iteration
if (tempArray[i] == tempArray[i + 1]) {
continue;
}
// swap two elements in array
char charTemp = tempArray[i];
tempArray[i] = tempArray[i + 1];
tempArray[i + 1] = charTemp;
charTemp = tempArray[i];
tempArray[i] = tempArray[i + 1];
tempArray[i + 1] = charTemp;
}
// substitution---------------------------------------------------------------
// Loop through alphabet
for (int i = 0; i < inputString.length(); i++) {
// create new String builder Object with inputString
StringBuilder sb = new StringBuilder(temp);
// iterate thru the whole String inserting the items.
for (int j = 0; j < 26; j++) {
// if the letter is the same as the one about to be
// inserted...
// skip the iteration
if (inputChar[i] == alphabet[j])
continue;
sb.setCharAt(i, alphabet[j]);
}
// restore the String to it's original form
sb.setCharAt(i, inputChar[i]);
}
// insertion--------------------------------------------------------------------
// Loop through alphabet
for (int i = 0; i < inputString.length() + 1; i++) {
// create new String builder Object with inputString
StringBuilder sb = new StringBuilder(temp);
// iterate thru the whole String inserting the items.
for (int j = 0; j < 26; j++) {
sb.insert(i, alphabet[j]);
// restore the String to it's original form
sb.delete(i, i + 1);
}
}
}
/**
* Generates random strings and puts them into an ArrayList
*
* @param arraySize
* - size of the array to be filled
* @return
* - a ArrayList full of strings
*/
public static ArrayList<String> fillArray(int arraySize){
// size of the array
int size = arraySize;
// fill a char array with the alphabet
char[] alphabet = { 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h',
'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's',
't', 'u', 'v', 'w', 'x', 'y', 'z' };
// initialize and ArrayList of Strings
ArrayList<String> stringsArrayList = new ArrayList<String>();
// set a Random as rand
Random rand = new Random();
// make rand's seed 456
rand.setSeed(456);
// create an empty string
String s = "";
// loop from zero to size - 1
for (int i = 0; i < size; i++) {
// randomly generate a String size no bigger than ten
int stringSize = rand.nextInt(10);
// if the string size is zero change it to 5
if (stringSize == 0)
stringSize = 5;
// loop through the string size and add to the string s
for (int t = 1; t <= stringSize; t++) {
int getChar = rand.nextInt(25);
s = s + alphabet[getChar];
}
// add s to a the array list
stringsArrayList.add(s);
// make s empty again
s = "";
}
// return the ArrayList
return stringsArrayList;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,008
|
\section{Introduction}
In the past decade rapid progress has been made in our theoretical
understanding of the QCD phase structure, both with continuum methods
and with the lattice. By now, functional continuum approaches
to QCD allow us to discuss the strongly correlated low-energy sector
within a first-principle approach. Then the couplings to be fixed are
simply the fundamental parameters of QCD: the strong coupling
$\alpha_s$ and the current quark masses $m_{\rm current}$, see
\cite{Braun:2009gm,Pawlowski:2010ht,Haas:2013qwp,Herbst:2013ufa}. In
principle, this approach also allows to pin down the coupling
constants in low-energy effective models of QCD as functions of
$\alpha_s$ and $m_{\rm current}$. These models are usually defined at
a (UV) momentum scale $\Lambda_\text{UV}$ of about \SI{1}{GeV} in terms of an
effective Lagrangian with a set of coupling parameters $\vec
\lambda$. As low-energy couplings of QCD they can be deduced uniquely
from QCD as $\vec \lambda(\alpha_s,m_{\rm current})$. Such an approach
is completed by determining the set of all relevant low-energy
coupling parameters (at the UV scale $\Lambda_\text{UV}$) which may have an
impact on the infrared physics at hand. In summary, this set-up
anchors low-energy models within first-principle QCD and their
independent couplings are only those of QCD. As a consequence, not
only qualitative but also quantitative physics questions of the
strongly correlated low-energy sector of QCD become accessible.
As described above, it is of chief importance in this set-up to pin
down the relevant couplings in low-energy models. Moreover, the
analysis of low-energy quantum, thermal and density fluctuations in
these models have to be brought to a quantitative level. The current
work does a further significant step in this direction in the context
of the quark-meson (QM) model. We study the frequency and momentum
dependence of two-point correlation functions, which is interesting for
several reasons. Most importantly, it gives us direct access to the
relevant question of physical observables such as pole masses and
decay constants, which are so far only indirectly accessible in the
Euclidean approach. Within this context it also allows us to
determine and clarify the relations between the physical observables
determined at the poles, e.g.\ at $p^2=-m_\pi^2$ and the low-energy
parameters of the models at $p^2=0$. In particular, we determine the
relations between pole masses corresponding to propagator poles at
$p_0^2=-m_{\rm pol}^2$, screening masses corresponding to poles at
$\vec p^2=-m_{\rm scr}^2$ and curvature masses $m_{\rm cur}^2$
evaluated at $p^2=0$. Only the latter are directly accessible within
Euclidean approaches and are the mass parameters in the effective
action.
So far, the above relation and its convergence with a given
approximation scheme has not been studied. However, this is chiefly
important for the relative weight of quantum, thermal and density
fluctuations: The characteristic scale of quantum fluctuations is the
curvature mass $m_{\rm cur}$. Below this mass scale the propagation of
quantum fluctuations is suppressed. In turn, the characteristic scale
of density fluctuations is the pole mass $m_{\rm pol}$ (of modes with
non-vanishing quark number). At finite temperature these scales have
some temperature dependence. Thus, the correct identification of
$m_{\rm pol},\,m_{\rm scr}$ and $m_{\rm cur}$ is an important issue in
particular for quantitative approaches towards QCD at finite
temperature and density, where potential mismatches can lead to large
systematic errors.
We discuss this issue at the example of the derivative expansion,
which is an expansion in momenta over mass-scale, $p^2/m^2$. The
derivative expansion is the most popular expansion scheme used in
low-energy effective models. In most applications the local potential
approximation (LPA) is used, where one employs classical
propagators. In contrast, the computation in the present work involves
fully momentum-dependent propagators. This advanced approximation can
be used to resolve apparent inconsistencies reported in the literature
within the LPA, see
\cite{Strodthoff:2011tz,Kamikado:2012bt,Svanes:2010we}. Finally, we
also evaluate the quality of the LPA$'$ scheme, which includes
momentum-independent wavefunction renormalisation factors, in
comparison to the full calculation.
The strength of our computational approach lies in the fact, that it
provides a stable numerical iteration procedure with only little
numerical overhead compared to the momentum-independent
calculation. From the technical point of view, the method is
applicable to a wide range of possible theories. Furthermore, it can
be extended to complex external momenta along the lines of
\cite{Strodthoff:2011tz,Floerchinger:2011sc,Kamikado:2012bt,Tripolt:2013jra,Tripolt:2014wra}
hence providing direct access to spectral functions in an Euclidean
framework without the need for analytical continuation of given
Euclidean data.
The article is organised as follows. In Sect.~\ref{sec:formalism} we
discuss the embedding of low-energy effective models, and specify the
effective action of the quark-meson model. We also elaborate on the
different mass definitions and their physics content, as well as
providing a brief introduction to the functional renormalisation group (FRG)
and the computational set-up. In Sect.~\ref{sec:results} we present
the results of mass calculations at vanishing and finite
temperature. We discuss the implications for relative fluctuation
scales in particular in view of the chiral phase
boundary. Furthermore, we compare LPA, LPA$'$ and the calculation with
fully momentum-dependent mesonic propagators in terms of quantitative
accuracy. The latter is henceforth referred to as full calculation.
\section{Low-energy QCD and fluctuations}
\label{sec:formalism}
\subsection{Low-energy effective models}
\label{sec:effectiveaction}
We aim at describing the low-energy sector of two-flavor QCD within a
quark-meson model \cite{Ellwanger:1994wy,Jungnickel:1995fp, Berges:1997eu,Schaefer:2004en}.
As already mentioned in the introduction,
low-energy effective models can be firmly anchored in first-principle
QCD, see
\cite{Braun:2009gm,Pawlowski:2010ht,Haas:2013qwp,Herbst:2013ufa}. The
key idea behind this embedding in full QCD is the functional RG
approach to QCD with dynamical hadronisation,
\cite{Gies:2001nw,Gies:2002hq,Pawlowski:2005xe,Floerchinger:2009uf}.
There, the flow is initiated at a large momentum scale $\Lambda_{\rm
UV}\gg \Lambda_{\rm QCD}$, where it starts with the effective action
of perturbative QCD with dynamical quarks and gluons. Then, by
lowering the momentum scale within this first-principle QCD framework,
the hadronic degrees of freedom get dynamical at the hadronisation
scale, while the quark and gluon degrees of freedom decouple. This is
most simply seen in the Landau gauge, where the gluon propagator is
infrared gapped, the gapping being directly related to the QCD mass
gap, see e.g. \cite{Fischer:2008uz}. Accordingly, the gluons can be
integrated out first, leading to an effective theory with quarks and
hadronic degrees of freedom in a gluonic background potential at a
momentum scale $\Lambda_\text{UV}\approx$ \SI{1}{GeV}, such as Polyakov loop
enhanced low-energy models. First results within such a QCD-enhanced
model approach have been presented in
\cite{Herbst:2013ufa,Haas:2013qwp}. This setting entails, that
first-principle QCD flows can be employed to provide initial
parameters and further glue input, such as background potentials, for
model calculations, thereby systematically removing ambiguities in
these approaches.
The QCD computation relies only on two input parameters, the strong
coupling $\alpha_s$ and the current quark mass $m_\text{current}$, and
allows to consistently include quantum and thermal fluctuations, where
hadronic correlations are captured via the dynamical hadronisation. In
this way model computations can profit directly from a systematic
improvement in predictive power. Conversely, the quantitative advances
in model calculations, such as put forward in this work, can be easily
carried over to full QCD computation. This allows to systematically
improve the approximation of the first-principle QCD flows in the
low-energy regime.
In this work we study a quark-meson model taking into account the
momentum dependence of mesonic two-point functions. At vanishing
temperature, the model in the presence of low-energy quantum
fluctuations is approximated by an effective action of the form
\begin{equation}
\begin{split}
\Gamma=&\int_x \Bigl\{ \bar\psi (\slashed \partial+h (\sigma+\text{i}
\gamma^5\vec \tau \cdot \vec \pi))\psi\\
&+\tfrac{1}{2}Z\partial_\mu \phi_i \partial_\mu
\phi_i+\tfrac{1}{8}Y\partial_{\mu}\rho\partial_{\mu}\rho
+ U(\rho) \Bigr\}\,.
\label{eq:effectiveaction}
\end{split}
\end{equation}
Here $\rho=\sigma^2+\vec\pi^2$, and we include momentum- and field-dependent
bosonic wavefunction renormalisation factors $Z=Z(p^2;\rho)$
and $Y=Y(p^2;\rho)$. The bosonic sector corresponds to a fluctuating
$O(N)$ model \cite{Wetterich:1991be,Berges:2000ew}, while the fermionic sector is classical, anticipating
the decoupling of quark fluctuations at low energies and
temperatures. In the following, we expand $Z$ and $Y$ about a field
value $\rho_0$, restricting ourselves to the zeroth order terms, i.e.\
$Z=Z(p^2;\rho_0)$ and $Y=Y(p^2;\rho_0)$. However, we take into account
the full momentum dependence of $Z,Y$, as well as computing a full
effective potential $U(\rho)$. The quantitative accuracy of a low
order in the field-expansion of $Z$ has been tested in
\cite{Pawlowski:2014zaa}, which is in this work extended by the
inclusion of $Y$. The mesonic two-point functions, evaluated at
a constant field configuration $\phi_i=(\sqrt{\rho},\vec 0)_i$, read
\begin{equation}
\begin{split}
\Gamma^{(2)}_{\pi_{i}\pi_{j}}&=\delta_{ij}Z_{\pi} p^{2}
+2U'(\rho)\delta_{ij}\,,\\
\Gamma^{(2)}_{\sigma\sigma}&=Z_{\sigma} p^{2}
+2U'(\rho)+4U''(\rho)\rho\,,
\end{split}
\end{equation}
where $Z_{\pi}=Z(p^2;\rho_0)$ and
$Z_{\sigma}=Z(p^2;\rho_0)+Y(p^2;\rho_0)\rho$.
\subsection{Pole-, screening- and curvature masses}
\label{sec:massdefinitions}
Particle masses can be extracted directly from the fully
momentum-dependent propagators. In this section, we review different
mass definitions based on the renormalised inverse two-point function
$\bar\Gamma^{(2)}(p_0,\vec p^2)=\Gamma^{(2)}(p_0,\vec p^2)/\bar
Z$. Here, the momentum-independent wavefunction renormalisation $\bar
Z$ relates the bare field $\phi$ to the renormalised field
\begin{equation}\label{eq:WFR}
\bar \phi=\bar Z^\frac{1}{2}\phi\,.
\end{equation}
At vanishing temperature the wavefunction renormalisation $\bar Z$ is
directly related to $Z=Z_\pi$ in \eq{eq:effectiveaction} evaluated at some
fixed external momentum. In turn, at finite
temperature the heat bath singles out a rest frame, and the wave
function renormalisation $Z$ splits into one component parallel,
$Z_\parallel$, and one perpendicular, $Z_\perp$, to the heat
bath. Accordingly, we parameterise the inverse propagator (at
$\rho=\rho_0$) as
\begin{equation}
\label{eq:AnsatzGamma2}
\Gamma^{(2)}(p_0,\vec p^2)=Z_\parallel(p_0,\vec p^2)p_0^2
+Z_\perp(p_0,\vec p^2)\vec p^2+m^2 \,,
\end{equation}
with $Z_\parallel, Z_\perp$ being finite for vanishing momentum and/or frequency.
While the decomposition in \eq{eq:AnsatzGamma2} into
$Z_\parallel(p_0,\vec p^2)$ and $Z_\perp(p_0,\vec p^2)$ is not unique for general momenta, it is for
vanishing $|\vec p|$ or $p_0$. Hence, we can define momentum-independent
wavefunction renormalisation factors parallel and perpendicular to the
heat bath within an evaluation at $p_0=0$ or
$\vec{p}=0$, respectively, i.e.\
\begin{equation}
\label{eq:wffactors}
\begin{split}
Z_\parallel&=\lim_{p_0\to 0}\frac{\Delta\Gamma^{(2)}(p_0,0)}{p_0^2} \,,\\[2ex]
Z_\perp&=\lim_{|\vec p|\to 0}\frac{\Delta\Gamma^{(2)}(0,\vec p^2)}{\vec p^2} \,,
\end{split}
\end{equation}
where $\Delta \Gamma^{(2)}(p_0,\vec p^2)\equiv \Gamma^{(2)}(p_0,\vec
p^2)- \Gamma^{(2)}(0,0)$. Then a standard choice at finite temperature
for the renormalisation of the field is $\bar Z=Z_\perp$. This
definition interpolates between the $O(4)$-symmetric definition at
$T=0$ with $\bar Z= Z_\perp =Z_\parallel$ to that of the dimensionally
reduced theory for $T\to \infty$, where the propagation is
perpendicular to the heat bath. This choice is based on the mass scale
of spatial quantum fluctuations. The ratio $Z_\|/Z_\bot$ gives the
relative weight of the temporal and spatial fluctuations, and hence is
susceptible to the difference of the scale of thermal and quantum
fluctuations.
Now we define pole ($m_{\text{pol}}$), screening ($m_{\text{scr}}$),
and curvature ($m_{\text{cur}}$) masses via
\begin{equation}
\begin{split}
\bar\Gamma^{(2)}(\text{i} m_{\text{pol}},0)&=0\,,\\[2ex]
\bar\Gamma^{(2)}(0,|\vec p|^2=- m_{\text{scr}}^2)&=0\,,\\[2ex]
\bar\Gamma^{(2)}(0,0)&=m_{\text{cur}}^2\,,
\end{split}
\label{eq:masses}
\end{equation}
assuming propagator poles at real $p^2$. The masses $m_{\rm pol}^2$,
$m_{\rm scr}^2$ are the solutions to \eq{eq:masses} with the minimal
distance to $p^2=0$, in general they have a minimal distance to the
Euclidean frequency axis. Accordingly, the pole and screening masses
are respective inverse temporal and spatial screening lengths. For
minimal distance poles at $\pm\text{i} m_{\text{pol}}$ and $\pm \text{i}
m_{\text{scr}}$, we find an exponential decay of the propagator in
position space,
\begin{equation}
\begin{split}
T \sum_{p_0} \left[\Gamma^{(2)}(p_0, 0)\right]^{-1}e^{\text{i} p_0 t}
&\sim e^{-m_{\text{pol}} |t|}\,,\\[2ex]
\int d^3 p \left[\Gamma^{(2)}(0,\vec p^2)\right]^{-1}e^{\text{i} \vec p
\cdot \vec x}&\sim e^{-m_{\text{scr}} |\vec{x}|}\,,
\end{split}
\end{equation}
for $|t|\to\infty$ and $|\vec{x}|\to\infty$, respectively. At
vanishing temperature $\Gamma^{(2)}$ is a function of the $O(4)$
invariant $p_0^2+\vec p^2$ and hence pole and screening masses agree
by definition, i.e.\ $m_{\text{pol}}=m_{\text{scr}}$. At finite
temperatures, the ratio of pole and screening masses is given by
\begin{equation}
\label{eq:ratiopolescr}
\frac{m_{\text{pol}}^2}{m_{\text{scr}}^2}=\frac{Z_\perp(0,\vec p^2=
- m_{\text{scr}}^2)}{Z_\parallel(\text{i} m_{\text{pol}}, 0)}\,.
\end{equation}
Due to the breaking of Euclidean $O(4)$-invariance via the heat bath,
the ratio \eq{eq:ratiopolescr} takes values different from unity. In
general it is also different from
$Z_\parallel/Z_\perp=Z_\parallel(0,\vec 0)/Z_\perp(0,\vec 0)$ which is
accessible in finite-temperature LPA$'$ calculations, e.g.\
\cite{Braun:2009si}. Naturally pole and screening masses take
different values at finite temperature. Moreover, these differences
due to the breaking of Euclidean $O(4)$-invariance extend to the
momentum and frequency dependence and to finite chemical potential.
Note that contrary to pole and screening masses, which are directly
physics observables, the curvature mass is not. This is already
obvious from the fact that it depends on the renormalisation
prescription. For $\bar Z = Z_\perp$ it relates to the screening mass,
but is not identical with the latter. A more detailed discussion is
presented below in the context of the finite temperature case.
Finally, in the vacuum in relativistic theories one also has the onset
mass $m_\text{ons}$. Its definition exploits the fact that the
critical chemical potential associated to the onset of a condensation
phenomenon is linked to the pole mass $m_{\rm pol}$ of the lightest
resonance with non-vanishing quark or baryon number via an exact
Silver Blaze argument \cite{Cohen:2003kd}. The onset mass
$m_\text{ons}$ coincides with the lightest pole mass in the quark
propagator by a Silver Blaze argument, which can be shown in any
diagrammatic expansion scheme in full propagators such as the
functional renormalisation group approach or a 2PI-expansion. The
chemical potential enters the propagator as an imaginary shift of the
zero momentum component. Hence, there is no dependence of the
frequency integrals on the chemical potential until the chemical
potentials exceeds the closest singularity to the real (Euclidean)
$p_0$-axis of the quark propagator in the complex $p_0$-plane. The
position of this singularity coincides with the pole mass of the
corresponding resonance. This agreement between pole and onset mass
was checked explicitly in \cite{Strodthoff:2011tz} in a numerical calculation
within the LPA. We emphasise that in such a
diagrammatic approach diagrams contributing to the meson propagators or
higher correlation functions with vanishing quark number are not
directly sensitive to the chemical potential up to the onset mass in
the quark propagator.
It remains to establish a connection between pole and curvature
mass. With our choice $\bar Z=Z_\bot$ we find
\begin{align}\label{eq:mpolmcur}
m_{\text{cur}}^2=\frac{Z_\parallel(\text{i} m_{\text{pol}},0)}{\bar{Z}}
m_{\text{pol}}^2= \frac{Z_\parallel(\text{i}
m_{\text{pol}},0)}{Z_\perp(0,0)}m_{\text{pol}}^2 \,.
\end{align}
In particular, at zero temperature the ratio
$m_{\text{cur}}^2/m_{\text{pol}}^2$ is given by the ratio $Z(-
m^2_\text{pol})/Z(0)$. This entails that pole and curvature masses
still agree approximately at vanishing temperature if the momentum
dependence of $Z(p^2)$ is rather mild for $|p^2|< m_{\rm
pol}^2$. Assuming a well-behaved analytic continuation for these
momenta this is tightly linked to a mild momentum dependence of
$Z(p^2)$ on Euclidean momenta $p^2\geq 0$.
We emphasise again that the curvature mass is not defined uniquely. In
particular, we may define $\bar Z= Z_\parallel(\text{i}
m_{\text{pol}},0)$, for which both definitions agree. This is to be
expected from the K{\"a}ll{\'e}n-Lehmann spectral representation which
relies on expanding about the particle pole.
At zero temperature the equality $m_\text{pol}=m_\text{cur}$ is best
achieved by parameterising the
inverse propagator as $\Gamma^{(2)}(p^2)=Z(p^2)(p^2+m_\text{pol}^2)$
and by renormalising the fields with $Z(\smash{p^2}=0)$. For
computational convenience, we choose $\bar Z=Z_\bot$ with
\eq{eq:AnsatzGamma2}, \eq{eq:wffactors}. Then the equality is not
guaranteed and hence all statements about the approximate equality of
these masses should be understood as statements about the mild
momentum dependence of the wavefunction renormalisation. As a final
remark, in truncation schemes with momentum-independent wavefunction
renormalisation factors, such as LPA or LPA$'$, pole and curvature
mass naturally agree if one chooses to renormalise with $\bar{Z}=Z$.
\subsection{Flow equations and momentum dependence}
\label{sec:frg}
For the computation of the effective potential $U(\rho)$ and the
momentum- and frequency-dependent two-point functions
$\Gamma^{(2)}(p_0,\vec p^2)$ we use the functional renormalisation group,
for QCD-related reviews see
\cite{Berges:2000ew,Pawlowski:2005xe,Gies:2006wv,Schaefer:2006sr,Braun:2011pp}. It
is based on the Wilsonian idea of integrating fluctuations momentum
shell by momentum shell. Technically this is achieved by
introducing an IR regulator function $R_k$ which suppresses quantum
fluctuations from momentum modes with momenta smaller than some RG
scale $k$ which is subsequently taken from some large UV scale to
zero. The evolution of the scale-dependent analogue of the effective
action $\Gamma$, the effective average action $\Gamma_k$, is described
by a simple 1-loop equation involving full field-dependent propagators \cite{Wetterich:1992yh},
\begin{equation}
\label{eq:floweq}
\partial_t\Gamma_k[\psi,\phi]=\frac{1}{2}\text{Tr}\,
\frac{1}{\Gamma_k^{(2)}[\psi,\phi]+R_k}\partial_t R_k,
\end{equation}
where $\Gamma_k^{(2)}$ denotes the second functional derivative of
$\Gamma_k$ with respect to the fields and $t =\log k/\Lambda$ with
some reference scale $\Lambda$. The trace $\mathrm{Tr}$ sums over momenta and
frequencies, internal indices as well as over field species including
the standard relative minus sign for the fermionic loop. The
corresponding functional equations can rarely be solved exactly. For
the present task we resort to the ansatz \eq{eq:effectiveaction},
which includes momentum- and frequency-dependent wavefunction
renormalisation factors $Z_k$ and $Y_k$ as well as a full effective
potential $U_k$. Approximations with full momentum and frequency
dependence have been applied since long within the FRG, see e.g.\
~\cite{Ellwanger:1995qf,Bergerhoff:1997cv,Pawlowski:2003hq,%
Blaizot:2005wd,Fischer:2008uz,Fister:2011uw}, applications in higher
orders of the derivative expansions are found in e.g.\
\cite{Canet:2003qd,Litim:2010tt}. In the present work we suggest a new
iterative procedure for the solution of fully momentum- and field-dependent
approximations with relatively small numerical costs.
Furthermore, although we do not include a genuine running of the
Yukawa coupling as in \cite{Pawlowski:2014zaa}, one either computes at
a fixed bare or renormalised Yukawa coupling. However, in full QCD
flows the renormalised Yukawa coupling stays approximately constant
\cite{Mitter:2014mat}. Hence we employ the latter choice, further
details can be found in App.~\ref{app:convergence}. The expressions
for the inverse propagators, see \eq{eq:AnsatzGamma2}, generalise at
finite $k$ similarly to the effective potential. The flow equations
for the inverse two-point functions can be obtained from
\eq{eq:floweq} by taking two functional derivatives and are
represented diagrammatically in Fig.~\ref{fig:flow2ptfn}. In general,
these flow equations involve three- and four-point vertices as input,
which are in our case computed from the effective potential, see
App.~\ref{app:flow_equations} for details on the truncation scheme.
\begin{figure}[t]
\centering
\includegraphics[width=0.97\columnwidth]{deltaGamma2Next}
\caption{Flow of the momentum-dependent part of the two-point
function $\Delta \Gamma^{(2)}$. Dashed (solid) lines denote full
mesonic (quark) propagators and crossed circles correspond to
insertions of $\partial_t R_k$ of the respective fields. Tadpole
contributions cancel in the present truncation with momentum-independent
mesonic vertices.}
\label{fig:flow2ptfn}
\end{figure}
\begin{figure}[b]
\centering
\includegraphics[width=0.8\columnwidth]{iterationScheme.png}
\caption[Iteration Scheme]{Illustration of the iteration procedure.}
\label{fig:iterationScheme}
\end{figure}
In our truncation the flow equation for the effective potential takes
the schematic form
\begin{align}
\partial_t U_k(\rho) = \mathcal{F}_U \left[ U_k,
\Delta \GammaTwo_k\right](\rho) \,
\label{eq:iter:flowEffPot_general}
\end{align}
where
$\Delta\Gamma_k^{(2)}(q^2)=\GammaTwo_k(q^2)-\GammaTwo_k(0)$
at $\rho=\rho_0$. Its flow is given by
\begin{align}
\partial_t \Delta\GammaTwo_k(p^2) = \mathcal{F}_{G}
\left[ U_k, \Delta \GammaTwo_k\right](\rho_0,p^2) \,.
\label{eq:iter:flowGammaTwo_approx}
\end{align}
The explicit form of the flow equations is specified in
App.~\ref{app:flow_equations}. In the following, we expand
\eq{eq:iter:flowEffPot_general} about a field value
$\rho_0$. Then \eq{eq:iter:flowEffPot_general} and
\eq{eq:iter:flowGammaTwo_approx} constitute an equation system which
can be solved by an iterative procedure, which is illustrated
pictorially in Fig.~\ref{fig:iterationScheme} and described in App.~\ref{app:iteration}.
In general, this
iterative method enjoys very good convergence properties. Concerning
the expansion in powers of the field, it has been shown in
\cite{Pawlowski:2014zaa} that an expansion about a fixed bare field
has the best convergence properties. For further details we
refer the reader to App.~\ref{app:convergence}.
\section{Results} \label{sec:results}
In the present section we discuss results within the converged
iterative method introduced above. The approach put forward in the
previous section allows us to discuss the relation between the
different fluctuation scales present in the mesonic sector of QCD:
despite its non-uniqueness the curvature mass relates to the
fluctuation scale of quantum fluctuations. The ratios of
$Z_\perp/Z_\parallel$, and $m_{\rm cur}/m_{\rm scr}$, are a measure
for the relative strength of thermal fluctuations while the pole mass
is the fluctuation scale of density fluctuations.
Additionally, we are interested in the question after the simplest
approximation that already includes all quantitative effects.
\subsection{Masses and fluctuation scales} \label{sec:results:massesandfluctuations}
\begin{table}[b]
\centering
\begin{tabular}{cccc}
\toprule
\symhspace{1mm}{step} & \symhspace{1mm}{$m_\text{cur}$ [MeV]} & \symhspace{1mm}{$m_\text{pol}$ [MeV]} & \symhspace{1mm}{$\sigma_\text{min}$ [MeV]}\\
\midrule
0 & 198.1 & 198.1 &58.2\\
1 & 135.2 & \num{133+-2}&92.5 \\
2 & 135.3 & \num{133+-2}&92.8\\
3 & 135.3 & \num{133+-2}&92.9 \\
4 & 135.3 & \num{133+-2}&92.8\\
5 & 135.3 & \num{133+-2}&92.9 \\
\bottomrule
\end{tabular}
\caption{Pion curvature and pole masses and the minimum of the effective potential for different iteration steps at $T=0$ for a physical parameter set at $\Lambda_\text{UV}=\SI{500}{MeV}$.
The UV parameters are tuned such that the physical pion mass emerges as the fully converged result.
Using the same parameters for a LPA$'$ calculation one obtains $m_\text{pol}=m_\text{cur}=\SI{135.0}{MeV}$.}
\label{tab:results:massesT0_tune_polLambda500}
\end{table}
At zero temperature, it is sufficient to numerically investigate the
relative size of pole and curvature mass. Screening and pole mass are
equal due to Euclidean $O(4)$ invariance. This holds true for all
cutoff scales and iteration steps, as the regulators used here
preserve $O(4)$ symmetry. As a first non-trivial finding we obtain
rather similar pole and curvature masses, with relative deviations of
less than one percent, see
Tabs.~\ref{tab:results:massesT0_tune_polLambda500}-\ref{tab:results:massesT0_tune_polLambda900}
for calculations at different UV cutoff scales. This result is tightly
linked to a mild momentum dependence of the wavefunction
renormalisation, see the discussion after \eq{eq:mpolmcur}.
The larger the UV cutoff, the more difficult the iteration procedure
gets from the numerical point of view and the slower the convergence
within the iteration procedure gets. This is illustrated in
Tab.~\ref{tab:results:massesT0_tune_polLambda500} and
Tab.~\ref{tab:results:massesT0_tune_polLambda900}, where the
calculation at the smaller cutoff scale converges practically after
the first iteration step, whereas the calculation at the larger cutoff
scale requires more than three iteration steps until approximate
convergence is reached. The reason for the slower convergence has to
be found in the fixed renormalised Yukawa coupling as a calculation
with a fixed bare Yukawa coupling, irrespective of the chosen UV
cutoff scale, shows similarly good convergence properties as our
calculation at the lowest cutoff scale, see
App.~\ref{app:convergence}. In a calculation with a fixed renormalised
coupling, as employed here, the fermionic contribution, as the
dominant contribution to the flow at large cutoff scales, is no longer
decoupled from the bosonic parts of the model but dependent on the
wavefunction renormalisation $Z$. In particular, there is a relevant
running of $Z$ at large cutoff scales, which makes the calculation
increasingly difficult from the numerical point of view with
increasing cutoff scales. However, the most important conclusion from
this section remains the approximate agreement of pole and curvature
masses at vanishing temperature in the fully iterated result.
\begin{table}[t]
\centering
\begin{tabular}{cccc}
\toprule
\symhspace{1mm}{step} & \symhspace{1mm}{$m_\text{cur}$ [MeV]} & \symhspace{1mm}{$m_\text{pol}$ [MeV]} & \symhspace{1mm}{$\sigma_\text{min}$ [MeV]}\\
\midrule
0 & 412.8 & 412.8 & 16.8 \\
1 & 144.8 & \num{142+-2} & 83.5 \\
2 & 136.4 & \num{135+-2} & 91.8 \\
3 & 135.1 & \num{134+-2} & 93.1 \\
4 & 134.9 & \num{133+-2} & 93.2 \\
5 & 134.9 & \num{133+-2} & 93.2 \\
\bottomrule
\end{tabular}
\caption{Similar to Tab.~\ref{tab:results:massesT0_tune_polLambda500} but for $\Lambda_\text{UV}=\SI{700}{MeV}$. LPA$'$ masses: \SI{135.2}{MeV}.}
\label{tab:results:massesT0_tune_polLambda700}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{cccc}
\toprule
\symhspace{1mm}{step} & \symhspace{1mm}{$m_\text{cur}$ [MeV]} & \symhspace{1mm}{$m_\text{pol}$ [MeV]} & \symhspace{1mm}{$\sigma_\text{min}$ [MeV]}\\
\midrule
0 & 817.0 & 817.0 &5.1\\
1 & 163.4 & \num{158+-2}&67.9 \\
2 & 138.5 & \num{137+-2}&89.9\\
3 & 136.5 & \num{135+-2}&92.4 \\
4 & 135.4 & \num{134+-2}&93.6\\
5 & 135.3 & \num{134+-2}&93.6 \\
\bottomrule
\end{tabular}
\caption{Similar to Tab.~\ref{tab:results:massesT0_tune_polLambda500} but for $\Lambda_\text{UV}=\SI{900}{MeV}$. LPA$'$ masses: \SI{134.1}{MeV}.}
\label{tab:results:massesT0_tune_polLambda900}
\end{table}
At finite temperature, pole and screening masses start to deviate as
expected from \eq{eq:ratiopolescr}, see
Fig.~\ref{fig:results:massesQMT_converged} for the pion masses. For a
similar observation in the NJL model see e.g.\
\cite{Florkowski:1993br}. All low-energy effective models have in
common that they have a {\it physical} UV scale above which they loose
predictive power. In the present formulation this scale is given by
the initial cutoff scale $\Lambda_{\rm UV}$. The thermal range
$\Lambda_{T}$ of the model is defined as the minimal cutoff scale
$\Lambda_{\rm UV}$ above which thermal fluctuations do not probe the
cutoff scale. This is investigated in App.~\ref{app:thermalrange}
and leads to $\Lambda_{T}\lesssim 7\, T$ for the regulators used, see
\eq{eq:R} with $m=2$.
Here we present results for temperatures $T\lesssim \SI{180}{MeV}$ which is
well covered by $\Lambda_{\rm UV}=\SI{1.4}{GeV}$. The fact that the
curvature mass stays close to the screening mass at all temperatures
is related to the fact that we chose $\bar Z=Z_\perp$ to renormalise
fields and is once again an expression for small non-trivial momentum
dependencies.
\begin{figure}[t]
\centering
\includegraphics[width=0.97\columnwidth]{massesTPhys_L1400.pdf}
\caption{Temperature dependence of different pion mass
definitions extracted from fully iterated result at $\Lambda_\text{UV}=\SI{1.4}{GeV}$.}
\label{fig:results:massesQMT_converged}
\end{figure}
The approximate agreement of pole and curvature masses demonstrated
above is a consequence of the momentum dependence obtained from the
converged iteration procedure. In particular, their difference allows
to pin down effects of scale mismatches in existing calculations in
simple but commonly used truncation schemes such as the LPA. Here we
aim at quantifying the systematic error which is inherent in these
calculations. For different cutoff scales we follow the usual
procedure in the literature and tune initial conditions such that we
obtain correct physical observables in the IR. Then we compute the LPA
onset mass by calculating the momentum-dependent meson propagator
using the given LPA solution, corresponding to the first half of the
first iteration step in Fig.~\ref{fig:iterationScheme}. The pole mass
extracted from this propagator equals, up to approximation effects,
the LPA onset mass, which can be probed directly by including a
coupling to (isospin) chemical potential, see
\cite{Strodthoff:2011tz,Kamikado:2012bt}. This LPA onset mass can now
be compared to the LPA curvature mass extracted from the curvature of
the effective potential as shown in
Tab.~\ref{tab:results:LPAcurvaturepole}. Whereas their ratio tends to
one for smaller UV scales, it increases with the UV scale and reaches
a value of 1.71 for $\Lambda_\text{UV}= \SI{1.4}{GeV}$.
Such large deviations are in line with earlier studies
\cite{Strodthoff:2011tz,Kamikado:2012bt} where deviations of
\SI{30}{\%} were observed for a 3d regulator function and a cutoff
scale $\Lambda_\text{UV}=\SI{900}{MeV}$. In the present work we use
4d exponential regulators, \eq{eq:R} with $m=2$. For different
regulators the {\it physical} cutoff scales, $k_{\rm phys}(k)$ do not
necessarily agree, for a detailed discussion and applications see
\cite{Pawlowski:2005xe,Marhauser:2008fz}. This entails that coinciding \label{cutoff_ratios}
physical cutoff scales are obtained for different regulator scales
$k$. A rough estimate for this ratio of cutoff scales is given by the
ratio of the (bosonic) flow of the masses. For the standard exponential
regulator \eq{eq:R} with $m=1$ we find $ k_{3\dd}/k_{4\dd,m=1}\approx
3/2$, for the exponential regulator \eq{eq:R} with $m=2$ we find $
k_{3\dd}/k_{4\dd,m=2}\approx 5/4$.
In summary this entails that the commonly used UV cutoff scales
$\Lambda_{\text{UV},3\dd}=\SI{700}{MeV}$ and \SI{900}{MeV} correspond to UV cutoff scales
$\Lambda_{\text{UV},4\dd,m=2}=\SI{560}{MeV}$ and \SI{720}{MeV}, respectively.
For the following numerical examples we will therefore focus on the case of $\Lambda_{\text{UV},4\dd,m=2}=\SI{700}{MeV}$.
\begin{table}[t]
\centering
\begin{tabular}{cccc}
\toprule
\symhspace{1mm}{$\Lambda_\text{UV}$ [GeV]} & \symhspace{1mm}{$m_\text{cur}$ [MeV]} & \symhspace{1mm}{$m_\text{ons}$ [MeV]} & \symhspace{1mm}{$m_\text{cur}/m_\text{ons}$} \\
\midrule
0.5 & 135.0 & \num{109+-2}& 1.24\\
0.7 & 135.2 & \num{98+-2}& 1.38\\
0.9 & 135.0 & \num{90+-2}& 1.50\\
1.1 & 135.4 & \num{85+-2} & 1.59\\
1.4 & 135.3 & \num{79+-2} & 1.71\\
\bottomrule
\end{tabular}
\caption{Comparison of LPA curvature and onset masses for fixed curvature masses in the IR and different UV cutoff scales $\Lambda_\text{UV}$.\vspace{-0.5em}}
\label{tab:results:LPAcurvaturepole}
\end{table}
At first sight, deviations between the curvature and the onset mass might seem irrelevant for studies at
a given expansion order such as the commonly employed zeroth order
derivative expansion or LPA. However, as already explained in the
beginning of this section, the curvature mass and the ratio
$Z_\perp/Z_\parallel$ sets the relevant scales for quantum and thermal
fluctuations while the pole/onset mass is that of density
fluctuations. In other words, an approximation scheme where these mass
scales differ by \SI{38}{\%} leads to a significant quantitative
change of the ratio of critical temperature $T_c$ over onset chemical
potential $\mu_c$. A rough estimate, assuming that the onset chemical potential/ critical temperature
measured in the respective mass scales stays constant in the different approximation schemes, provides
\begin{align}\label{eq:ratioTmu}
\left[\0{ \mu_c}{T_c}\right]_{\rm full} / \left[\0{ \mu_c}{T_c}
\right]_{\rm LPA}\approx \left[\0{m_{\rm cur}}{m_{\rm ons}}\right]_{\rm LPA}\approx 1.38\,,
\end{align}
where the subscript ${}_{\rm full}$ refers to the present
momentum-dependent approximation.
This mismatch of scales has consequences for the scale setting procedure
in LPA. The commonly used procedure is to
fix the physical parameters at the initial UV scale $\Lambda_\text{UV}=\SI{700}{MeV}$ such,
that the pion (pole) mass of approximately \SI{135}{MeV} is the
curvature mass $m_{\pi,\rm cur}$, as it agrees with the pole mass in
this order of the derivative expansion (classical dispersion). And
indeed the full results of the last section justify this
identification. As the scale of quantum and thermal fluctuations is
identical in this approximation due $Z_\perp/Z_\parallel=1$, and
$m_{\rm cur}/m_{\rm scr}=1$ all these fluctuations are treated
self-consistently. In turn, the scale for density fluctuations in LPA is set by the onset mass, see
Tab.~\ref{tab:results:LPAcurvaturepole}, which is approximately \SI{98}{MeV} at a UV cutoff scale $\Lambda_\text{UV}=\SI{700}{MeV}$.
This entails that the strength of density fluctuations is overestimated by \SI{28}{\%}.
Alternatively, one can identify the pion mass with the onset mass \cite{Strodthoff:2011tz,Kamikado:2012bt}. With hindsight this comes at the expense of
having a too large scale for quantum and thermal fluctuations of about
\SI{186}{MeV} instead of \SI{135}{MeV}, see
Tab.~\ref{tab:results:LPAcurvaturepole}. In other words, quantum and
thermal fluctuations are underestimated by \SI{38}{\%}.
To summarise, there is no way of circumventing a mismatch of fluctuation scales
in a truncation scheme with vastly different curvature and onset masses such as the LPA.
Its implications for the chiral phase boundary at finite density are discussed in
Sec.~\ref{sec:FiniteDensity}. Apart from the quantitative change of
the phase boundary this mismatch may also inflict qualitative changes
at large chemical potential $\mu>\mu_c$, for example if the phase
structure at nuclear densities and beyond involves competing order
effects.
\subsection{Momentum dependence and initial conditions}
In order to study the impact of fully momentum-dependent propagators
on FRG calculations in low-energy QCD, we compare the temperature
dependence of the quark condensate $\chevron{\sigma}$ in different
orders of the truncation scheme. $\chevron{\sigma}(T)$ is sensitive to
a correct relative inclusion of thermal and quantum fluctuations as well as
the absolute scale. We compare three different
truncations, namely LPA, LPA$'$ and the fully momentum-dependent
calculation.
As already discussed in Sec.~\ref{sec:effectiveaction}, there are two
possibilities to fix the initial conditions. Firstly, one can derive
the initial conditions from computing QCD-flows for the model's
parameters in the UV. In the following we evaluate the consequences of this
set-up by comparing different simpler truncation schemes to the momentum-dependent
calculation put forward in this work, which is expected to lie closest to the full QCD flow.
Secondly, one can tune the model parameters such
that the vacuum physics of QCD is reproduced within the respective model and approximation
scheme. In low-energy
QCD without inherent approximations these two sets of initial
conditions agree. In turn, within approximations, they are different.
Hence, we shall consider both approaches in our investigation separately.
\subsubsection{Fixed microphysics} \label{subsec:results:sameUV}
We first study the effects of including fully momentum-dependent
propagators while keeping the UV-physics of the model fixed at
$\Lambda_\text{UV}=\SI{900}{MeV}$. In such a QCD-embedded approach the input
parameters at $\Lambda_\text{UV}$ are derived within QCD flows. However, in this work, we employ
a parameter set which leads to correct physical observables in the IR
for the full calculation to mimic the effect of fixing initial conditions
from full QCD. Our findings for the chiral crossover are summarised in
Figs.~\ref{fig:results:sameUVcrossover}.
On the one hand, note that for a cutoff scale of $\Lambda_\text{UV}=\SI{900}{MeV}$ no comparison
to the LPA is possible because the LPA calculation with initial
conditions from the full calculation only shows chiral symmetry
breaking for $\Lambda_\text{UV}<\Lambda^*_{\text{LPA}}\approx\SI{600}{MeV}$, see
App.~\ref{app:uvcutofflpa}. Therefore, we restrict ourselves in the LPA
case to a comparison at vanishing temperature for a UV cutoff scale
$\Lambda_\text{UV}=\SI{500}{MeV}$, which can be inferred from
Tab.\ref{tab:results:massesT0_tune_polLambda500}. Note that with the
thermal range $\Lambda_T\lesssim 7 \,T$ we only have access to temperatures
$T\lesssim \SI{70}{MeV}$ anyway, see App.~\ref{app:thermalrange}.
However, even for $\Lambda_\text{UV}=\SI{500}{MeV}$ LPA is not quantitatively consistent with
the full calculation with deviations of \SI{50}{\%} in $\chevron{\sigma}$ and
$m_\pi$.
\begin{figure}[t]
\centering
\includegraphics[width=0.97\columnwidth]{crossoverFixedUV_L0900}
\caption[Crossover with fixed UV parameters]{Chiral condensate
vs.~temperature for fixed UV parameters using different
truncations and $\Lambda_\text{UV}=\SI{900}{MeV}$.}
\label{fig:results:sameUVcrossover}
\end{figure}
On the other hand, Fig.~\ref{fig:results:sameUVcrossover} shows that the LPA$'$
calculation is even quantitatively consistent with the full result. The largest relative
deviations of about \SI{3}{\%} arise in the vicinity of the
pseudo-critical temperature $T_c$ and are certainly related to
pseudo-critical fluctuations. At about $T_c$, the correlation length
is large, and the system changes its dynamical degrees of freedom from
quarks to mesons. Both properties imply that this region is most
sensitive to momentum fluctuations. This is also in line with the
expectation that including the full momentum dependence or higher
orders of the derivative expansion is crucial around the critical
temperature $T_c$, e.g.~ in order to calculate critical exponents at
high numerical precision, see e.g.\
\cite{Litim:2010tt,Benitez:2011xx}.
The generally very good agreement of LPA$'$ and the fully momentum-dependent
truncation in Fig.~\ref{fig:results:sameUVcrossover} is intimately related to
the use of a cutoff function $R_k$, which regularises both
frequencies $p_0$ and spatial momenta $\vec{p}$, see
\cite{Fister:2011uw,Boettcher:2013kia}. Thus, the associated RG
flows are local both in $\absVal{\vec{p}}$- and in $p_0$-space and
the argument from the zero temperature discussion in App.~\ref{app:lpaprime}
applies. Conversely, if we employed a regulator which only affects
spatial momenta, such as is commonly done in finite-temperature FRG
calculations, the entire Matsubara summation would be required to
compute a given RG flow. Put differently, the flows at every scale
$k$ would receive contributions from both very small and very large
Matsubara frequencies, thereby at least partly invalidating the
$T=0$ reasoning presented in App.~\ref{app:lpaprime}. Accordingly, calculations based on
three-dimensional cutoff functions are anticipated to give rise to
deviations which are larger than the ones we observe in
Fig.~\ref{fig:results:sameUVcrossover}.
\subsubsection{Fixed vacuum physics} \label{subsec:results:sameIR}
\begin{table}[b]
\centering
\begin{tabular}{ccc||cc}
\toprule
step & $m_\text{cur}$ [MeV] & $m_\text{pol}$ [MeV] & $m_\text{cur}$ [MeV]&$m_\text{pol}$ [MeV]\\
\midrule
0 & 135.2 & 135.2 & 135.2 & 135.2\\
1 & 96.5 & \num{96+-2} & 135.5& \num{134+-2}\\
2 & 96.8 & \num{96+-2} & 135.5& \num{134+-2}\\
3 & 96.8 & \num{96+-2} & 135.6& \num{134+-2}\\
4 & 96.8 & \num{96+-2} & 135.6& \num{134+-2}\\
5 & 96.8 & \num{96+-2} & 135.6& \num{134+-2}\\
\bottomrule
\end{tabular}
\caption{Pion curvature and pole masses for different
iteration steps at $T=0$ applying the iteration procedure on
top of a LPA (left) and a LPA$'$ (right) parameter set for a UV cutoff scale $\Lambda_\text{UV} = \SI{700}{MeV}$ . The UV
parameters are tuned such that the physical pion mass emerges
in the respective zeroth step.}
\label{tab:results:massesT0_tune_cur}
\end{table}
In the previous section we have discussed the different truncations in
view of the direct connection of the QM model to first-principle
QCD. Then the UV initial conditions can in principle, be calculated
from QCD flows. Within such a combined approach the QM model can be
systematically upgraded to the full low-energy effective
action of QCD. This technically challenging programme is well under
way, and eventually will give quantitative reliability to enhance
low-energy effective model computations.
However, we might also disregard the direct connection to QCD and fix
the initial conditions in the infrared by adjusting the correct vacuum
physics: choose some generic set of (renormalised) IR observables
$(\bar{f}_\pi,\bar{m}_\pi,\bar{m}_\psi)$ and then tune the
microphysics separately in each truncation scheme such that the given
mass scales emerge in the limit $k\to 0$ at vanishing temperature and density. On
the basis of this adjustment one then can study finite-temperature or
finite-density physics. This approach is the standard effective model
approach to low-energy QCD. The discussion in the present section is
meant to evaluate and improve the reliability of this set-up.
Note first that such a procedure falls short of a direct connection to
first-principle QCD in the UV. Moreover, in particular in low-order
approximations such as LPA, some fluctuation physics is simply stored
in the initial condition. For example, in LPA for the model at hand,
we have to change the UV initial conditions such that they effectively
take care of the missing momentum effects. In order to assess the
impact of adding the full momentum dependence on top of a given LPA or
LPA$'$ solution, we show in Tab.~\ref{tab:results:massesT0_tune_cur}
the iteration procedure applied to a given LPA or LPA$'$ solution,
referred to as zeroth iteration step in
Tab.~\ref{tab:results:massesT0_tune_cur}. On the one hand, as
expected from Sec.~\ref{sec:results:massesandfluctuations}, the
iteration on top of the given LPA result shows a large deviation of
over \SI{40}{\%} in the masses after the first iteration step,
illustrating again the large mismatch of fluctuation scales in the
LPA. On the other hand, with a deviation of less than one percent, the
LPA$'$ solution is already very close to the full result. Even more
conveniently for practical purposes, the first iteration step deviates
less than one per mill from the full result. Hence, evaluating the
momentum dependence on the basis of a given LPA$'$ solution, already
provides a simple but very good approximation to the full
momentum-dependent solution.
\begin{figure}[t]
\centering
\includegraphics[width=0.97\columnwidth]{crossoverFixedIR_L0900}
\caption[Crossover with fixed IR parameters]{Chiral condensate
vs. temperature for fixed physical IR parameters in the
vacuum using different truncations and $\Lambda_\text{UV}=\SI{900}{MeV}$.}
\label{fig:results:sameIRcrossover}
\end{figure}
Our findings on the chiral crossover are summarised in
Fig.~\ref{fig:results:sameIRcrossover}. We again observe that the LPA$'$
scheme approximates the full flow very well. To be more precise, the relative deviation of the chiral condensate never exceeds $\SI{3}{\%}$. Note, however, that the
relative deviation again exhibits a peak centered at the
pseudo-critical temperature, which indicates accuracy issues in the
presence of pseudo-critical fluctuations. Possible reasons for the generally good agreement between the two schemes were already extensively studied
in the preceding paragraph as well as in App.~\ref{app:lpaprime}.
Accordingly, we will concentrate on the remaining
comparison between the LPA and the full calculation.
Here, Fig.~\ref{fig:results:sameIRcrossover} reveals that despite
large deviations of about \SI{19}{\%} in $T_c$ measured in absolute scales,
which is to large parts caused by different sigma mass ranges which
can be reached in the different approximation schemes, the deviations
in terms of relative scales are rather small as well, at least for
temperatures outside the critical regime. This presumably reflects the
fact, that the propagators' non-trivial momentum dependencies enter
the computation of the chiral condensate from the effective potential
only indirectly. Put differently, the crossover's shape seems to be
mainly fixed by the infrared mass scales in the vacuum.
\begin{figure}[t]
\centering
\includegraphics[width=0.97\columnwidth]{massesRGScale.pdf}
\caption{Scale dependence of the pion mass at $T=0$ in LPA and in the full calculation for $\Lambda_\text{UV}=\SI{900}{MeV}$ for fixed physical IR parameters.}
\label{fig:results:scaledepmasses}
\end{figure}
The quantitative discrepancy between LPA and the full calculation can be understood
most easily from Fig.~\ref{fig:results:scaledepmasses}, where we compare
the scale dependence of the pion mass for fixed initial
conditions in the IR as an illustration of cutoff regions where the LPA calculation gives qualitatively or quantitatively correct results compared to the
full calculation. Remarkably, both is only the case for cutoff scales below \SI{200}{MeV} whereas for larger scales the results are not even qualitatively
correct as it is clearly visible from the different slopes as a result of the fixed renormalised Yukawa coupling in the full calculation.
\subsection{Finite density}\label{sec:FiniteDensity}
Let us finally evaluate the consequences of the results in the last
two sections for finite density computations. These observations point
at a mismatch of scales in LPA between quantum/thermal and density
fluctuations with a factor as given in \Eq{eq:ratioTmu}. As before
we employ the cutoff scale $\Lambda_\text{UV}=\SI{700}{MeV}$ as
numerical example.
To illustrate its consequences we consider a simple application to the
physics of the phase diagram in LPA, where we resort to the simple
rescaling argument that has worked so successfully for quantum and
thermal fluctuations in Section~\ref{subsec:results:sameIR}. Within
this reasoning, the overestimation of density fluctuations can be
approximately undone by simply rescaling the chemical potential axis
with a factor $1.38$. This weakens the curvature of the chiral phase
boundary $T_c(\mu)/T_c(0)$ at finite chemical potential. At small
chemical potential the phase boundary can be expanded in powers of
$\mu^2$ as follows:
\begin{align}\label{eq:curvf}
\frac{T_c(\mu)}{T_c(0)} = 1-\kappa_\mu \left(\frac{\mu}{\pi T_c(0)}
\right)^2+\mathcal{O}\biggl(\!\left(\frac{\mu}{\pi
T_c(0)}\right)^4\biggr)\,,
\end{align}
for a discussion of the phase structure of the $N_f=2$ quark-meson
model in LPA and LPA$'$ as well as with higher order quark-meson scattering
processes see \cite{Pawlowski:2014zaa}. \Eq{eq:curvf} entails that a
stretching of the chemical potential axis with a factor $1.38$ weakens
the curvature $\kappa_\mu$ by a factor $0.53$. In
\cite{Pawlowski:2014zaa} the curvature was computed from the chiral
susceptibility as $\kappa_{\mu,\rm LPA}\approx 1.4$. Applying the reduction factor, this reduces to $\kappa_\mu\approx 0.74$,
which influences the result in the direction of the lattice curvature $\kappa_\mu\approx 0.5$ \cite{deForcrand:2002ci}.
The above discussion suggests that taking into account the full
momentum dependence of the propagators in the quark-meson model may
account for the mismatch between the curvature of the phase boundary
computed in the models and the lattice result. However,
\cite{Pawlowski:2014zaa} also contains a computation with constant
wavefunction renormalisations for meson and quark fields (LPA$'$) and a fully
(meson-)field-dependent Yukawa coupling (corresponding to higher order
quark-mesonic scattering processes). The
curvature $\kappa_\mu $ for this computation agrees surprisingly well
with the LPA result. This means that either the higher order
quark-meson scatterings counterbalance the momentum effects on the
curvature, or the LPA$'$ computation with a 3d regulator in
\cite{Pawlowski:2014zaa} does not cover the full momentum dependence.
The results for this investigation will be presented elsewhere.
\section{Conclusion}
In the present work we have discussed the relation between different
mesonic mass scales in low-energy QCD within a $N_f=2$ quark-meson
model. To that end we have computed fully momentum-dependent two-point
functions of pions and the $\sigma$-meson as well as a full mesonic
potential within a functional renormalisation group approach. This
allows us to compare pole, screening and curvature masses both at
vanishing and finite temperature; respective definitions are discussed
in detail in Section~\ref{sec:massdefinitions}. Whereas pole and screening mass coincide by
definition for vanishing temperature they start to deviate at finite
temperatures. Moreover, we find that the fluctuation scales for
density fluctuations, related to the pole masses at vanishing
temperature, and that for quantum and thermal fluctuations, related to
the curvature masses, almost agree.
The present momentum-dependent set-up has also been used to evaluate
the reliability of lower order approximations. In the present work we
considered the two standard approximations to the QM model: the local
potential approximation (LPA), where only classical propagators and
the full mesonic potential are considered, and LPA$'$, which
additionally involves constant wavefunction renormalisations for the
mesonic fields. Our results show a very good agreement between the
fully momentum-dependent calculation and the LPA$'$ calculation with
relative deviations of at most \SI{3}{\%} in a narrow region around
the pseudo-critical temperature, which justifies the use of the LPA$'$
as simple but very reliable truncation which includes already a large
part of the momentum dependence of the full propagator.
In turn, we observe a large mismatch between the
pion onset and curvature masses in LPA at vanishing temperature which
reaches \SI{38}{\%} for typical UV cutoff scales $\Lambda_\text{UV}=\SI{700}{MeV}$,
which has important implications for the relative fluctuation scales
for vacuum/thermal and density fluctuations. Neglecting
these effects leads to large systematic errors at finite chemical
potential. Moreover, even at very low UV cutoff scales the LPA truncation for fixed
initial conditions in the UV does not lead to quantitatively correct
results compared to the outcome of the full calculation.
In the line of these findings we estimated the effect of this mismatch
of quantum/thermal versus density fluctuation scales in LPA within a
simple rescaling the chemical potential axis accordingly, see
Section~\ref{sec:FiniteDensity}. This simple argument leads to a
result for the curvature of the chiral phase boundary, which lies
reasonably close to the lattice result, but remains to be checked in
larger truncation schemes.
An investigation of the combination of the approximation in
\cite{Pawlowski:2014zaa} with $O(4)$-symmetric regulators and full
momentum dependence at finite density will be presented elsewhere.
This requires the extension of the present 4d regulator classes to
finite chemical potential, which is also tightly linked to the
computation of real time quantities such as spectral functions in a
fully $O(4)$ and Minkowski-invariant set-up, which will be discussed in
a future publication. The present findings
strongly emphasise the necessity of such a symmetry-preserving
approach.
\acknowledgments We thank L.~Fister, F.~Rennecke and L.~von Smekal for
discussions. JMP thanks the Yukawa Institute for Theoretical Physics,
Kyoto University. Discussions during the YITP workshop YITP-T-13-05 on
'New Frontiers in QCD' were useful to complete this work. This work is
supported by the Helmholtz Alliance HA216/EMMI and the grant
ERC-AdG-290623.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,060
|
Q: Создание одной сессии на всё время существования сервиса Корректно ли в aiohttp создать сессию на всё время жизни сервиса? Какую роль в aiohttp играют сессии? Хранят ли они пул коннектов?
A: Да, так можно и нужно делать именно потому что сессии хранят пул коннектов.
aiohttp.ClientSession should be created once for the lifetime of the
server in order to benefit from connection pooling.
Sessions save cookies internally. If you don't need cookie processing,
use aiohttp.DummyCookieJar. If you need separate cookies for different
http calls but process them in logical chains, use a single
aiohttp.TCPConnector with separate client sessions and
own_connector=False.
proof
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,593
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\section{Introduction}
Let $G$ be a finite graph, loops and parallel edges permitted. This article continues a series of papers
\cite{AJG05}, \cite{CCC07} using elementary finite
Fourier analysis to derive evaluations of the Tutte polynomial
$T(G;x,y)$ at certain points $(x,y)=(a,b)$ where
$(a-1)(b-1)\in\{2,3,4\}$. Letting $H_q$ denote the hyperbola
$\{(x,y):(x-1)(y-1)=q\}$,
the evaluations of the Tutte polynomial on $H_2$ and $H_4$ are expressed in terms of eulerian subgraphs of
$G$ and the size of subgraphs modulo $2,3,4$ or $6$, the evaluations
on $H_3$ in terms of directed
eulerian subgraphs and size modulo $3$.
The theorems
obtained in this article are analogous to the well-known fact that the
eulerian subgraphs of $G$ all have the same size modulo $2$
if $G$ is bipartite and otherwise the number
of eulerian subgraphs of odd size is equal to the number of even size. They seem however to be more
elusive of explanation than this straightforward example.
In particular, Theorem~\ref{sum cubes with q=6} states that a
graph $G$ has a nowhere-zero
$4$-flow if and only if there is a correlation between (i) the event that
three subgraphs $A,B,C$ chosen uniformly at random have
pairwise eulerian symmetric differences and (ii) the event that
$\lfloor\frac{|A|+|B|+|C|}{3}\rfloor$ is even. A companion to this
theorem is Theorem~\ref{squares with q=4} that $G$ is eulerian (has a
nowhere-zero $2$-flow) if and only if there is a correlation between (i)
the event that subgraphs $A,B$ chosen uniformly at random have
eulerian symmetric difference and (ii) the event that
$\lfloor\frac{|A|+|B|}{2}\rfloor$ is even.
In this language, a graph $G$ is bipartite (has a nowhere-zero $2$-tension) if
and only if there is a correlation between
the event that a subgraph $A$ chosen uniformly at random is
eulerian and the event that
$|A|$ is even.
In section~\ref{poset} we develop a result of Onn \cite{Onn04}
which states a parity criterion for the existence of nowhere-zero
$q$-flows of a graph.
This leads to another correlation between a parity event and an
event involving eulerian subgraphs of $G$, this time connected to
whether or not $G$ has a proper vertex $4$-colouring.
Onn in his paper uses
the algebraic method used by Alon and Tarsi \cite{AT92}, \cite{AT97}, \cite{A99} in their proof of a parity criterion for the existence of
proper $q$-colourings of $G$ in terms of eulerian subdigraphs of an
orientation of $G$.
In section~\ref{cubic graphs} we consider graphs with a cycle
double cover by triangles (such as plane triangulations) and derive a
criterion for the existence of a proper vertex $4$-colouring of $G$ in
terms of the correlation between (i) the event that a pair of subgraphs
$A,B$ of $G$ are eulerian and between them cover all the edges of $G$
and (ii) the event that $|A|\equiv|B|\pmod 3$.
In section~\ref{4 regular} we prove theorems of a similar
character to the cited results of Alon and Tarsi, involving evaluations of
the Tutte polynomial on $H_3$, this time confining our attention to
$4$-regular graphs (such as line graphs of cubic graphs). These results stem also from the work of Matiyasevich \cite{M04}, whose probabilistic restatements of the Four
Colour Theorem inspired the mode of expression for the Tutte
polynomial evaluations throughout this paper.
The method of proof throughout is to use identities from
elementary Fourier analysis from which the
interpretations of the Tutte polynomial evaluations can be extracted.
The results of section~\ref{H2 H4} (the main theorems of which are to
be found in sections~\ref{square root of unity}
to \ref{sixth root of unity}) ultimately derive from Lemma~\ref{sum
cubes}, which comprises a set of identities which can be found in \cite{E98} and
\cite{Mi05}. In the language of coding theory, these identities relate the sum of powers of coset weight enumerators of a
binary code to the Hamming weight enumerator of the code. In the
context of this paper, the code is the cycle space of a graph. In
sections~\ref{poset}, \ref{cubic graphs} and \ref{4 regular} the main
tool is the discrete version of the Poisson summation formula (or, in the context of
coding theory, the MacWilliams duality
theorem for complete weight enumerators). A
simultaneous generalisation of Lemma~\ref{sum cubes} and the Poisson
summation formula is presented in Lemma~\ref{cwe sum}: this is used in
Section~\ref{4 regular}.
A more expansive
exposition of the material in
sections~\ref{sec:fourier} and \ref{hwes} of the present article can be found in
\cite{CCC07} but is included here for convenience. All the facts quoted
without reference in section~\ref{sec:fourier} on Fourier transforms can be found for example in
\cite{Terras99}.
Eulerian subgraphs of $G$ are cycles in the graphic matroid
underlying $G$. In \cite{Kung07} the results of \cite{AJG05} are extended from graphs
to matrices. In a similar way, the results of the present paper depend only on
the cycle space of $G$ and could be extended to binary matroids.
\subsection{Notation and definitions}
Let $G=(V,E)$ be a graph. In section~\ref{H2 H4} we consider $\mathcal{C}_2\subseteq\mathbb{F}_2^E$ the subspace of
$2$-flows (eulerian subgraphs, cycles) of $G$, and
$\mathbb{F}_2^E/\mathcal{C}_2$ the set of cosets of $\mathcal{C}_2$ in the additive group
$\mathbb{F}_2^E$.
The quotient space $\mathbb{F}_2^E/\mathcal{C}_2$ is isomorphic to the orthogonal subspace
$\mathcal{C}_2^\perp$ of $2$-tensions (cutsets, cocycles) of $G$.
The rank of $G$ is defined by $r(G)=|V|-k(G)$, where $k(G)$ is the
number of components of $G$, and the nullity by $n(G)=|E|-r(G)$. The
subspace $\mathcal{C}_2$ has dimension $n(G)$ over $\mathbb{F}_2$, and
$\mathcal{C}_2^\perp$ dimension $r(G)$.
To each subset $A\subseteq E$ there is a subgraph $(V,A)$ of $G$
obtained by deleting the edges in $E\setminus A$ from $G$. For short
this subgraph will be referred to just by its edge set $A$,
``the
subgraph $A$'' meaning the graph $(V,A)$.
A subgraph $A$ is {\em eulerian} if all its vertex degrees are even.
The subspace $\mathcal{C}_2$ of $2$-flows of $G$ may be identified with the
set of eulerian subgraphs of $G$.
For two subgraphs $A,B$ of $G$ the
symmetric difference $A\bigtriangleup B$ corresponds to
addition of the indicator vectors of $A$ and $B$ in $\mathbb{F}_2^E$. The size $|A|$ of
$A$ is equal to the Hamming weight of the indicator vector of $A$.
Two subsets $A,B\subseteq E$ belong to the same coset of $\mathcal{C}_2$
if and only if $A\bigtriangleup B$ is eulerian, and this is the case
if and only if the subgraphs $A,B$ have the same degree sequence
modulo $2$.
The space of $\mathbb{F}_4$-flows of $G$ will be denoted by
$\mathcal{C}_4$. The space $\mathcal{C}_4$ is isomorphic to $\mathcal{C}_2\times\mathcal{C}_2$. From this observation a graph
$G=(V,E)$ has a nowhere-zero $4$-flow if and only if there are two eulerian
subgraphs which together cover $E$, whence the well-known
equivalence of the Four Colour Theorem with existence of an edge
covering of any given planar graph by two of its eulerian subgraphs.
For $A\subseteq E$ the rank $r(A)$ of $A$
is defined to be the rank of the subgraph $(V,A)$. The {\em Tutte polynomial} of
$G$ is defined by
$$T(G;x,y) =\sum_{A\subseteq E}(x-1)^{r(E)-r(A)}(y-1)^{|A|-r(A)}.$$
The hyperbolae $H_q=\{(x,y)\,:\,(x-1)(y-1)=q\}$ for $q\in\mathbb{N}$ play a special role in the theory
of the Tutte polynomial, summarised for example in \cite[\S
3.7]{DW93}.
In particular, $(-1)^{r(G)}q^{k(G)}T(G;1-q,0)=P(G;q)$ is
the number of proper vertex $q$-colourings of $G$ and
$(-1)^{n(G)}T(G;0,1-q)=F(G;q)$ is the number of nowhere-zero $\mathbb{Z}_q$-flows of
$G$.
On $H_2$ there are the evaluations
$P(G;2)=(-1)^{r(G)}2^{k(G)}T(G;-1,0)$ and $F(G;2)=(-1)^{n(G)}T(G;0,-1)$. The Tutte
polynomial on $H_2$ is the partition function of the Ising model of
statistical physics, or, what is the same thing, the Hamming weight
enumerator of the subspace $\mathcal{C}_2$ of $2$-flows (this is Van
der Waerden's eulerian expansion of the Ising model~\cite{vdW41}). In
section~\ref{H2 H4} the Tutte polynomial at the points
$(-2,\frac{1}{3})$ and $(-\frac{1}{2},-\frac{1}{3})$ on $H_2$, in addition to $(-1,0)$ and $(0,-1)$, are
given an interpretation in terms of eulerian subgraphs of $G$.
On $H_4$ we have $P(G;4)=(-1)^{r(G)}4^{k(G)}T(G;-3,0)$ and
$F(G;4)=(-1)^{n(G)}T(G;0,-3)$. Also, $T(G;-1,-1)=(-1)^{|E|}(-2)^{\dim(\mathcal{C}_2\cap\mathcal{C}_2^\perp)}$, where
$\mathcal{C}_2\cap\mathcal{C}_2^\perp$ is the bicycle space of $G$~\cite{RR78}.
The Tutte
polynomial on $H_4$ is the partition function of the $4$-state Potts
model, which coincides with the Hamming weight
enumerator of the subspace $\mathcal{C}_2\times\mathcal{C}_2$ of
$\mathbb{F}_4$-flows. Evaluations of the Tutte polynomial at the point
$(-2,-\frac{1}{3})$ on $H_4$ as well as $(-3,0)$, $(0,-3)$ and
$(-1,-1)$ are also given an interpretation in section~\ref{H2 H4} in terms of eulerian
subgraphs of $G$.
By MacWilliams duality (see section~\ref{hwes} below), the
interpretions that we give for evaluations of the Tutte polynomial at points $(a,b)$ in terms of eulerian
subgraphs (cycles, $\mathcal{C}_2$) become interpretations for points $(b,a)$ in terms of
cutsets (cocycles, $\mathcal{C}_2^\perp$). In the same way, for
example, there is the expansion of the
Ising model of a graph over its cutsets (bipartitions)~\cite[\S 4.3]{DW93}
corresponding to Van der Waerden's expansion over
eulerian subgraphs.
\subsection{The Fourier transform}\label{sec:fourier}
In this section we summarise the facts about the Fourier transform on rings such as
$\mathbb{F}_q$ and $\mathbb{Z}_q$ (the integers modulo $q$) that the reader needs to be aware of
in this paper.
Let $Q$ be a commutative ring (either $\mathbb{F}_q$ or
$\mathbb{Z}_q$ in the sequel) and
$Q^E$ the set of all vectors $x=(x_e:e\in E)$ with entries in $Q$
indexed by $E$. The indicator function $1_k$ for $k\in
Q$ is defined
by $1_k(\ell)=1$ if $k=\ell$ and $0$ otherwise. A subset $S\subseteq
Q$ has indicator function defined by $1_S=\sum_{k\in S}1_k$.
A {\em character} of $Q$ is a homomorphism $\chi:Q\rightarrow\mathbb{C}^\times$ from the additive
group $Q$ to the multiplicative group of $\mathbb{C}$. The set of
characters form a group $\widehat{Q}$ under pointwise
multiplication isomorphic to $Q$ as an abelian group. For $Q^E$ the group of characters
$\widehat{Q^E}$ is isomorphic to $\widehat{Q}^E$.
For each $k\in Q$, write $\chi_k$ for the image of $k$ under a fixed
isomorphism of $Q$ with $\widehat{Q}$. In particular, the principal
(trivial) character $\chi_0$ is defined by $\chi_0(\ell)=1$ for all $\ell\in
Q$, and $\chi_{-k}(\ell)=\overline{\chi_k(\ell)}$ for all $k,\ell\in Q$, where the bar
denotes complex conjugation.
A character $\chi\in\widehat{Q}$ is a {\em generating character} for $Q$ if
$\chi_k(\ell)=\chi(k\ell)$ for
each character $\chi_k\in\widehat{Q}$. The ring $\mathbb{Z}_q$
has a generating character $\chi$ defined by $\chi(k)=e^{2\pi i
k/q}$. (The fixed
isomorphism $k\mapsto \chi_k$ of $\mathbb{Z}_q$ with $\widehat{\mathbb{Z}_q}$ in this case is
given by taking $\chi_k(\ell)= e^{2\pi ik\ell/q}$.)
The field $\mathbb{F}_q$ with $q=p^m$ has a generating
character $\chi$ defined by $\chi(k)=e^{2\pi i
\mbox{\rm \tiny Tr}(k)/p}$, where ${\rm Tr}(k)=k+k^p+\cdots +
k^{p^{m-1}}$ is the trace of $k$. (Here the isomorphism
$\mathbb{F}_q\rightarrow\widehat{\mathbb{F}_q},\; k\mapsto\chi_k$ is given
by taking $\chi_k(\ell)=e^{2\pi i\mbox{\rm \tiny Tr}(k\ell)/p}$.)
If $\chi$ is a
generating character for $Q$ then $\chi^{\otimes E}$, defined by
$\chi^{\otimes{E}}(x)=\prod_{e\in E}\chi(x_e)$ for $x=(x_e:e\in E)\in Q^E$, is a generating character for $Q^E$. The euclidean
inner product (dot product) on $Q^E$ is defined by $x\cdot y=\sum x_ey_e$. Since
$\prod\chi(x_e)=\chi(\sum x_e)$, it follows that
$\chi^{\otimes E}$ satisfies $\chi^{\otimes E}_x(y)=\chi(x\cdot y)$ for $x,y\in
Q^E$.
The vector space
$\mathbb{C}^{Q^E}$ over $\mathbb{C}$ of all functions from $Q^E$ to
$\mathbb{C}$ is an inner product space with Hermitian
inner product $\langle\, ,\,\rangle$ defined for
$f,g:Q^E\rightarrow\mathbb{C}$ by
$$\langle f,g\rangle=\sum_{x\in Q^E}f(x)\overline{g(x)}.$$
Fix an isomorphism $k\mapsto \chi_k$ of $Q$ with $\widehat{Q}$ and let
$\chi$ be a generating character for $Q$ such that
$\chi_k(\ell)=\chi(k\ell)$.
For $f\in\mathbb{C}^{Q}$ the {\em Fourier transform}
$\widehat{f}\in\mathbb{C}^{Q}$ is defined for $k\in Q$ by
\[\widehat{f}(k)=\langle f,\chi_k\rangle=\sum_{\ell\in
Q}f(\ell)\chi(-k\ell).\]
The Fourier transform of a function $g:Q^E\rightarrow\mathbb{C}$ is
then given by
$$\widehat{g}(x)=\sum_{y\in Q^E}g(y)\chi(-x\cdot y).$$
In the space $\mathbb{C}^{Q^E}$, the Fourier inversion
formula is $\widehat{\widehat{f}}(x)=q^{|E|}f(-x)$, or
$$f(y)=q^{-|E|}\langle \widehat{f},\chi_{-y}\rangle=q^{-|E|}\sum_{x\in
Q^E}\widehat{f}(x)\chi(x\cdot y).$$
Plancherel's or Parseval's
identity is $\langle f,g\rangle=q^{-|E|}\langle\widehat{f},\widehat{g}\rangle.$
For a subset $\mathcal{S}$ of $Q^E$, the annihilator
$\mathcal{S}^{\sharp}$ of
$\mathcal{S}$ is defined by
$\mathcal{S}^{\sharp}=\{y\in Q^E:\forall_{x\in
\mathcal{S}}\,\,\chi_x(y)=1\}.$
The annihilator $\mathcal{S}^\sharp$ is a subgroup of $Q^E$
isomorphic to $Q^E/\mathcal{S}$. When $\mathcal{S}$ is a
$Q$-submodule of $Q^E$ and $Q$ has a generating character,
the annihilator of $\mathcal{S}$ is equal to the orthogonal $\mathcal{S}^\perp$
to $\mathcal{S}$ (with respect
to the euclidean inner product), defined by
$\mathcal{S}^\perp=\{y\in Q^E:\forall_{x\in \mathcal{S}}\,\, x\cdot y=0\}.$
A key property of the Fourier transform is that for a $Q$-submodule
$\mathcal{S}$ of $Q^E$
$$\widehat{1_\mathcal{S}}(y)=\sum_{x\in \mathcal{S}}\chi_x(y)=|\mathcal{S}|1_{\mathcal{S}^\sharp}(y),$$
and the Poisson summation formula is that
\[\sum_{x\in \mathcal{S}}f(x+z)=\frac{1}{|\mathcal{S}^\sharp|}\sum_{x\in
\mathcal{S}^\sharp}\widehat{f}(x)\chi_{z}(x).\]
For $Q$-submodule $\mathcal{S}$ of $Q^E$ the coset
$\{x+z:x\in\mathcal{S}\}$ of $\mathcal{S}$ in the additive group $Q^E$
is denoted by $\mathcal{S}+z$, an element
of the quotient module $Q^E/\mathcal{S}$.
\subsection{Flows, tensions and Hamming weight enumerators}\label{hwes}
For the moment we continue with $Q$ a commutative ring on $q$ elements with a generating
character. Let $\mathcal{S}$ be a subset of $Q^E$, the set
of vectors with entries in $Q$ indexed by edges of $G$.
The Hamming weight of a vector $x=(x_e:e\in E)\in Q^E$ is defined by
$|x|=\#\{e\in E:x_e\neq 0\}$. The Hamming weight enumerator of
$\mathcal{S}$ is defined by
$${\rm hwe}(\mathcal{S};t)=\sum_{x\in\mathcal{S}}t^{|E|-|x|},$$
the exponent being the number of zero entries in $x$.
When $\mathcal{S}$ is a $Q$-submodule of $Q^E$ the MacWilliams
duality theorem states that
\begin{equation}\label{MacWilliams}{\rm
hwe}(\mathcal{S};t)=\frac{(t-1)^{|E|}}{|\mathcal{S}^{\perp}|}{\rm
hwe}\left(\mathcal{S}^\perp;\frac{t+q-1}{t-1}\right),\end{equation}
which follows from the Poisson summation formula with
$f=(t1_0+1_{Q\setminus 0})^{\otimes E}$.
A {\em $Q$-flow} of $G$ is defined with reference to a ground
orientation $\gamma$ of $G$; the number of $Q$-flows of a given Hamming
weight is independent of $\gamma$. A vector $x\in Q^E$ is a $Q$-flow of $G$ if, for each
vertex $v\in V$,
$$\sum_{e\in E}\gamma_{v,e}x_e=0,$$
where $\gamma_{v,e}=+1$ if $e$ is directed into $v$, $\gamma_{v,e}=-1$
is $e$ is directed out of $v$, and $\gamma_{v,e}=0$ if $e$ is not
incident with $v$. The $Q$-flows form a $Q$-submodule of $Q^E$ whose
orthogonal $Q$-submodule is the set of $Q$-tensions of
$G$. The latter comprise the set
of $y\in Q^E$ such that there exists a
vertex $Q$-colouring $z\in Q^V$ with $y_e=z_u-z_v$ for
all edges $e=(u,v)$ (directed by the orientation $\gamma$). To each
$Q$-tension $y$ there correspond $q^{k(G)}$ vertex $Q$-colourings for
which $y_e=z_u-z_v$.
A {\em nowhere-zero} $Q$-flow has Hamming weight $|E|$, and likewise a
nowhere-zero $Q$-tension. To a nowhere-zero $Q$-tension $y$
corresponds a set of $q^{k(G)}$ proper vertex $Q$-colourings of $G$, i.e., $z\in Q^V$ such that $z_u\neq z_v$ whenever $u$ is adjacent to $v$ in $G$.
The Tutte polynomial on the hyperbola $H_q$ is related to the Hamming weight
enumerator of the set of $Q$-flows via the identity
\begin{equation}\label{qflows tutte}
{\rm hwe}(\mbox{\rm $Q$-flows of
$G$};t)=(t-1)^{n(G)}T\left(G;t,\frac{t+q-1}{t-1}\right).
\end{equation}
By \eqref{MacWilliams} the set of $Q$-tensions has Hamming weight enumerator
$${\rm hwe}(\mbox{\rm
$Q$-tensions of $G$};t)=(t-1)^{r(G)}T\left(G;\frac{t+q-1}{t-1},t\right).$$
\section{The Tutte polynomial on $H_2$ and $H_4$}\label{H2 H4}
From equation \eqref{qflows tutte}, the Tutte polynomial specialises on $H_2$ to the Hamming weight
enumerator of the space $\mathcal{C}_2$ of $\mathbb{F}_2$-flows,
$${\rm
hwe}(\mathcal{C}_2;t)=(t-1)^{n(G)}T\left(G;t,\frac{t+1}{t-1}\right).$$
If $G$ is eulerian then the all $1$ vector belongs to $\mathcal{C}_2$
with the consequence that ${\rm hwe}(\mathcal{C}_2;t)=t^{|E|}{\rm
hwe}(\mathcal{C}_2;t^{-1})$, whence if $G$ is eulerian then
\begin{equation}\label{eul reciprocal}T\left(G;t,\frac{t+1}{t-1}\right)=(-1)^{n(G)}t^{r(G)}T\left(G;\frac{1}{t},\frac{1+t}{1-t}\right).\end{equation}
Dually, if $G$ is bipartite then the all $1$ vector belongs to
$\mathcal{C}_2^\perp$ and in this case
$$T\left(G;\frac{t+1}{t-1},t\right)=(-1)^{r(G)}t^{n(G)}T\left(G;\frac{1+t}{1-t},\frac{1}{t}\right).$$
Using the MacWilliams duality theorem~\eqref{MacWilliams},
$$(t-1)^{n(G)}T\left(G;t,\frac{t+1}{t-1}\right)=\sum_{\mbox{\tiny
eulerian}\,A\subseteq E}t^{|E|-|A|}=2^{-r(G)}(t-1)^{|E|}\sum_{\mbox{\tiny cutsets}\, A\subseteq E}\left(\frac{t+1}{t-1}\right)^{|E|-|A|}.$$
In particular
$$\sum_{\mbox{\tiny eulerian}\,A\subseteq E}(-1)^{|A|}=2^{|E|-|V|}P(G;2).$$
Likewise, the Tutte polynomial on $H_4$ is the Hamming weight
enumerator of the space $\mathcal{C}_4\cong\mathcal{C}_2\times\mathcal{C}_2$
of $\mathbb{F}_4$-flows,
$${\rm hwe}(\mathcal{C}_2\times\mathcal{C}_2;t)=(t-1)^{n(G)}T\left(G;t,\frac{t+3}{t-1}\right).$$
The MacWilliams duality theorem~\eqref{MacWilliams} here is that
$${\rm hwe}(\mathcal{C}_2\times\mathcal{C}_2;t)=\frac{(t-1)^{|E|}}{|\mathcal{C}_2^{\perp}|^2}{\rm
hwe}\left(\mathcal{C}_2^\perp\times\mathcal{C}_2^\perp;\frac{t+3}{t-1}\right),$$
which in terms of eulerian subgraphs of $G$ says that
$$\sum_{\mbox{\tiny
eulerian}\,A,B\subseteq E}t^{|E|-|A\cup B|}=4^{-r(G)}(t-1)^{|E|}\sum_{\mbox{\tiny
cutsets}\, A, B\subseteq
E}\left(\frac{t+3}{t-1}\right)^{|E|-|A\cup B|}.$$
In particular
$$\sum_{\mbox{\tiny
eulerian}\,A,B\subseteq E}(-3)^{|E|-|A\cup B|}=(-1)^{|E|}4^{|E|-|V|}P(G;4).$$
\subsection{Coset weight enumerators}
In a previous article~\cite{CCC07} the following specialisations of
the Tutte polynomial to the hyperbola $H_2$ and
the hyperbola $H_4$ were derived as an illustration of the techniques
afforded by elementary Fourier analysis. See also \cite{E98} (quoted in
\cite{Mi05}) for these identities in the context of coding theory. In
this section we give a combinatorial interpretation of
these identities for particular values of $t$ and derive evaluations
of the Tutte polynomial on $H_2$ and $H_4$.
When writing $\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2$ in the range
of summations below, we assume that $z\in\mathbb{F}_2^E$ ranges
over a transversal of the cosets, each coset $\mathcal{C}_2+z$
appearing only once
\begin{lem}\label{sum cubes}
Let $G=(V,E)$ be a graph and let $\mathcal{C}_2$ be the subspace of
$\mathbb{F}_2$-flows of $G$. Then, for $t\in\mathbb{C}$,
$$\sum_{\mathcal{C}_2+z\in \mathbb{F}_2^E/\mathcal{C}_2}|{\rm
hwe}(\mathcal{C}_2+z;t)|^2=(t+\overline{t})^{r(G)}|t-1|^{2n(G)}T\left(\!G;\frac{|t|^2+1}{t+\overline{t}},\left|\frac{t+1}{t-1}\right|^2\right),$$
$$\sum_{\mathcal{C}_2+z\in \mathbb{F}_2^E/\mathcal{C}_2}{\rm
hwe}(\mathcal{C}_2+z;t)^2=(2t)^{r(G)}(t-1)^{2n(G)}T\left(\!G;\frac{t^2+1}{2t},\left(\frac{t+1}{t-1}\right)^2\right),$$
and
$$\sum_{\mathcal{C}_2+z\in \mathbb{F}_2^E/\mathcal{C}_2}{\rm hwe}(\mathcal{C}_2+z;t)^3=(t+1)^{|E|} t^{r(G)}(t-1)^{2n(G)}T\left(\!G;\frac{t^2-t+1}{t},\left(\frac{t+1}{t-1}\right)^2\right).$$
\end{lem}
Note that $\left(\frac{t+1}{t-1}\right)^2$ is real if and only if
$t\in\mathbb{R}$ or $|t|=1$, since
$\frac{t+1}{t-1}=\frac{\overline{t}-t+|t|^2-1}{|t|^2+1-t-\overline{t}}$.
By putting $t=e^{i\theta}$ in the identities of Lemma~\ref{sum cubes}, routine
calculations yield the following.\footnote{From~\cite[Corollary 7.5]{CCC07} it is readily seen
that for $r\in\mathbb{N}$ the sum of $r$th powers of the coset weight enumerators of $\mathcal{C}_2$ is a specialisation of the {\em complete weight
enumerator} (see section~\ref{poset} for a definition) of the $(r-1)$-fold direct product
$\mathcal{C}_2^\perp\times\cdots\times\mathcal{C}_2^\perp$ (isomorphic to the space of $\mathbb{F}_{2^{r-1}}$-tensions of
$G$).
Only for
$r\leq 3$ is this specialisation a Hamming weight enumerator,
and so the sum of $r$th powers of coset Hamming weight enumerators of
$\mathcal{C}_2$ is a specialisation of the Tutte polynomial only for
$r\leq 3$. For example, the sum of fourth powers of the coset weight
enumerators ${\rm hwe}(\mathcal{C}_2+z;t)$ turns out to be equal to
$$2^{-3r(G)}(t^2-1)^{2|E|}\sum_{\mbox{\rm \tiny eulerian }\, A,B,C\subseteq E}s^{|E|-|A\cup
B\cup C|-|A\cap B\cap C|}$$
where $s=\left(\frac{t+1}{t-1}\right)^2$.
\begin{cor}\label{real sums} If $t=e^{i\theta}$ for some
$\theta\in(0,2\pi)$ then
\begin{equation}\label{abs sum squares theta}2^{-|E|}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E}|{\rm hwe}(\mathcal{C}_2+z;e^{i\theta})|^2=(\cos\theta)^{r(G)}(1-\cos\theta)^{n(G)}T\left(\!G;\frac{1}{\cos\theta},\frac{1+\cos\theta}{1-\cos\theta}\right),\end{equation}
\begin{equation}\label{sum squares real}(2e^{i\theta})^{-|E|}\,\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}{\rm
hwe}(\mathcal{C}_2+z;e^{i\theta})^2=(\cos\theta-1)^{n(G)}T\left(\!G;\cos
\theta,\frac{\cos\theta+1}{\cos\theta-1}\right),\end{equation}
and
$$\hspace{-5cm} (2e^{i\theta})^{-\frac{3}{2}|E|}\,\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}{\rm
hwe}(\mathcal{C}_2+z;e^{i\theta})^3=$$
\begin{equation}\label{sum cubes real}2^{-r(G)}(1+\cos\theta)^{\frac{1}{2}|E|}(\cos\theta-1)^{n(G)}T\left(\!G;2\cos \theta-1,\frac{\cos\theta+1}{\cos\theta-1}\right).\end{equation}
\end{cor}
If $\theta\in\mathbb{Q}$ then $2 \cos \theta =e^{i\theta}+e^{-i\theta}$ is an algebraic
integer, and hence an ordinary integer if $\cos\theta\in\mathbb{Q}$.
The only rational values of $\cos \theta$ for $\theta\in\mathbb{Q}$
are thus
$0,\pm\frac{1}{2},\pm 1$, corresponding to $\theta=\pm\frac{\pi}{2},
\pm\frac{\pi}{3}, \pm\frac{2\pi}{3},\pi,0$.
Thus when $e^{i\theta}$ is a $q$th root of unity for $q\in\{2,3,4,6\}$ the evaluations
of the Tutte polynomial in Corollary~\ref{real sums} are at rational
points (see Table~\ref{table} below).
\begin{table}[ht]
\begin{center}
\caption{{\bf \small Evaluations of the Tutte polynomial in the
identities of Corollary~\ref{real sums}}}\label{table}
$$\begin{array}{c|c|c|c|c}
\theta & \cos \theta & \mbox{\rm eq. \eqref{abs sum squares theta}, point on $H_2$} & \mbox{\rm eq. \eqref{sum
squares real}, point on $H_2$} & \mbox{\rm eq. \eqref{sum cubes real}, point on $H_4$}\\
\hline
\pi & -1 & (-1,0) & (-1,0) & (-3,0)\\
2\pi/3 & -\frac{1}{2} & (-2,\frac{1}{3}) & (-\frac{1}{2},-\frac{1}{3}) &
(-2,-\frac{1}{3})\\
\pi/2 & 0 & * & (0,-1) & (-1,-1)\\
\pi/3 & \frac{1}{2} & (2,3) & (\frac{1}{2},-3) & (0,-3)\end{array}$$
{\small * For $\theta=\pi/2$, the right-hand side of equation (\ref{abs sum
squares theta}) is equal to $1$ independent of $G$.
For $\theta=\pi$ the factor
$(1+\cos\theta)^{\frac{1}{2}|E|}$ on the right-hand side of (\ref{sum cubes real}) is equal
to zero.}
\end{center}
\end{table}
In order to give combinatorial interpretations of identity~\eqref{abs sum
squares theta} (and identity~\eqref{sum squares real}) we shall be interested in the correlation between two types of
event when choosing $A,B\subseteq E$ uniformly at random. First, the event that $A\bigtriangleup B$ is eulerian. Second,
for various integers $q$,
the event that $|A|-|B|$ (respectively $|A|+|B|$) belongs to a certain subset of congruence
classes modulo $q$.
Similarly, in order to interpret identity~\eqref{sum cubes real} we
choose $A,B,C\subseteq E$ uniformly at random and look at the correlation between
the event that $A\bigtriangleup B, B\bigtriangleup C,
C\bigtriangleup A$ are each
eulerian and the event that $|A|+|B|+|C|$ belongs to a certain
subset of congruence classes modulo $q$.
\subsection{Bias}
If $q$ is not a power of two, for fixed $k\in\{0,1,\ldots, q-1\}$
none of the events $|A|-|B|\equiv k (\bmod\, q)$, $|A|+|B|\equiv k (\bmod\, q)$ or $|A|+|B|+|C|\equiv
k (\bmod\, q)$ can have
probability $\frac{1}{q}$ when $A,B,C\subseteq E$ are taken
uniformly at random. As observed in Remark~\ref{equidistributed}
below, amongst powers of two only for $q=2$ is it true
that the values of $|A\pm|B|(+|C|)$ are equidistributed modulo $q$ (although it remains possible that for some values
of $k$ the event $|A\pm|B|(+|C|)\equiv k (\bmod\, q)$ has probability $\frac1q$).
Let $\Sigma$ be an event in the uniform probability space on pairs
$A,B\subseteq E$ or triples $A,B,C\subseteq E$, and $\overline{\Sigma}$ is its
complement. In the sequel the event $\Sigma$ takes the form $|A|\pm
|B|\in S (\bmod\, q)$ or
$|A|\!+\!|B|\!+\!|C|\in S (\bmod\, q)$ for a subset $S$ of the integers $\{0,1\ldots, q-1\}$ modulo
$q$.
Define the {\em bias} towards $\Sigma$ by
$${\rm Bias}(\Sigma)=\mathbb{P}(\Sigma)-\mathbb{P}(\overline{\Sigma}).$$
Note that ${\rm Bias}(\overline{\Sigma})=-{\rm Bias}(\Sigma)$ and the event $\Sigma$ has probability
$\frac{1}{2}[1+{\rm Bias}(\Sigma)]$.
For example, if $q=2$ and $S=\{0\}$ then
$\Sigma=\{A,B\subseteq E:|A|+|B|\in S (\bmod\, q)\}$ is the event that $|A|+|B|$
is even and this has the same probability as the event
$\overline{\Sigma}=\{A,B\subseteq E:|A|+|B|\not\in S (\bmod\, q)\}$ that
$|A|+|B|$ is odd. For $q>2$, the bias of an event of the form $|A|\pm
|B|\in S (\bmod\, q)$ or
$|A|+|B|+|C|\in S (\bmod\, q)$ is usually not equal to zero. This is excepting the case
when $q$ is even and $S=\{0,2,\ldots, q-2\}$ or $S=\{1,3,\ldots, q-1\}$ for
which the event $\Sigma$ is about the parity of $|A|\pm|B|(+|C|)$
again. However, in Theorem~\ref{A + B + C with q=4} it is found that
${\rm Bias}(|A|\!+\!|B|\!+\!|C|\equiv 0,1 (\bmod\, 4))=0$ when
$|E|\equiv 1 (\bmod\, 4)$
\begin{lem}\label{bias S}
Suppose $A,B,C\subseteq E$ are chosen uniformly at
random and that $S\subseteq\{0,1\ldots, q-1\}$. Then
$${\rm Bias}(\Sigma)=2q^{-1}\langle\widehat{g},\widehat{1_S}\rangle-1,$$
where $1_S$ is the indicator function of $S$ and
$$\widehat{g}(k)=\begin{cases} 2^{-|E|}(1+\cos\frac{2\pi k}{q})^{|E|}\\
2^{-|E|}e^{-2\pi ik|E|/q}(1+\cos\frac{2\pi k}{q})^{|E|} \\
2^{-\frac{3}{2}|E|}e^{-2\pi ik\frac{3}{2}|E|/q}(1+\cos\frac{2\pi k}{q})^{\frac{3}{2}|E|}\end{cases}$$
according as
$$\Sigma=\begin{cases}\{A,B\subseteq E:|A|-|B|\in S(\bmod\, q)\}\\
\{A,B\subseteq E:|A|+|B|\in S(\bmod\, q)\}\\
\{A,B,C\subseteq E:|A|+|B|+|C|\in S(\bmod\,q)\}\end{cases}.$$
\end{lem}
\begin{pf} We prove the lemma for the event
$\Sigma=\{A,B\subseteq E:\,|A|-|B|\in S(\bmod\, q)\}$. The
other cases are similar.
Define
$$g(\ell)=\mathbb{P}\big(\,|A|-|B|\equiv\ell(\bmod\, q)\,\big).$$
Then
$$\mathbb{P}(\Sigma)=\sum_{\ell\in
S}g(\ell)=\langle g,1_S\rangle=q^{-1}\langle\widehat{g},\widehat{1_S}\rangle,$$
using Plancherel's formula at the end.
By definition, ${\rm
Bias}(\Sigma)=2\mathbb{P}(\Sigma)-1$, and the first part of
the lemma is proved. It remains to calculate $\widehat{g}(k)$ for $k\in\{0,1,\ldots, q-1\}$:
\begin{align*}
\widehat{g}(k)=\sum_{0\leq \ell\leq q-1}e^{-2\pi i k\ell/q}g(\ell) & =2^{-2|E|}\sum_{A, B\subseteq E}e^{-2\pi ik(|A|-|B|)/q}\\
& =2^{-2|E|}\sum_{A\subseteq
E}e^{-2\pi ik|A|/q}\sum_{B\subseteq
E}e^{2\pi ik|B|/q}\\
& = 2^{-2|E|}(1+e^{-2\pi ik/q})^{|E|}(1+e^{2\pi ik/q})^{|E|}\\
& =2^{-2|E|}(e^{2\pi ik/q}+2+e^{-2\pi ik/q})^{|E|}\\
& =2^{-|E|}(1+\cos\frac{2\pi k}{q})^{|E|}.\end{align*}
This completes the proof.
\end{pf}
\begin{rem}\label{equidistributed}
In the notation of the proof of Lemma~\ref{bias S}, the probabilities
$g(\ell)=\mathbb{P}(|A|-|B|=\ell(\bmod\, q))$ are equal for all
$\ell\in\{0,1,\ldots, q-1\}$ if and only if
$$\sum_{0\leq\ell\leq q-1}g(\ell)e^{-2\pi i\ell/q}=0.$$
But the left-hand sum is equal to
$\widehat{g}(1)=2^{-|E|}(1+\cos\frac{2\pi}{q})^{|E|}$, and this is equal to
zero if and only if $q=2$. Hence the values $|A|\!-\!|B|$ for $A,B\subseteq E$ are
equidistributed modulo $q$ if and only if $q=2$. Similarly, the values
$|A|\!+\!|B|$ for $A,B\subseteq E$ and the values of
$|A|\!+\!|B|\!+\!|C|$ for $A,B,C\subseteq E$ are only equidistributed
modulo $q$ when $q=2$.
\end{rem}
Let $\Delta$ be an event in the uniform probability space on pairs
$A,B\subseteq E$ or triples $A,B,C\subseteq E$.
We define the
conditional bias of $\Sigma$ given $\Delta$ by
$${\rm Bias}(\Sigma\,|\,\Delta)=\mathbb{P}(\Sigma\,|\,\Delta)-\mathbb{P}(\overline{\Sigma}\,|\,\Delta).$$
In what follows, $\Delta$ is either the event that $A\bigtriangleup B$ is
eulerian (where $\Sigma$ is one of the events $|A|\pm|B|\in
S\,(\mbox{\rm mod}\, q)$ for
some $S\subseteq\{0,1\ldots, q-1\}$) or the event
that $A\bigtriangleup B$ and $B\bigtriangleup C$ are both eulerian
(where $\Sigma$ is the event that $|A|+|B|+|C|\in S\,(\mbox{\rm mod}\,
q)$ for some $S\subseteq\{0,1\ldots, q-1\}$).
The covariance of (the indicator functions of) the events $\Sigma$ and
$\Delta$ is defined by the difference
$\mathbb{P}(\Sigma\cap\Delta)-\mathbb{P}(\Sigma)\mathbb{P}(\Delta)$.
We define the {\em correlation}\footnote{The {\em correlation
coefficient} of the indicator functions of the events $\Sigma$
and $\Delta$ is another normalisation of the
covariance, namely $$\frac{\mathbb{P}(\Sigma\cap
\Delta)-\mathbb{P}(\Sigma)\mathbb{P}(\Delta)}{\sqrt{\mathbb{P}(\Sigma)\mathbb{P}(\Delta)(1-\mathbb{P}(\Sigma))(1-\mathbb{P}(\Delta))}}.$$}
between the events $\Sigma$
and $\Delta$ by dividing the covariance through by
$\mathbb{P}(\Delta)$,
$${\rm Correlation}(\Sigma\,|\,\Delta)=\mathbb{P}(\Sigma\,|\,\Delta)-\mathbb{P}(\Sigma).$$
Correlation is related to bias via the relation
$${\rm Bias}(\Sigma\,|\,\Delta)-{\rm Bias}(\Sigma)=2\,{\rm
Correlation}(\Sigma\,|\,\Delta).$$
When ${\rm Bias}(\Sigma)\neq 0$ we shall have occasion to also measure correlation via the ratio
\begin{equation}\label{ratio corr}\frac{{\rm Bias}(\Sigma\,|\,\Delta)}{{\rm Bias}(\Sigma)}=\frac{2{\rm
Correlation}(\Sigma\,|\,\Delta)}{{\rm Bias}(\Sigma)}+1.\end{equation}
This can be viewed as the scale factor from the existing
bias towards $\Sigma$ to the bias towards $\Sigma$ given the event $\Delta$.
When the ratio~\eqref{ratio corr} is greater than $1$ the existing bias towards $\Sigma$ is magnified,
when the ratio is less than $1$ the existing bias is diminished.
The ratio~\eqref{ratio corr} is equal to
$1$ if and only if there is no correlation between the events
$\Sigma$ and $\Delta$.
In the next section we derive general expressions for ${\rm
Bias}(\Sigma)$ and ${\rm
Bias}(\Sigma\,|\,\Delta)$, for any choice of $q$ and $S$ in the definition of $\Sigma$,
the latter expressed in terms of the Tutte
polynomial evaluations of Corollary~\ref{real sums} for
$\theta\in\{2\pi k/q:k=1,\ldots, q-1\}$.
By taking $q\in\{2,3,4,6\}$ we obtain interpretations for various
evaluations of the Tutte polynomial at the points listed in Table~\ref{table}. The reason why
we limit ourselves to $q\in\{2,3,4,6\}$ is not specifically on account of these
corresponding to evaluations at rational points, but rather that only for
these values of $q$ do we obtain evaluations of the Tutte
polynomial at a single point rather than a sum of evaluations at two
or more distinct points.
An evaluation of the Tutte
polynomial at a single point is of interest as such an evaluation is a {\em
Tutte-Grothendieck invariant} on graphs, i.e., satisfies a ``linear''
deletion-contraction recurrence relation.
Specifically, if $\setminus$ denotes deletion and $/$ contraction, a function $f$ on graphs satisfying
$$f(G)=\begin{cases}
af(G\setminus e)+bf(G/e) & \mbox{ if $e$ is neither an isthmus nor a loop,}\\
xf(G\setminus e) & \mbox{ if $e$ is an isthmus,}\\
yf(G\setminus e) & \mbox{ if $e$ is a loop,}\end{cases}$$
and with value $c^{|V|}$ on the edgeless graph $(V,\emptyset)$ is
given by the evaluation\footnote{An {\em isthmus} or bridge is an edge
forming a cutset of size $1$ and a {\em loop} is an edge forming a
cycle of size $1$. If $b=0$ then
$$f(G)=c^{|V|}a^{n(G)-\ell(G)}x^{r(G)}y^{\ell(G)},$$
and if $a=0$ then $$f(G)=c^{k(G)+i(G)}b^{r(G)-i(G)}x^{i(G)}y^{n(G)},$$
where $i(G)$ is the number of isthmuses and $\ell(G)$ the number of
loops in $G$.} \cite{B98}, \cite{DW93},
$$f(G)=c^{k(G)}a^{n(G)}b^{r(G)}T(G;cx/b, y/a).$$
\label{fn}
Many of the evaluations of the Tutte polynomial in sections~\ref{square root of unity} to \ref{sixth root of unity} have
been highlighted as theorems either because they have other well-known combinatorial meanings (such
as the number of nowhere-zero $4$-flows in Theorem~\ref{sum cubes with
q=6}) and the opaqueness of the connection to these other
interpretations is intriguing,
or because they on the contrary do not have such other well-known interpretations
(such as $T(G;-2,-\frac{1}{3})$ in Theorem~\ref{cube root A
B C}).
Direct proofs of the corresponding
deletion-contraction recurrences do not seem straightforward in many cases.
\subsection{Evaluations of coset weight enumerators at $q$th roots of
unity}\label{evaluation at qth}
In this section the identities in
Corollary~\ref{real sums} are given interpretations in terms of ${\rm
Bias}(\Sigma|\Delta)$ where $\Sigma$ takes the form
$|A|\pm|B|(+|C|)\in S\,(\mbox{\rm mod}\,q)$ and $\Delta$ is the event
that $A\bigtriangleup B$ (and $B\bigtriangleup C$) is eulerian. These interpretations in their
general form make for rather dull reading, but their particular cases for $q\in\{2,3,4,6\}$ are the more
interesting theorems that follow as corollaries. The latter are
presented in sections~\ref{square root of unity} to
\ref{sixth root of unity} to which the reader might wish to turn
before referring back to this section
\begin{lem}\label{interpret h2|} Suppose $A,B\subseteq E$ are chosen uniformly at
random and $\Delta$ is the event that $A\bigtriangleup B$ is
eulerian. Then, for any $k\in\{0,1,\ldots, q-1\}$,
$$2^{-|E|}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}|{\rm
hwe}(\mathcal{C}_2+z;e^{2\pi ik/q})|^2=2^{n(G)}\sum_{0\leq\ell\leq
q-1}e^{2\pi i k\ell/q}\,\mathbb{P}(|B|-|A|\equiv \ell(\bmod{q})\,\mid\,\Delta).$$
\end{lem}
\begin{pf}
Given a coset $\mathcal{C}_2+z$ and $\ell\in\{0,\ldots,q-1\}$, define
$$p_\ell=p_\ell(\mathcal{C}_2+z)=\#\{x\in\mathcal{C}_2+z:|E|-|x|\equiv
\ell(\bmod q)\},$$
so that
$$\sum_{0\leq
\ell\leq q-1}e^{2\pi ik\ell/q}p_\ell={\rm hwe}(\mathcal{C}_2+z;e^{2\pi ik/q}).$$
Going on to define
$$P_\ell=P_\ell(\mathcal{C}_2+z)=\sum_{j-k\equiv \ell(\bmod
q)}p_jp_k,$$
we have
\begin{equation}\label{|hwe|^2 to Pl}|{\rm hwe}(\mathcal{C}_2+z;e^{2\pi ik/q})|^2 = \big|\sum_{0\leq
\ell\leq q-1}e^{2\pi ik\ell/q}p_\ell\big|^2= \sum_{0\leq\ell\leq q-1} e^{2\pi ik\ell/q}P_\ell.\end{equation}
Let $C\subseteq E$ have indicator vector $z\in\mathbb{F}_2^E$ and suppose $A\subseteq
E$ is chosen uniformly at random with indicator vector $x\in\mathbb{F}_2^E$.
Then $|E|-|x|=|E\setminus A|$ and $x\in\mathcal{C}_2+z$ if and only if
$x+z\in\mathcal{C}_2$, i.e., $A\bigtriangleup C$ is eulerian. Hence
$$2^{-|E|}p_\ell(\mathcal{C}_2+z)=\mathbb{P}(|E\setminus
A|\equiv\ell(\bmod\, q)\cap A\bigtriangleup C\,\mbox{\rm eulerian}).$$
Similarly, if $A,B\subseteq E$ are chosen uniformly at random then
$$2^{-2|E|}P_\ell(\mathcal{C}_2+z) =$$
$$\sum_{j-k\equiv \ell(\bmod q)}\mathbb{P}(|E\setminus A|\equiv j(\!\bmod q)\,\cap\,A\bigtriangleup C\,\mbox{\rm eulerian})\,\mathbb{P}(|E\setminus B|\equiv k(\!\bmod
q)\,\cap B\bigtriangleup C\,\mbox{\rm eulerian})$$
$$ = \mathbb{P}(|E\setminus A|-|E\setminus B|\equiv\ell(\bmod q)\,\cap
A\bigtriangleup C\,\mbox{\rm eulerian}\cap B\bigtriangleup
C\,\mbox{\rm eulerian}).$$
Given $C,C'\subseteq E$, the events $\{A\bigtriangleup C\,\mbox{\rm
eulerian}\cap B\bigtriangleup C\,\mbox{\rm eulerian}\}$ and $\{A\bigtriangleup C'\,\mbox{\rm
eulerian}\cap B\bigtriangleup C'\,\mbox{\rm eulerian}\}$ are
either equal (when $C\bigtriangleup C'$ is eulerian) or
disjoint. For suppose that $C$ has indicator vector
$z$ and $C'$ indicator vector $z'$. If
$x+z\in\mathcal{C}_2$ and $x+z'\in\mathcal{C}_2$ then
$z+z'\in\mathcal{C}_2$, i.e., $C\bigtriangleup C'$ is eulerian. Conversely, if $z+z'\in\mathcal{C}_2$ and $x+z\in\mathcal{C}_2$ then $x+z+(z+z')=x+z'\in\mathcal{C}_2$.
Letting $C\subseteq E$ range over a collection of subsets no
two of which have eulerian symmetric difference, the union of events $\{A\bigtriangleup C\,\mbox{\rm
eulerian}\cap B\bigtriangleup C\,\mbox{\rm eulerian}\}$ is thus a
disjoint union and equal
to the event $\{A\bigtriangleup B\,\mbox{\rm
eulerian}\}=\Delta$. Hence, letting $z$ range over a transversal of cosets of $\mathcal{C}_2$,
$$2^{-2|E|}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}P_\ell(\mathcal{C}_2+z)=\mathbb{P}(|E\setminus
A|-|E\setminus B|\equiv \ell(\bmod\, q)\cap\Delta).$$
From equation~\eqref{|hwe|^2 to Pl} it follows that
$$2^{-2|E|}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}|{\rm hwe}(\mathcal{C}_2+z;e^{2\pi ik/q})|^2=\sum_{0\leq\ell\leq
q-1}e^{2\pi i k\ell/q}\,\mathbb{P}(|B|-|A|\equiv\ell\,(\bmod\,q)\,\cap\,\Delta).$$
Dividing through by $\mathbb{P}(\Delta)=2^{-r(G)}$ gives the result.
\end{pf
\begin{lem}\label{interpret h2} Suppose that $A,B\subseteq E$ are chosen uniformly at
random and $\Delta$ is the event that $A\bigtriangleup B$ is eulerian. Then, for any $k\in\{0,1,\ldots, q-1\}$,
$$2^{-|E|}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}{\rm
hwe}(\mathcal{C}_2+z;e^{2\pi ik/q})^2=2^{n(G)}\sum_{0\leq\ell\leq
q-1}e^{2\pi i k\ell/q}\,\mathbb{P}(|A|+|B|\equiv \ell\,(\mbox{\rm
mod}\, q)\,\mid\,\Delta).$$
\end{lem}
\begin{pf}
The same mutatis mutandis as the proof of Lemma~\ref{interpret
h2|} with
$$P_\ell=P_\ell(\mathcal{C}_2+z)=\sum_{j+k\equiv\ell(\bmod q)}p_jp_k$$
replacing the definition of $P_\ell$ given in the proof of that lemma.
Note that since $A,B\subseteq E$ are chosen uniformly at
random and $(E\setminus A)\bigtriangleup(E\setminus B)=A\bigtriangleup
B$, by symmetry we have $\mathbb{P}(|E\setminus
A|+|E\setminus B|\equiv\ell\,(\mbox{\rm mod}\,
q)\,\cap\,\Delta)=\mathbb{P}(|A|+|B|\equiv\ell\,(\mbox{\rm mod}\,
q)\,\cap\,\Delta)$.
\end{pf
\begin{lem}\label{interpret h3} Suppose that $A,B,C\subseteq E$ are chosen uniformly at
random and $\Delta$ is the event that $A\bigtriangleup B,
B\bigtriangleup C$ are both eulerian. Then, for any $k\in\{0,1,\ldots, q-1\}$,
$$2^{-|E|}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}{\rm
hwe}(\mathcal{C}_2+z;e^{2\pi ik/q})^3=4^{n(G)}\sum_{0\leq\ell\leq
q-1}e^{2\pi i k\ell/q}\,\mathbb{P}(|A|+|B|+|C|\equiv \ell\,(\mbox{\rm
mod}\, q)\,\mid\,\Delta).$$
\end{lem}
\begin{pf} The same as the proof of Lemma~\ref{interpret h2|} with
the following being the main alterations.
Set $$P_\ell=P_\ell(\mathcal{C}_2+z)=\sum_{i+j+k\equiv \ell\,(\mbox{\tiny mod}\,
q)}p_ip_jp_k.$$
Then
$${\rm hwe}(\mathcal{C}_2+z;e^{2\pi i k/q})^3=\sum_{0\leq\ell\leq
q-1} e^{2\pi ik\ell/q}P_\ell$$
and
$$2^{-3|E|}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}P_{\ell}(\mathcal{C}_2+z)=\mathbb{P}(|E\setminus
A|+|E\setminus B|+|E\setminus C|\equiv\ell\,(\bmod\,q)\,\cap\,\Delta).$$
As in the proof of Lemma~\ref{interpret h2}, we can by symmetry replace $E\setminus A, E\setminus B, E\setminus C$
by $A,B,C$. Hence
$$2^{-3|E|}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}{\rm
hwe}(\mathcal{C}_2+z;e^{2\pi ik/q})^3=\sum_{0\leq\ell\leq
q-1}e^{2\pi i k\ell/q}\,\mathbb{P}(|A|+|B|+|C|\equiv\ell\,(\bmod\,q)\,\cap\,\Delta).$$
Dividing through by $\mathbb{P}(\Delta)=2^{-2r(G)}$ gives the result.
\end{pf
\begin{lem}\label{Bias Sigma Delta} Suppose that $A,B\,(,C)\subseteq E$ are chosen uniformly at
random and $\Delta$ is the event that $A\bigtriangleup B$ (and
$B\bigtriangleup C$) is eulerian. Then
$${\rm Bias}(\Sigma\,\mid\,\Delta)=2q^{-1}\langle\widehat{f},\widehat{1_S}\rangle-1,$$
where $1_S$ is the indicator function of $S$ and, for
$k\in\{0,1,\ldots, q-1\}$,
$$\widehat{f}(k)=\begin{cases} 2^{-n(G)}\left(\cos\frac{2\pi
k}{q}\right)^{r(G)}\left(1\!-\!\cos\frac{2\pi
k}{q}\right)^{n(G)}T\left(\!G;\frac{1}{\cos\frac{2\pi
k}{q}},\frac{1+\cos\frac{2\pi k}{q}}{1-\cos\frac{2\pi k}{q}}\right)\\
2^{-n(G)}e^{-2\pi i k|E|/q}\left(\cos\frac{2\pi
k}{q}\!-\!1\right)^{n(G)}T\left(\!G;\cos\frac{2\pi
k}{q},\frac{\cos\frac{2\pi k}{q}+1}{\cos\frac{2\pi
k}{q}-1}\right)\\
2^{-n(G)-\frac{1}{2}|E|}e^{-2\pi i k\frac{3}{2}|E|/q}\left(1\!+\!\cos\frac{2\pi
k}{q}\right)^{\frac{1}{2}|E|}\left(\cos\frac{2\pi
k}{q}\!-\!1\right)^{n(G)}T\left(\!G;2\cos\frac{2\pi
k}{q}\!-\!1,\frac{\cos\frac{2\pi k}{q}+1}{\cos\frac{2\pi
k}{q}-1}\right)\end{cases}$$
according as
$$\Sigma=\begin{cases}\{A,B\subseteq E:|A|-|B|\in S\,(\bmod\,q)\}\\
\{A,B\subseteq E:|A|+|B|\in S\,(\bmod\,q)\}\\
\{A,B,C\subseteq E: |A|\!+\!|B|\!+\!|C|\in S\,(\bmod\,q)\}\end{cases}.$$
\end{lem}
\begin{pf} Again we prove the result for $\Sigma=\{A,B\subseteq
E:|A|-|B|\in S\,(\bmod\,q)\}$ and $\Delta$ the event that
$A\bigtriangleup B$ is eulerian, the other cases being entirely
similar.
Define $f(\ell)=\mathbb{P}(|A|-|B|\equiv \ell\,\mid\,\Delta)$. Then
$\mathbb{P}(\Sigma)=\langle
f,1_S\rangle=q^{-1}\langle\widehat{f},\widehat{1_S}\rangle$, by Parseval's
formula. By Lemma~\ref{interpret h2|},
$$\widehat{f}(k)=2^{-|E|-n(G)}\sum_{\mathcal{C}_2+z\in\mathbb{F}_2^E/\mathcal{C}_2}|{\rm
hwe}(\mathcal{C}_2+z;e^{-2\pi i k/q})|^2$$
and identity~\eqref{abs sum squares theta} gives the result.
\end{pf}
Lemma~\ref{Bias Sigma Delta} shows that ${\rm
Bias}(\Sigma\,\mid\,\Delta)$ is given in terms of evaluations of the Tutte
polynomial at one or more points. The remainder of this section is
spent establishing when an evaluation at just one point is involved
and a Tutte-Grothendieck invariant results.
Note that $\widehat{f}(0)=1$ for each $\widehat{f}$ defined in Lemma~\ref{Bias
Sigma Delta} ($f$ defines a probability distribution on
$\mathbb{Z}_q$ and $\widehat{f}(0)=\sum f(k)=1$). Recall the definition
of $g$ from
Lemma~\ref{bias S}, where Bias$(\Sigma)$ is expressed in terms of the
inner product $\langle\widehat{g},\widehat{1_S}\rangle$.
Since $f$ and $g$ are
real-valued, $\widehat{g}(-k)=\overline{\widehat{g}(k)}$ and $\widehat{f}(-k)=\overline{\widehat{f}(k)}$. Remark
also that if $q$ is even and $\Sigma=\{A,B,C\subseteq
E:|A|\!+\!|B|\!+\!|C|\in S\,(\bmod\,q)\}$ then
$\widehat{f}(q/2)=0$ (for all graphs $G$), but that $\widehat{f}(k)\neq 0$ for
some graph $G$ when $k\neq q/2$.
The {\em support} of a function $h:Q\rightarrow\mathbb{C}$ is defined
by ${\rm supp}(h)=\#\{k\in Q:h(k)\neq 0\}$. Thus ${\rm
supp}(\widehat{g})=\mathbb{Z}_q\setminus\{q/2\}$, and ${\rm
supp}(\widehat{f})=\mathbb{Z}_q$ (or possibly
$\mathbb{Z}_q\setminus\{q/2\}$ as we have just seen).
From Lemma~\ref{Bias Sigma Delta}, ${\rm
Bias}(\Sigma\,|\,\Delta)=2q^{-1}\langle\widehat{f},\widehat{1_S}\rangle-1$ will involve an evaluation of the Tutte
polynomial at a single point (valid for all graphs $G$) only if
${\rm supp}(\widehat{f}\cdot \widehat{1_S})\subseteq\{0,\ell,-\ell\}$ for some
$\ell$, or ${\rm supp}(\widehat{f}\cdot \widehat{1_S})\subseteq\{0,\ell,-\ell,q/2\}$ if $q$ is even and $\Sigma=\{A,B,C\subseteq
E:|A|\!+\!|B|\!+\!|C|\in S\,(\bmod\,q)\}$. This is so that the
only non-zero terms contributing to the expression
$2q^{-1}\langle\widehat{f},\widehat{1_S}\rangle-1$ are $\widehat{f}(\ell)\overline{\widehat{1_S}(\ell)}$ and
its complex conjugate $\widehat{f}(-\ell)\overline{\widehat{1_S}(-\ell)}$.
Suppose $\Sigma'$ is an event defined just as $\Sigma$ except with
$S'\subseteq\mathbb{Z}_q\setminus S$ replacing $S$. Then the bias towards $S$ can be compared to
the bias toward $S'$ by considering the difference ${\rm
Bias}(\Sigma\,|\,\Delta)-{\rm
Bias}(\Sigma'\,|\,\Delta)$. When $S'=\mathbb{Z}_q\setminus S$
this difference is simply $2{\rm Bias}(\Sigma\,|\,\Delta)$. The
criterion for ${\rm Bias}(\Sigma\,|\,\Delta)-{\rm Bias}(\Sigma'\,|\,\Delta)$ to be an evaluation of the Tutte polynomial at a
single point valid for all graphs is that ${\rm supp}(\widehat{f}\cdot
\widehat{1_S-1_{S'}})\subseteq\{0,\ell,-\ell\, (,q/2)\}$ (with $q/2$
included under the same conditions as before).
The multiplicative group
of units of $\mathbb{Z}_q$ is denoted by $\mathbb{Z}_q^\times$ and has
order $\phi(q)$, where $\phi(q)=\#\{1\leq k\leq q:(k,q)=1\}$ is
Euler's totient function. For $S\subseteq\mathbb{Z}_q$ and
$\ell\in\mathbb{Z}_q$ we write $\ell S=\{\ell s:s\in S\}$
\begin{lem}\label{support} If $h:\mathbb{Z}_q\rightarrow\mathbb{Q}$ and $\widehat{h}(k)\neq
0$ then ${\rm supp}(\widehat{h})\supseteq k\mathbb{Z}_q^\times$.
If ${\rm supp}(\widehat{h})\subseteq d\mathbb{Z}_q$ for some divisor $d$ of $q$
then $h$ is constant on cosets of $(q/d)\mathbb{Z}_q$.
Thus if $\widehat{h}(k)\neq 0$ for a unit $k$ of $\mathbb{Z}_q$ then
$\widehat{h}(\ell)\neq 0$ for all $\ell\in\mathbb{Z}_q^\times$, while if
there is no unit $k$ in the support of $\widehat{h}$ then $h$ is constant on cosets of a proper subgroup of $\mathbb{Z}_q$.
\end{lem}
\begin{pf}
The first statement depends on the fact that $h$ takes rational
values. Suppose
$j\mapsto \sigma_j$ is the natural isomorphism of the group of Galois automorphisms of $\mathbb{Q}(e^{2\pi
i/q})$ with $\mathbb{Z}_q^\times$, i.e.,\ $\sigma_j:e^{2\pi
i/q}\mapsto e^{2\pi
i j/q}$. Then
\begin{align*}\sigma_j(\widehat{h}(k)) &
=\sigma_j\big(\sum_{0\leq\ell\leq q-1}h(\ell)e^{2\pi i k\ell/q}\big)\\
& =\sum_{0\leq \ell\leq q-1}h(\ell)\sigma_j(e^{2\pi i k\ell/q})\\
& =\widehat{h}(jk).\end{align*}
Hence $\widehat{h}(k)\neq 0$ implies $\widehat{h}(jk)\neq 0$ for all $j\in\mathbb{Z}_q^\times$.
For the second statement, the assumption is that
$\widehat{h}=\sum_{0\leq \ell\leq q/d-1}\widehat{h}(d\ell)1_{d\ell}$. Taking Fourier transforms, for each
$k\in\mathbb{Z}_q$ we have
$$qh(k)=\sum_{0\leq \ell\leq q/d-1}\widehat{h}(d\ell)e^{2\pi i k\ell/(q/d)}.$$
Then
$$h(k+q/d)=q^{-1}\sum_{0\leq\ell\leq q/d-1}\widehat{h}(d\ell)e^{2\pi i
k\ell/(q/d)+2\pi i\ell}=h(k),$$
so that $h$ has period $q/d$ and is constant on additive cosets of $(q/d)\mathbb{Z}_q$.
\end{pf}
Whether ${\rm Bias}(\Sigma\,|\, \Delta)-{\rm
Bias}(\Sigma'\,|\, \Delta)$ involves an evaluation of the Tutte
polynomial at a single point depends on how many how zero terms
there are in its expression as $2q^{-1}\langle\widehat{f},\widehat{1_S}-\widehat{1_{S'}}\rangle$ obtained from
Lemma~\ref{Bias Sigma Delta} applied to $\Sigma$ and $\Sigma'$. We require $|S|=|S'|$ for
the sets $S, S'$ defining the events $\Sigma, \Sigma'$ since
$\widehat{1_S}(0)-\widehat{1_{S'}}(0)=|S|-|S'|$ and $\widehat{f}(0)=1$.
\begin{cor}\label{when} Suppose that $\Sigma$ is one of the events $\{A,B\subseteq
E:|A|\pm|B|\in S(\bmod q)\}$, $\Sigma'$ is similarly
defined with $S'\subseteq\mathbb{Z}_q\setminus S$ in place of
$S$, and $\Delta$ is the event
that $A\bigtriangleup B$ is eulerian. Then ${\rm
Bias}(\Sigma\,|\, \Delta)-{\rm
Bias}(\Sigma'\,|\, \Delta)$ is up to a factor depending only on
$|E|$ and $r(G)$ an evaluation of the Tutte
polynomial of $G$ at a single point only if $|S|=|S'|$ and
$q\in\{2,3,4\}$ or $S, S'$ are each unions of additive cosets of $d\mathbb{Z}_q$ for $d\in\{2,3,4\}$ a divisor
of $q$.
If $\Sigma$ is the event $\{A,B,C\subseteq E:|A|+|B|+|C|\in S\,(\mbox{\rm mod}\,
q)\}$, $\Sigma'$ the same event with
$S'\subseteq\mathbb{Z}_q\setminus S$ in place of $S$, and $\Delta$ the event
that $A\bigtriangleup B, B\bigtriangleup C$ are both eulerian, then ${\rm
Bias}(\Sigma\,|\, \Delta)-{\rm
Bias}(\Sigma'\,|\, \Delta)$ involves an evaluation of the Tutte
polynomial at a single point only if $|S|=|S'|$ and $q\in\{2,3,4,6\}$ or $S$
and $S'$ are each unions of additive cosets of $d\mathbb{Z}_q$ for $d\in\{2,3,4,6\}$ a divisor
of $q$.
\end{cor}
Note that if $S$ is a union of additive cosets of $d\mathbb{Z}_q$ then the
event $\Sigma$ is a congruence condition modulo $d$ so these choices for
$S$ are herewith ignored.
\begin{pf}
Let $h=1_S-1_{S'}$. The only integers $q\geq 2$ for which $\phi(q)\leq 2$ are
$2,3,4,6$. By Lemma~\ref{support} either we are in the case where
$\widehat{h}$ is supported on an additive subgroup $d\mathbb{Z}_q$ or
$\widehat{h}(k)\neq 0$ for all units of $\mathbb{Z}_q$, of which there
are $\phi(q)$. In the latter case only if $\phi(q)\leq 2$ is it the
case that $\widehat{f}(k)\widehat{h}(k)=0$ for $k\not\in\{1,-1\}$. The
former case by Lemma~\ref{support} reduces to the latter with $q$ replaced by $d$.
\end{pf}
The only choices for $q\ge 3$ and $S, S'\subseteq\mathbb{Z}_q$ are up to
exceptions trivial by Corollary~\ref{when}
given by the following
theorem, whose proof is a simple matter of substituting in the
expressions provided by Lemma~\ref{bias S} and Lemma~\ref{Bias Sigma
Delta}
\begin{thm}\label{general} Let $q\in\{3,4,6\}$. Suppose $A,B\,(,C)\subseteq E$ are chosen uniformly at
random and $\Delta$ is the event that $A\bigtriangleup B$ (and
$B\bigtriangleup C$) is eulerian. Suppose further that $S, S'\subseteq
\mathbb{Z}_q$ and ${\rm supp}(\widehat{1_S}-\widehat{1_{S'}})=\{1,-1\}$
(or possibly $\{1,-1, q/2\}$ for the third case of the following statement).
If $\Sigma$ is the event
$|A|\pm|B|\,(+|C|)\in S\,(\bmod\,q)$, $\Sigma'$ is the event
$|A|\pm|B|\,(+|C|)\in S'\,(\bmod\,q)$ and ${\rm
Bias}(\Sigma)\neq{\rm Bias}(\Sigma')$, then
$$\frac{{\rm Bias}(\Sigma\,|\,\Delta)-{\rm Bias}(\Sigma'\,|\,\Delta)}{{\rm Bias}(\Sigma)-{\rm Bias}(\Sigma')}$$
$$=\begin{cases} 2^{r(G)}\left(1+\cos\frac{2\pi
}{q}\right)^{-|E|}\left(\cos\frac{2\pi}{q}\right)^{r(G)}\left(\cos\frac{2\pi
}{q}-1\right)^{n(G)}T\left(\!G;\frac{1}{\cos\frac{2\pi
}{q}},\frac{1+\cos\frac{2\pi }{q}}{1-\cos\frac{2\pi}{q}}\right)\\
2^{r(G)}\left(1+\cos\frac{2\pi
}{q}\right)^{-|E|}\left(\cos\frac{2\pi}{q}-1\right)^{n(G)}T\left(\!G;\cos\frac{2\pi}{q},\frac{\cos\frac{2\pi}{q}+1}{\cos\frac{2\pi}{q}-1}\right) \\
2^{r(G)}\left(1+\cos\frac{2\pi}{q}\right)^{-|E|}\left(\cos\frac{2\pi}{q}-1\right)^{n(G)}T\left(\!G;2\cos\frac{2\pi}{q}-1,\frac{\cos\frac{2\pi}{q}+1}{\cos\frac{2\pi}{q}-1}\right)\end{cases}$$
according as
$$\Sigma=\begin{cases}\{A,B\subseteq E:|A|\!-\!|B|\in S\,(\bmod\,q)\}\\
\{A,B\subseteq E:|A|\!+\!|B|\in S\,(\bmod\,q)\}\\
\{A,B,C\subseteq E:|A|\!+\!|B|\!+\!|C|\in S\,(\bmod\,q)\}\end{cases},$$
$$\Sigma'=\begin{cases}\{A,B\subseteq E:|A|\!-\!|B|\in S'\,(\bmod\,q)\}\\
\{A,B\subseteq E: |A|\!+\!|B|\in S'\,(\bmod\,q)\}\\
\{A,B,C\subseteq E: |A|\!+\!|B|\!+\!|C|\in S'\,(\bmod\,q)\}\end{cases}.$$
\end{thm}
Taking $q$ even, $|S|=q/2$ and $S'=\mathbb{Z}_q\setminus S$, Theorem~\ref{general} gives ${\rm Bias}(\Sigma\,|\,\Delta)/{\rm Bias}(\Sigma)$ as a Tutte
polynomial evaluation.
So what choices of $S$ and $S'$ fulfil the conditions of Theorem~\ref{general}?
The answer is to be found in the theorems of sections~\ref{square root of unity} to \ref{sixth root of unity}, which are immediate
corollaries of Lemmas~\ref{bias S} and \ref{Bias Sigma Delta} and Theorem~\ref{general}.
\subsubsection{Evaluations for $q=2$}\label{square root of unity}
\begin{prop} \label{squares with q=2}
Suppose that $A,B\subseteq E$ are subgraphs of $G$ chosen uniformly at
random. Then the event that $|A|+|B|$ is even is correlated with the
event $\Delta$ that $A\bigtriangleup B$ is eulerian as follows:
$${\rm Bias}(\,|A|\!+\!|B|\equiv 0\,(\mbox{\rm mod}\, 2)\,\mid\,\Delta)=(-1)^{r(G)}T(G;-1,0)=2^{-k(G)}P(G;2).$$
\end{prop}
\begin{pf} In Lemma~\ref{Bias Sigma Delta} take $S=\{0\}$, $\Sigma=\{A,B\subseteq E:|A|+|B|\equiv
0(\bmod 2)\}$ and $\Delta$ the event that $A\bigtriangleup B$ is
eulerian. Then, by the result of that lemma, ${\rm
Bias}(\Sigma\,|\,\Delta)=\widehat{f}(0)+\widehat{f}(1)-1$, where
$\widehat{f}(0)=1$ and $\widehat{f}(1)=2^{-n(G)}(-1)^{r(G)}2^{r(G)}T(G;-1,0)=2^{-k(G)}P(G;2)$.
\end{pf}
Of course the correlation between parity and eulerian symmetric difference in
Proposition~\ref{squares with q=2} can be seen immediately by considering the identity
$$|A\bigtriangleup B|+2|A\cap B|=|A|+|B|.$$
Eulerian subgraphs are all of even size if and
only if $G$ is bipartite (no odd cycles).
Given the event $\Delta$ that $A\bigtriangleup B$
is eulerian the parity of $|A|+|B|$ must be even when $G$ is
bipartite.
Otherwise, if $G$ is not bipartite half the eulerian
subgraphs are even, half odd, and so the parity of $|A|+|B|$ is
equally likely to be even or odd given $\Delta$.
For three subgraphs $A,B,C\subseteq
E$ of $G$, if $A\bigtriangleup B, B\bigtriangleup C$ are eulerian then so is $C\bigtriangleup A$.
From the identity
$$|A\bigtriangleup B|+|B\bigtriangleup C|+|C\bigtriangleup A|=2(|A|+|B|+|C|)-2(|A\cap
B|+|B\cap C|+|C\cap A|),$$
it seems difficult to tell whether there might be any correlation between the
event that $A\bigtriangleup B, B\bigtriangleup C$ are eulerian and
some condition on $|A|+|B|+|C|$
\begin{thm}\label{three uncorr even}
Suppose $A,B,C\subseteq E$ are subgraphs of $G$ chosen uniformly at
random. Then the event
that $|A|+|B|+|C|$ is even is uncorrelated with the event $\Delta$ that
$A\bigtriangleup B$, $B\bigtriangleup C$ are eulerian, i.e., $${\rm
Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0\!\!\pmod 2\,|\,\Delta)=0.$$
\end{thm}
\begin{pf} Take $\Sigma=\{A,B,C\subseteq E:|A|+|B|+|C|\equiv 0(\bmod\, 2)\}$ and $\Delta$ the event that $A\bigtriangleup B$ and
$B\bigtriangleup C$ are both
eulerian. By Lemma~\ref{Bias Sigma Delta}, ${\rm
Bias}(\Sigma\,|\,\Delta)=\widehat{f}(0)+\widehat{f}(1)-1$, where
$\widehat{f}(0)=1$ and $\widehat{f}(1)=0$.
\end{pf}
However, we shall see that the residue of $|A|+|B|+|C|$ modulo $3, 4$ and
$6$ does have a
bearing on the event that $A\bigtriangleup B, B\bigtriangleup C$ are
eulerian.
\subsubsection{Evaluations for $q=3$}\label{cube root of unity}
The evaluations of the Tutte polynomial obtained for $q=3$ are, unlike
the cases $q=2, 4$ and $6$, at
points without other more familiar combinatorial interpretations
\begin{thm}\label{A - B q=3}
Let $A,B\subseteq E$ be chosen uniformly at random and let $\Delta$
be the event that $A\bigtriangleup B$ is eulerian. Then
$${\rm Bias}(\,|A|\equiv |B|\!+\!1\,(\bmod\,
3)\,\mid\,\Delta)={\rm Bias}(\,|A|\equiv|B|\!+\!2\,(\bmod\, 3)\,\mid\,\Delta)$$
and
$$\frac{{\rm Bias}(\,|A|\equiv |B|\,(\bmod\,
3)\,\mid\,\Delta)-{\rm Bias}(\,|A|\equiv |B|\!+\!1\,(\bmod\, 3)\,\mid\,\Delta)}{{\rm Bias}(\,|A|\equiv
|B|\,(\bmod\, 3)\,)-{\rm Bias}(\,|A|\equiv
|B|\!+\!1\,(\bmod\, 3)\,)}$$
$$=(-2)^{r(G)}3^{n(G)}T(G;-2,\frac{1}{3}).$$
\end{thm}
\begin{pf}
Lemma~\ref{Bias Sigma Delta} with $S=\{1\}$ yields
$${\rm Bias}(|A|-|B|\equiv 1(\bmod\, 3)\,|\,\Delta)=\frac23(1+\widehat{f}(1)e^{2\pi
i/3}+\widehat{f}(2)e^{4\pi i/3})-1,$$
where $f(\ell)=\mathbb{P}(|A|-|B|\equiv\ell (\bmod\,3)\,|\,\Delta)$. (We define a
character $\chi$ on $\mathbb{Z}_3$ by setting $\chi(1)=e^{2\pi i/3}$, and the Fourier transform is
defined by $\widehat{f}(k)=f(0)+e^{4k\pi i/3}f(1)+e^{2k\pi i/3}f(2)$. Thus
$\widehat{1_1}=1_0+e^{4\pi i/3}1_1+e^{2\pi i/3}1_2$ and
$\widehat{1}_2=\overline{\widehat{1}_1}$.)
With $S=\{2\}$ the same lemma yields
$${\rm Bias}(|A|-|B|\equiv 2(\bmod 3)\,|\,\Delta) = \frac23(1+\widehat{f}(1)e^{4\pi
i/3}+\widehat{f}(2)e^{2\pi i/3})-1.$$
Since Lemma \ref{Bias Sigma Delta} also tells us that $\widehat{f}(1)=\widehat{f}(2)$, the first statement of the theorem
is established.
By Theorem~\ref{general} with $S=\{0\}, S'=\{1\}$ (for which
$\widehat{1_S}-\widehat{1_{S'}}=(1-e^{4\pi i/3})1_1+(1-e^{2\pi i/3})1_2$),
$\Sigma=\{A,B\subseteq E:|A|-|B|\equiv 0(\bmod 3)\}$ and
$\Sigma'=\{A,B\subseteq E: |A|-|B|\equiv 1 (\bmod 3)\}$,
$$\frac{{\rm Bias}(\,|A|-|B|\equiv 0\,(\mbox{\rm mod}\,
3)\,\mid\,\Delta)-{\rm Bias}(\,|A|-|B|\equiv 1\,(\mbox{\rm
mod}\, 3)\,\mid\,\Delta)}{{\rm Bias}(\,|A|-|B|\equiv 0\,(\mbox{\rm mod}\, 3)\,)-{\rm Bias}(\,|A|-|B|\equiv
1\,(\mbox{\rm mod}\, 3)\,)}$$
$$=2^{r(G)}(\frac12)^{-|E|}(-\frac12)^{r(G)}(-\frac32)^{n(G)}T(G;-2,\frac13).$$
\end{pf}
\begin{thm}\label{cube root A B}
Let $A,B\subseteq E$ be chosen uniformly at random and let $\Delta$
be the event that $A\bigtriangleup B$ is eulerian. Then
$${\rm Bias}(\,|A|\!+\!|B|\equiv |E|\!+\!1\,(\mbox{\rm mod}\,
3)\,\mid\,\Delta)={\rm Bias}(\,|A|\!+\!|B|\equiv |E|\!+\!2\,(\mbox{\rm mod}\, 3)\,\mid\,\Delta)$$
and
$$\frac{{\rm Bias}(\,|A|\!+\!|B|\equiv |E|\,(\mbox{\rm mod}\,
3)\,\mid\,\Delta)-{\rm Bias}(\,|A|\!+\!|B|\equiv |E|\!+\!1\,(\mbox{\rm
mod}\, 3)\,\mid\,\Delta)}{{\rm Bias}(\,|A|\!+\!|B|\equiv
|E|\,(\mbox{\rm mod}\, 3)\,)-{\rm Bias}(\,|A|\!+\!|B|\equiv
|E|\!+\!1\,(\mbox{\rm mod}\, 3)\,)}$$
$$=4^{r(G)}(-3)^{n(G)}T(G;-\frac{1}{2},-\frac{1}{3}).$$\end{thm}
\begin{pf} Lemma~\ref{Bias Sigma Delta} with $S=\{|E|\!+\!1\}$ yields
$${\rm Bias}(|A|+|B|\equiv |E|\!+\!1(\bmod 3)\,|\,\Delta)=\frac23(1+\widehat{f}(1)e^{2\pi
i(|E|+1)/3}+\widehat{f}(2)e^{4\pi i(|E|+1)/3})-1,$$
where $f(\ell)=\mathbb{P}(|A|+|B|\equiv\ell (\bmod
3)\,|\,\Delta)$. With $S=\{|E|\!+\!2\}$ the same lemma yields
$${\rm Bias}(|A|+|B|\equiv |E|\!+\!2(\bmod 3)\,|\,\Delta) = \frac23(1+\widehat{f}(1)e^{2\pi i(|E|+2)/3}+\widehat{f}(2)e^{4\pi i(|E|+2)/3})-1.$$
Lemma~\ref{Bias Sigma Delta} tells us that $\widehat{f}(1)=e^{2\pi
i|E|/3}\widehat{f}(2)$, and the first statement of the theorem follows
with both biases equal to $\widehat{f}(1)(e^{2\pi i(|E|+1)/3}+e^{2\pi i(|E|+2)/3})-\frac13$.
The second statement of the theorem follows from Theorem~\ref{general}
upon taking $S=\{|E|\}, S'=\{|E|\!+\!1\}$,
$\Sigma=\{A,B\subseteq E:|A|\!+\!|B|\equiv |E|(\bmod{3})\,\}$ and
$\Sigma'=\{A,B\subseteq E: |A|\!+\!|B|\equiv |E|\!+\!1(\bmod{3})\,\}$.
\end{pf}
Note that by equation~(\ref{eul reciprocal}) at the beginning of this
section, if $G$ is eulerian then the evaluations of the Tutte polynomial in Theorem~\ref{A -
B q=3} and Theorem~\ref{cube
root A B} are equal. Indeed, $|E\setminus A|+|B|\equiv\pm
1\,(\mbox{\rm mod}\, 3)$ if and only if $|A|-|B|\equiv|E|\mp
1\,(\mbox{\rm mod}\, 3)$, and if $G$ is eulerian
then a subgraph $A$ is eulerian if and only if its complement
$E\setminus A$ is eulerian.
\begin{thm}\label{cube root A B C} Let $A,B,C\subseteq E$ be chosen uniformly at random and let $\Delta$ be the event that $A\bigtriangleup B, B\bigtriangleup
C$ are both eulerian. Then
$${\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 1\,(\bmod 3)\,\mid\,\Delta)={\rm
Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 2\,(\bmod 3)\,\mid\,\Delta),$$ and
$$\frac{{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0\,(\bmod 3)\,\mid\,\Delta)-{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 1\,(\bmod 3)\,\mid\,\Delta)}{{\rm
Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0\,(\bmod 3)\,)-{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 1\,(\bmod 3)\,)}$$
$$=4^{r(G)}(-3)^{n(G)}T(G;-2,-\frac{1}{3}).$$
\end{thm}
\begin{pf}
Lemma~\ref{Bias Sigma Delta} with $S=\{1\}$ yields
$${\rm Bias}(|A|\!+\!|B|\!+\!|C|\equiv
1(\bmod{3})\,|\,\Delta)=\frac23(\widehat{f}(1)e^{2\pi
i/3}+\widehat{f}(2)e^{4\pi i/3})-\frac13$$
where $f(\ell)=\mathbb{P}(|A|\!+\!|B|\!+\!|C|\equiv\ell (\bmod
3)\,|\,\Delta)$. Lemma~\ref{Bias Sigma Delta} with $S=\{2\}$ yields
$${\rm Bias}(|A|\!+\!|B|\!+\!|C|\equiv
2(\bmod{3})\,|\,\Delta)=\frac23(\widehat{f}(1)e^{4\pi
i/3}+\widehat{f}(2)e^{2\pi i/3})-\frac13.$$
From Lemma~\ref{Bias Sigma Delta} it is also found that
$\widehat{f}(1)=\widehat{f}(2)$ and the first statement of the theorem follows.
Clearly then ${\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0(\bmod\,3)\,)\neq
{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 1(\bmod\,3)\,)$ and the second
statement of the theorem results from Theorem~\ref{general} upon taking $S=\{0\},$ $S'=\{1\}$,
$\Sigma=\{A,B,C\subseteq E:|A|\!+\!|B|\!+\!|C|\equiv
0(\bmod\,3)\}$ and $\Sigma'=\{A,B,C\subseteq E:|A|\!+\!|B|\!+\!|C|\equiv
1(\bmod\,3)\}$.
\end{pf}
\subsubsection{Evaluations for $q=4$}\label{fourth root of unity}
\begin{thm}\label{A - B with q=4}
Choosing $A,B\subseteq E$ uniformly at random, the event that $|A|-|B|\equiv
0\,\mbox{\rm or}\, 1\,(\bmod\, 4)$ (i.e.,
$\left\lfloor\frac{|A|-|B|}{2}\right\rfloor$ is even) is correlated with the event $\Delta$ that $A\bigtriangleup
B$ is eulerian:
$$\frac{{\rm Bias}(\,|A|\!-\!|B|\equiv
0,1\,(\bmod 4)\,\mid\,\Delta)}{{\rm
Bias}(|A|\!-\!|B|\equiv 0,1\,(\bmod\, 4)\,)}=2^{r(G)}.$$
\end{thm}
\begin{pf}
In Lemma~\ref{bias S} take $S=\{0,1\}$, for which
$\widehat{1_S}(k)=1+i^{-k}$, and calculate
$${\rm Bias}(|A|\!-\!|B|\equiv 0,1(\bmod{4}))=2^{-1-|E|}\big[2^{|E|}\cdot
2+1\cdot(1+i)+0\cdot 0+1\cdot (1-i)\big]-1=2^{-|E|}.$$
That this is non-zero allows us to apply Theorem~\ref{general}, in which we take $S=\{0,1\}$,
$S'=\{2,3\}=\mathbb{Z}_4\setminus S$, $\Sigma=\{A,B\subseteq
E:|A|\!-\!|B|\equiv 0,1(\bmod\,4)\}$ and $\Sigma'=\{A,B\subseteq
E:|A|\!-\!|B|\equiv 2,3(\bmod\,4)\}$. This yields (recalling from the footnote on page~\pageref{fn} how to
deal with division by zero in Tutte polynomial evaluations)
$$\frac{{\rm Bias}(\,|A|\!-\!|B|\equiv
0,1(\bmod\,4)\,\mid\,\Delta)-{\rm Bias}(|A|\!-\!|B|\equiv 2,3(\bmod{4})\,|\,\Delta)}{{\rm
Bias}(|A|\!-\!|B|\equiv 0,1(\bmod\,4)\,)-{\rm
Bias}(|A|\!-\!|B|\equiv 2,3(\bmod\,4)\,)}=2^{r(G)}.$$
\end{pf}
\begin{thm} \label{squares with q=4}
Choosing $A,B\subseteq E$ uniformly at random, the event that $|A|+|B|\equiv
0\,\mbox{\rm or}\, 1\,(\mbox{\rm mod}\, 4)$ (i.e.,
$\left\lfloor\frac{|A|+|B|}{2}\right\rfloor$ is even) is correlated with the event $\Delta$ that $A\bigtriangleup
B$ is eulerian in the following way:
$$\frac{{\rm Bias}(\,|A|\!+\!|B|\equiv
0,1(\bmod\,4)\,\mid\,\Delta)}{{\rm
Bias}(|A|\!+\!|B|\equiv 0,1(\bmod\, 4))}=2^{r(G)}F(G;2).$$
\end{thm}
\begin{pf}
In Lemma~\ref{bias S} take $S=\{0,1\}$, for which
$\widehat{1_S}(k)=1+i^{-k}$, and calculate
\begin{align*}
{\rm Bias}(|A|+\!|B|\equiv 0,1(\bmod\,4)) & =
2^{-1-|E|}\big[2^{|E|}\cdot 2+i^{-|E|}\cdot(1+i)+0\cdot 0+i^{|E|}\cdot (1-i)\big]-1\\
& =(-1)^{\lfloor|E|/2\rfloor}2^{-|E|}.
\end{align*}
We can now apply Theorem~\ref{general}, taking $S=\{0,1\}$,
$S'=\{2,3\}=\mathbb{Z}_4\setminus S$, $\Sigma=\{A,B\subseteq
E:|A|\!+\!|B|\equiv 0,1(\bmod\,4)\}$ and $\Sigma'=\{A,B\subseteq
E:|A|\!+\!|B|\equiv 2,3(\bmod\,4)\}$. The formula in Theorem~\ref{general} yields
$$\frac{{\rm Bias}(\,|A|\!+\!|B|\equiv
0,1(\bmod{4})\,\mid\,\Delta)-{\rm Bias}(|A|\!+\!|B|\equiv 2,3(\bmod\,4)\,|\,\Delta)}{{\rm
Bias}(|A|\!+\!|B|\equiv 0,1(\bmod\,4))-{\rm
Bias}(|A|\!+\!|B|\equiv
2,3(\bmod\,4))}$$
$$=2^{r(G)}(-1)^{n(G)}T(G;0,-1).$$
\end{pf}
Theorem~\ref{squares with q=4} says that if $G$ is not eulerian then the event that $A\bigtriangleup
B$ is eulerian removes from the
parity of $\lfloor\frac{|A|+|B|}{2}\rfloor$ its original bias of
$(-1)^{\lfloor|E|/2\rfloor}2^{-|E|}$ towards being even. Otherwise, when $G$ is eulerian, this bias is accentuated by a
factor of $2^{r(G)}$.
Theorem~\ref{squares with q=4} has a counterpart in Theorem~\ref{sum cubes with q=6} in section~\ref{sixth root of unity} below
\begin{thm} \label{A + B + C with q=4}
Suppose $|E|\not\equiv 1\pmod{4}$. Then, for $A,B,C\subseteq E$ chosen
uniformly at random, the event that $|A|\!+\!|B|\!+\!|C|\equiv
0\,\mbox{\rm or}\, 1\,\pmod 4$ (i.e.,
$\left\lfloor\frac{|A|+|B|+|C|}{2}\right\rfloor$ is even) is correlated with the event $\Delta$ that $A\bigtriangleup
B$ and $B\bigtriangleup C$ are eulerian as follows:
$$\frac{{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv
0,1\,(\bmod\, 4)\,\mid\,\Delta)}{{\rm
Bias}(|A|\!+\!|B|\!+\!|C|\equiv 0,1\,(\bmod\, 4))}=2^{r(G)}(-1)^{n(G)}T(G;-1,-1).$$
If $|E|\equiv 1\pmod 4$ then
$${\rm Bias}(|A|\!+\!|B|\!+\!|C|\equiv
0,1(\bmod\, 4)\,)=0={\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv
0,1\,(\bmod\, 4)\,\mid\,\Delta).$$
\end{thm}
\begin{pf}
In Lemma~\ref{bias S} take $q=4$ and $S=\{0,1\}$, for which
$\widehat{1_S}(k)=1+i^{-k}$, and calculate
\begin{align*}
{\rm Bias}(|A|\!+\!|B|\!+\!|C|\equiv 0,1(\bmod{4})) & = 2^{-1-\frac32|E|}\big[2^{\frac32|E|}\cdot
2+i^{-\frac32|E|}\cdot(1\!+\!i)+i^{\frac32|E|}\cdot
(1\!-\!i)\big]-1\\
& =2^{-\frac32|E|}{\rm Re}\big[i^{\frac32|E|}(1-i)\big]\\
& ={\begin{cases} 2^{-3|E|/2} & |E|\equiv 0,6\pmod 8\\
0 & |E|\equiv 1,5\pmod 8\\
-2^{-3|E|/2} & |E|\equiv 2,4\pmod 8\\
2^{(1-3|E|)/2} & |E|\equiv 3\pmod 8\\
-2^{(1-3|E|)/2} & |E|\equiv 7\pmod 8\end{cases}.}\end{align*}
When $|E|\not\equiv 1 (\bmod\, 4)$ we can apply Theorem~\ref{general} and
the result follows by a straightforward calculation.
When ${\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0,1 (\bmod\,4) )=0$ we use
Lemma~\ref{Bias Sigma Delta} with $q=4, S=\{0,1\}$, to calculate that
\begin{align*}
{\rm Bias}(|A|\!+\!|B|\!+\!|C|\equiv 0,1(\bmod\,4)\,|\,\Delta) & =
2{\rm Re}\big[2^{-n(G)-\frac12|E|}i^{\frac32|E|}(-1)^{n(G)}T(G;-1,-1)(1-i)\big]\\
& =0\hspace{1cm}\mbox{\rm when }|E|\equiv 1 (\bmod\, 4).\end{align*}
\end{pf}
From~\cite{RR78},
$2^{r(G)}(-1)^{n(G)}T(G;-1,-1)=(-2)^{r(G)+\dim(\mathcal{C}_2\cap\mathcal{C}_2^\perp)}$,
where $\mathcal{C}_2\cap\mathcal{C}_2^\perp$ is the bicycle space of
$G$, comprising subgraphs which are both eulerian and bipartite.
\subsubsection{Evaluations for $q=6$}\label{sixth root of unity}
When $q=6$ and $S=\{0,1,2\}$ the expression ${\rm
Bias}(\Sigma\,\mid\,\Delta)/{\rm Bias}(\Sigma)$ is only equal to an evaluation of the Tutte
polynomial at a single point when $\Sigma=\{A,B,C\subseteq E:|A|\!+\!|B|\!+\!|C|\equiv 0,1,2\,(\bmod\, q)\}$. This is due to the formula for ${\rm
Bias}(\Sigma\,\mid\,\Delta)$ given by Lemma~\ref{Bias Sigma Delta}
and the fact that ${\rm supp}(\widehat{1_S})=\{0,\pm 1, 3\}$.
For example, when $\Sigma$ is the event
$\{A,B\subseteq E:|A|\!+\!|B|\equiv 0,1,2\,(\bmod\, 6)\,\}$,
evaluations of the Tutte polynomial at the two points
$(\frac{1}{2},-3)$ and $(0,-1)$ would be involved
\begin{thm}\label{sum cubes with q=6} Suppose that $A,B,C\subseteq E$ are
chosen uniformly at random and $\Delta$ is the event that $A,B,C$ have
pairwise eulerian differences. Then the event that
$|A|+|B|+|C|\equiv 0,1,2\,(\bmod\, 6)$ (i.e., $\left\lfloor\frac{|A|+|B|+|C|}{3}\right\rfloor$ is
even) is correlated with $\Delta$ as follows:
$$\frac{{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0,1,2\,(\bmod\, 6)\,\mid\,\Delta)}{{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0,1,2\,(\bmod\, 6)\,)}=3^{-|E|}4^{r(G)}F(G;4).$$
\end{thm}
\begin{pf}
In Lemma~\ref{bias S} take $q=6$ and $S=\{0,1,2\}$, for which we calculate that
$\widehat{1_S}=31_0+1_3-2e^{2\pi i/3}1_1-2e^{-2\pi i/3}1_5$. By
Lemma~\ref{bias S},
\begin{align*}
{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0,1,2\,(\bmod\,6)\,) & =\frac{2}{6}\big[\widehat{g}(1)\overline{\widehat{1_S}(1)}+\widehat{g}(5)\overline{\widehat{1_S}(5)}\big]\\
& = \frac23{\rm Re}\big[\widehat{g}(1)\overline{\widehat{1_S}(1)}\big]
\end{align*}
where $\widehat{g}(1)=2^{-\frac32|E|}e^{-2\pi
i\frac32|E|/6}(1+\cos\frac{2\pi}{6})^{|E|}=2^{-\frac32|E|}e^{-|E|\pi i /2}(1+\frac12)^{\frac32|E|}$.
Hence
\begin{align*}
{\rm Bias}(\,|A|\!+\!|B|\!+\!|C|\equiv 0,1,2\,(\bmod\,6)\,) & =
\frac23{\rm Re}\big[e^{-\pi i|E|/2}\big(\frac34\big)^{\frac32|E|}(2e^{-2\pi i/3})\big]\\
& =\big(\frac34\big)^{\frac32|E|-1}{\rm Re}\big[i^{-|E|}e^{-2\pi
i/3}\big]\neq 0
\end{align*}
and we can apply Theorem~\ref{general} from which the result follows
by routine calculation.
\end{pf}
\section{A parity criterion for proper vertex colourings}\label{poset}
In the final three sections of this article we
need some further identities from finite Fourier analysis for complete weight enumerators
(which include Hamming weight enumerators as specialisations).
Let $Q$ be a commutative ring with a generating character such as $\mathbb{Z}_q$ or
$\mathbb{F}_q$, and let $f$ be a function
$f:Q\rightarrow\mathbb{C}$.
The {\em complete weight enumerator} of
a subset $\mathcal{S}$ of $Q^E$ is defined by
$${\rm cwe}(\mathcal{S};f)=\sum_{x\in\mathcal{S}}\prod_{e\in
E}f(x_e).$$
When $f=t1_0+1_{Q\setminus 0}$ the complete weight enumerator is
the Hamming weight enumerator ${\rm hwe}(\mathcal{S};t)$.
The MacWilliams duality theorem for complete weight enumerators is a
consequence of the Poisson summation formula and states
that when $\mathcal{S}$ is a $Q$-submodule of $Q^E$
\begin{equation}\label{McW cwe}{\rm cwe}(\mathcal{S};f)=\frac{1}{|\mathcal{S}^\perp|}{\rm
cwe}(\mathcal{S}^\perp;\widehat{f}).\end{equation}
The following generalises the first two identities of Lemma~\ref{sum cubes} and is proved for example in \cite{CCC07}
\begin{lem}\label{cwe sum}
Let $Q^E$ be a commutative ring with a generating
character. For $Q$-submodule $\mathcal{S}$ of $Q^E$ and functions
$f,g:Q\rightarrow\mathbb{C}$,
$$\sum_{\mathcal{S}+z\in Q^E/\mathcal{S}}{\rm
cwe}(\mathcal{S}+z;f)\overline{{\rm
cwe}(\mathcal{S}+z;g)}=\frac{1}{|\mathcal{S}^\perp|}{\rm
cwe}(\mathcal{S}^\perp;\widehat{f}\cdot\overline{\widehat{g}}).$$
\end{lem}
Let $\mathcal{C}$ be the set of
$Q$-flows of $G$ and its orthogonal $\mathcal{C}^\perp$ the set of
$Q$-tensions of $G$. A partial order $\leq$ on $Q^E$ is defined by
$x\leq y$ if and only if $x_e\in\{0,y_e\}$ for all $e\in E$. (For
$Q=\mathbb{F}_2$ the order $\leq$ is set inclusion.) This makes the
poset on $Q^E$ the direct product of the poset $P$ on $Q$ defined by setting
$0$ below all the non-zero elements of $Q$ and all pairs of non-zero
elements incomparable. Thus the M\"{o}bius function of the poset $P^E=(Q^E,\leq)$ is
defined by $\mu(x,y)=(-1)^{|y|-|x|}$. (See for example \cite{St97} for
background on posets.) For a function
$f:Q^E\rightarrow\mathbb{C}$, define $\mu f:Q^E\rightarrow\mathbb{C}$ by
$$\mu f(y)=\sum_{x\leq y}\mu(x,y)f(x)=\sum_{x\leq
y}(-1)^{|y|-|x|}f(x).$$
\begin{lem}\label{mu1C not zero}
Let $Q$ be a ring with a generating character $\chi$.
If $\mathcal{C}$ is a $Q$-submodule of $Q^E$ and
$\mathcal{C}^\perp$
its orthogonal space then
$$\mu 1_\mathcal{C}(y)=\frac{1}{|\mathcal{C}^\perp|}\sum_{x\in\mathcal{C}^\perp}\prod_{e\in E}(\chi(x_ey_e)-1).$$
\end{lem}
\begin{pf}
\begin{align*}\mu 1_{\mathcal{C}}(y) &
=\sum_{x\in\mathcal{C}}\prod_{e\in E}(1_{y_e}-1_0)(x_e)\\
& = \frac{1}{|\mathcal{C}^\perp|}\sum_{x\in\mathcal{C}^\perp}(\overline{\chi}_{y_e}-1)(x_e),\end{align*}
the latter equality by identity \eqref{McW cwe}, and since the
left-hand side is real $\overline{\chi}_{y_e}(x_e)$ can be replaced by its
conjugate $\chi_{y_e}(x_e)=\chi(x_ey_e)$.
\end{pf}
\begin{lem} \label{sum nz C perp}
$$\sum_{y: \forall_{e\in E}\, y_e\neq 0}\mu
1_\mathcal{C}(y)=(-1)^{|E|}|\mathcal{C}|{\rm
hwe}(\mathcal{C}^\perp;0).$$\end{lem}
\begin{pf}
\begin{align*}\sum_{y:\, \forall_{e\in E}\, y_e\neq 0}\mu
1_\mathcal{C}(y) & = \sum_{y:\, \forall_{e\in E}\, y_e\neq 0}\sum_{x\leq y,\, x\in\mathcal{C}}(-1)^{|y|-|x|}\\
&
=\sum_{x\in\mathcal{C}}(q-1)^{|E|-|x|}(-1)^{|E|-|x|},\end{align*}
reversing the order of summation and using $\#\{y\in(Q\setminus
0)^E:x\leq y\}=(q-1)^{|E|-|x|}$, whence
\begin{align*}\sum_{y:\, \forall_{e\in E}\, y_e\neq 0}\mu
1_\mathcal{C}(y) & =\sum_{x\in\mathcal{C}}(1-q)^{|E|-|x|}\\
& ={\rm hwe}(\mathcal{C};1-q)=\frac{(-q)^{|E|}}{|\mathcal{C}^\perp|}{\rm
hwe}(\mathcal{C}^\perp;0),\end{align*}
and finally $q^{|E|}/|\mathcal{C}^\perp|=|\mathcal{C}|$.
\end{pf}
The following is a variation on, and mild generalisation of, Theorem~1.2 in \cite{Onn04}
\begin{cor}\label{cond Qtensions}
Suppose $G$ is a graph and $Q$ is a ring of order $q$ with a
generating character. Let $\mathcal{C}$ be the set
of $Q$-flows of $G$ and
$\mathcal{C}^\perp$ the set of $Q$-tensions of $G$. Then $P(G;q)\neq
0$ if and only if there exists
$y\in (Q\setminus 0)^E$ such that $\mu 1_\mathcal{C}(y)\neq 0$,
i.e., such that
$$\sum_{x\leq y, \,
x\in\mathcal{C}}(-1)^{|x|}\neq 0.$$
\end{cor}
\begin{pf} From Lemma~\ref{mu1C not zero}, if $\chi(x_ey_e)\neq 1$ for all $e\in E$ then
$x_e\neq 0$ for all $e\in E$. The converse follows from Lemma~\ref{sum nz C perp}.
\end{pf}
A dual to Corollary~\ref{cond Qtensions} giving a criterion for
$F(G;q)\neq 0$ results by taking $\mathcal{C}$ to be the set of
$Q$-tensions. Corollary~\ref{cond Qtensions} was proved for $q=3$ and
generalised in a different direction by Alon and Tarsi~\cite{AT92} by
considering $\mathbb{Z}_q$-flows taking values in $\{0,\pm 1\}$ only. The latter
for $q$ greater than the maximum degree of $G$ are in bijective
correspondence with partial eulerian orientations
of $G$ (and for $q=3$ the same is true for $4$-regular graphs). See
also \cite{Tarsi} and
section~\ref{4 regular} below.
From Corollary~\ref{cond Qtensions} comes the familiar fact that $P(G;2)\neq 0$ if and only if the number of eulerian
subgraphs of $G$ of even size differs from those with odd size. More
interestingly, $P(G;4)\neq 0$ if and only if there is
$y\in\mathbb{F}_4^E$ such that $\mu 1_\mathcal{C}(y)\neq 0$, i.e., the difference between the number of
$\mathbb{F}_4$-flows $\leq y$ of even support size and those $\leq y$
of odd support size is non-zero, where in this case
\begin{align*}\mu
1_\mathcal{C}(y) & =4^{-r(G)}(-2)^{|E|}\#\{x\in\mathcal{C}^\perp:\forall_{e\in
E}\; x_e\not\in\{0, y_e\}\}\\
& = (-2)^{|E|-2|V|}\#\{z\in \mathbb{F}_4^V:\forall_{uv\in E}\; z_u+z_v\not\in
\{0,y_e\}\}.\end{align*}
It may be that $\mu 1_\mathcal{C}(y)=0$ for some $y\in
(\mathbb{F}_4^\times)^E$ even though $P(G;4)\neq 0$, since it may be
impossible to avoid hitting the value $y_e$ for some edge $e$ in any
nowhere-zero $\mathbb{F}_4$-tension $x$ of $G$. (For example, the
triangle $K_3$ and $y_e=1$ for each edge $e$).
Similarly $P(G;4)\neq 0$ if and only if for some
$y\in(\mathbb{Z}_4\setminus 0)^E$ there is a disparity between the
number of $\mathbb{Z}_4$-flows $\leq y$ of even support size and those
$\mathbb{Z}_4$-flows $\leq y$ of odd support size, and here
$$\mu 1_\mathcal{C}(y)=4^{-r(G)}\sum_{x\in\mathcal{C}^\perp}\prod_{e\in
E}(i^{x_ey_e}-1).$$
If $y_e=2$ then in order for $x\in\mathcal{C}^\perp$ to contribute
a non-zero term it is necessary that $x_e\not\in\{0, 2\}$. This too may not be
possible for some $y$ (consider $K_3$ again with $y_e=2$ for each edge).
A translation of Corollary~\ref{cond Qtensions} for $Q=\mathbb{F}_4$
into the language of correlations between events involving parity and
eulerian subgraphs runs as follows
\begin{thm}\label{tripart} Suppose $X,Y,Z\subseteq E$ partition the edges of
$G$ into three sets (not all of which need be non-empty).
Choosing $A\subseteq X,
B\subseteq Y$ and $C\subseteq Z$ uniformly at random, let $\Sigma$ be the event that
$|A|+|B|+|C|\equiv 0\,(\mbox{\rm mod}\, 2)$ and $\Gamma$ the
event that $A\cup C$ and $C\cup B$ are both
eulerian.
Then ${\rm Bias}(\Sigma\,\mid\, \Gamma)\neq 0$ for some tripartition
$\{X,Y,Z\}$ of $E$ if and only if $P(G;4)\neq
0$.
\end{thm}
Note that in contrast to the event $\Delta$ of section
\ref{H2 H4}, it does not follow that if $A\cup C,
C\cup B$ are eulerian then $A\cup
B$ is eulerian. Also, note that ${\rm Bias}(\Sigma)=0$ for any choice of $X,Y,Z$.
\begin{pf}
We use Corollary~\ref{cond Qtensions} to show the auxilary result that $P(G;4)\neq 0$ if and only if there exist $X,Y\subseteq E$ with $X\cup Y=E$ and
\begin{equation}\label{aux}\mathop{\sum_{\mbox{\rm \tiny eulerian } A\subseteq X,\, B\subseteq
Y}}_{A\bigtriangleup B\subseteq X\bigtriangleup Y}(-1)^{|A\cup
B|}\neq 0.\end{equation}
We then take $A\setminus B$, $B\setminus A$ in \eqref{aux} for the $A$ and $B$ of the
theorem, $X\setminus Y, Y\setminus X$ in \eqref{aux} for the $X$ and $Y$ of the
theorem, and finally set $C=A\cap B$ and $Z=X\cap Y$. This is enough to
prove the theorem as stated, for \eqref{aux}, with $|(A\cup C)\cup (C\cup
B)|=|A|+|B|+|C|$, is now the assertion that
$$\mathbb{P}(\Sigma\cap\Gamma)-\mathbb{P}(\overline{\Sigma}\cap\Gamma)=2^{-3|E|}\sum_{\mbox{\rm
\tiny eulerian }\, A\subseteq X, B\subseteq Y, C\subseteq
Z}(-1)^{|A|+|B|+|C|}\neq 0.$$
Let $x,y\in\mathbb{F}_2^E$ be the indicator vectors of $X, Y$ and
$z=(x,y)\in\mathbb{F}_2^E\times\mathbb{F}_2^E\cong\mathbb{F}_4^E$.
Define a partial
order $\leq$ on
$\mathbb{F}_4^E$ by setting $d\leq z$ if and only if $d_e\in\{0,
z_e\}$. Then $d=(a,b)\leq z=(x,y)$ if and only if $A\subseteq X,
B\subseteq Y$ and $A\bigtriangleup B\subseteq X\bigtriangleup Y$.
Note that $z_e\neq 0$ for all $e\in E$ if and only if $X\cup Y=E$. Denote
by $|d|$ the Hamming weight of $d\in\mathbb{F}_4^E\cong\mathbb{F}_2^E\times\mathbb{F}_2^E$. If $d=(a,b)$ for
$a,b\in\mathbb{F}_2^E$ the indicator vectors of $A,B\subseteq E$ then
$|d|=|A\cup B|$.
Then an equivalent statement to \eqref{aux} in terms of the space of
$\mathbb{F}_4$-flows $\mathcal{C}_4\cong\mathcal{C}_2\times\mathcal{C}_2$ is that $P(G;4)\neq 0$ if and only if there
exists $z\in(\mathbb{F}_4^\times)^E$ such that
$$\sum_{d\in\mathcal{C}_4,\, d\leq z}(-1)^{|d|}\neq
0.$$
This is the assertion of Corollary~\ref{cond Qtensions}.\end{pf}
Theorem~\ref{tripart} is related to the criterion for $P(G;4)\neq 0$ that $G$ be
covered by two bipartite subgraphs $X\cup Y, Y\cup Z$. Given the
latter are bipartite, if
$A\cup C\subseteq X\cup Z$ is eulerian and $C\cup B\subseteq Z\cup Y$
is eulerian then $|A\cup C|$ and
$|C\cup B|$ are both even so that $|A|+|B|$ is also even.
However, a bias in $|A|\!+\!|B|\,(\mbox{\rm mod}\, 2)$ does not imply
a bias in $|A|\!+\!|B|\!+\!|C|\,(\mbox{\rm mod}\, 2)$.
\section{Cubic graphs and triangulations} \label{cubic graphs}
In this section we use MacWilliams duality theorem~\eqref{McW cwe} for
complete weight enumerators to derive a correlation criterion for the existence of
a proper vertex
$4$-colouring of a triangulation.
\begin{thm}\label{4flows tensions} Let $\omega=e^{2\pi i/3}$ and
let $\psi:\mathbb{F}_4\rightarrow\{0,1,\omega,\omega^2\}$ be a non-trivial
Dirichlet character (multiplicative, and $\psi(0)=0$). For a graph $G$ with space of
$\mathbb{F}_4$-flows $\mathcal{C}_4$ and space of $\mathbb{F}_4$-tensions $\mathcal{C}_4^\perp$,
\begin{equation}\label{first psi}\sum_{z\in\mathcal{C}_4}\prod_{e\in E}\psi(z_e)=2^{n(G)-r(G)}\sum_{z\in\mathcal{C}_4^\perp}\prod_{e\in E}\psi(z_e).\end{equation}
In other words
\begin{equation}\label{second omega}\mathop{\sum_{\mbox{\rm \tiny eulerian}\, A,B\subseteq E}}_{A\cup
B=E}\omega^{|A|-|B|}=2^{n(G)-r(G)}\mathop{\sum_{\mbox{\rm \tiny cutsets}\, A,B\subseteq E}}_{A\cup
B=E}\omega^{|A|-|B|}.\end{equation}
In particular, if $G=(V,E)$ is a cubic graph
then
\begin{equation}\label{third cubic}2^{r(G)-n(G)}F(G;4)=\#\{z\in\mathcal{C}_4^\perp:\,\prod_{e\in E}
z_e=1\}-\frac{1}{2}\#\{z\in\mathcal{C}_4^\perp:\,\prod_{e\in E}
z_e\in\{\omega,\omega^2\}\}.\end{equation}
\end{thm}
\begin{pf} We begin by noting that, since
$z\in\mathcal{C}_4$ if and only if $\overline{z}\in\mathcal{C}_4$ and $\psi(\overline{z_e})=\overline{\psi}(z_e)$,
the equations~\eqref{first psi} and
\eqref{second omega} are between real numbers (in fact rational integers). Also, since $\prod_{e\in E}\psi(z_e)\neq 0$ if
and only if $z$ is nowhere-zero, i.e., $z_e\neq 0$ for all $e\in E$,
the range of the summations in equation~\eqref{first psi} is restricted to nowhere-zero
$\mathbb{F}_4$-flows on the left and nowhere-zero
$\mathbb{F}_4$-tensions on the right.
Identify $\mathbb{F}_4$ with its image
$\{0,1,\omega,\omega^2\}$ under
$\psi:\mathbb{F}_4\rightarrow\mathbb{C}$, i.e., $\psi$ is defined by
$\psi=1_1+\omega 1_\omega+\omega^21_{\omega^2}$. It is easily calculated
that $\widehat{\psi}=2\overline{\psi}$.
By the MacWilliams duality formula~\eqref{McW cwe},
\begin{align*}\sum_{z\in\mathcal{C}_4}\prod_{e\in E}\psi(z_e)={\rm
cwe}(\mathcal{C}_4;\psi) & =\frac{1}{|\mathcal{C}_4^\perp|}{\rm
cwe}(\mathcal{C}_4^\perp;\widehat{\psi})\\
& =4^{-r(G)}2^{|E|}{\rm
cwe}(\mathcal{C}_4^\perp,\overline{\psi}).\end{align*}
Since the
sums in this equation are real, the function $\overline{\psi}$ can be
replaced by its conjugate $\psi$, and this establishes equation~\eqref{first psi} of the theorem.
The second
statement~\eqref{second omega} is a straight translation of
\eqref{first psi} into
different language. Using the isomorphism of
additive groups $\mathbb{F}_4^E\cong\mathbb{F}_2^E\times\mathbb{F}_2^E$,
an element $z\in\mathbb{F}_4^E$ may be written $z=(x,y)$
with $x,y\in\mathbb{F}_2^E$ indicator vectors for subsets
$A,B\subseteq E$ respectively. The property that $z$ is nowhere-zero
translates to the property that $A\cup B=E$ and the condition
$z\in\mathcal{C}_4$ translates to the condition that $A$ and $B$ are
both eulerian. Similarly, the condition
$z\in\mathcal{C}_4^\perp$ translates to the condition that $A$ and $B$
are cutsets.
For the final statement~\eqref{third cubic} we use the property that a
cubic graph has a cutset double cover comprising the three-edge stars
at each vertex. For vertex $v\in V$, the three edges $\{e,f,g\}$ incident with $v$
form a star, and each edge occurs exactly twice amongst the $|V|$
stars, since each edge is adjacent to two
distinct vertices. It follows that $\psi(z_ez_fz_g)\in\{0,1\}$ for
each star $\{e,f,g\}$ and $z\in\mathcal{C}_4$, due to the
fact that if a sum of three non-zero elements of $\mathbb{F}_4$ is
equal to $0$ then their product is $1$.
Thus we see that
$$\sum_{z\in\mathcal{C}_4}\;\prod_{\mbox{\rm \tiny
stars}\,\{e,f,g\}}\psi(z_ez_fz_g)=F(G;4).$$
On the other hand, by the double cover property of the collection of stars,
$$\sum_{z\in\mathcal{C}_4}\;\prod_{\mbox{\rm \tiny stars}\,\{e,f,g\}}\psi(z_ez_fz_g)=\sum_{z\in\mathcal{C}_4}\prod_{e\in E}\psi(z_e)^2,$$
and $\psi(z_e)^2=\overline{\psi}(z_e).$
Using equation~\eqref{first psi}, in which $\psi$ is interchangeable
with its conjugate $\overline{\psi}$, this establishes that
$$F(G;4)=2^{n(G)-r(G)}\sum_{z\in\mathcal{C}_4^\perp}\prod_{e\in
E}\psi(z_e),$$
and equation~\eqref{third cubic} is just another way of writing this.
\end{pf}
We finish this section by interpreting identity~\eqref{third cubic} in
its dual form in terms of the bias of events in a uniform probability
space. The dual notion of a cutset double cover is a cycle double
cover. If $G$ is a plane cubic graph then its planar dual $G^*$ is a plane
triangulation and just as $G$ has a cutset
double cover by three-edge stars (at vertices) so $G^*$ has a
cycle double cover by triangles (faces).
\begin{thm} \label{triang} Suppose $G$ is a graph that has a cycle double cover by
triangles and suppose that $\Gamma$ is the event that $A,B\subseteq E$ are eulerian and
$A\cup B=E$. Then, choosing $A,B\subseteq E$
uniformly at random,
$${\rm Bias}(|A|\equiv |B|\!+\!1\,(\bmod 3)\,\mid\,\Gamma)={\rm
Bias}(|A|\equiv|B|\!+\!2\,(\bmod 3)\,\mid\,\Gamma)$$
and
$$\frac{{\rm Bias}(|A|\equiv |B|\,(\bmod 3)\,\mid\,\Gamma)-{\rm
Bias}(|A|\equiv |B|\!+\!1\,(\bmod 3)\,\mid\,\Gamma)}{{\rm Bias}(|A|\equiv |B|\,(\bmod 3)\,)-{\rm
Bias}(|A|\equiv|B|\!+\!1\,(\bmod 3)\,)}=\frac{2^{3|E|-2|V|}P(G;4)}{F(G;4)}.$$
\end{thm}
\begin{pf}
The left-hand sum in equation~\eqref{second omega} of Theorem~\ref{4flows tensions} has
the following interpretation:
\begin{equation}\label{as probability}2^{-2|E|}\mathop{\sum_{\mbox{\rm \tiny eulerian}\, A,B\subseteq E}}_{A\cup
B=E}\omega^{|A|-|B|}\end{equation}
\begin{align*} & =\mathbb{P}(|A|\equiv|B|\,(\bmod\,3)\,\cap\,\Gamma)+\omega\mathbb{P}(|A|\equiv|B|\!+\!1\,(\bmod\, 3)\,\cap\,\Gamma)+\omega^2\mathbb{P}(|A|\equiv|B|\!+\!2\,(\bmod\, 3)\,\cap\,\Gamma)\\
& = \mathbb{P}(|A|\equiv|B|\,(\bmod\,
3)\,\cap\,\Gamma)+\omega^2\mathbb{P}(|A|\equiv|B|\!+\!1\,(\bmod\,
3)\,\cap\,\Gamma)+\omega\mathbb{P}(|A|\equiv|B|\!+\!2\,(\bmod\, 3)\,\cap\,\Gamma),\end{align*}
the latter equality since, as remarked in the proof of
Theorem~\ref{4flows tensions}, the sum we started with is
real.
Hence
\begin{equation}\label{1 equals 2}\mathbb{P}(|A|\equiv|B|\!+\!1\,(\bmod\,
3)\,\cap\,\Gamma)=\mathbb{P}(|A|\equiv|B|\!+\!2\,(\bmod\, 3)\,\cap\,\Gamma).\end{equation}
By definition of $\Gamma$ and since $G$ has a cycle double cover by triangles,
$\mathbb{P}(\Gamma)=2^{-2|E|}F(G;4)\neq 0$.
Dividing equation~\eqref{1 equals 2} by $\mathbb{P}(\Gamma)$ yields the first statement of the theorem.
Equation~\eqref{1 equals 2} together with the identity developed in
\eqref{as probability} has the consequence that, in the notation of
Theorem~\ref{4flows tensions},
\begin{equation}\label{lhs as prob}2^{-2|E|}\sum_{z\in\mathcal{C}_4}\prod_{e\in
E}\psi(z_e)=\mathbb{P}(|A|\equiv|B|\,(\bmod\,
3)\,\cap\,\Gamma)-\mathbb{P}(|A|\equiv|B|\!+\!1\,(\bmod\, 3)\,\cap\,\Gamma).\end{equation}
By equation~\eqref{first
psi} of Theorem~\ref{4flows tensions}, in which $\prod_{e\in
E}\psi(z_e)\in\{0,1\}$ for $z\in\mathcal{C}_4^\perp$ since $G$ has a
cycle double cover by triangles, and equation~\eqref{lhs as prob},
\begin{align*}\mathbb{P}(|A|\equiv|B|\,(\bmod 3)\,\cap\,\Gamma)-\mathbb{P}(|A|\equiv|B|\!+\!1\,(\bmod
3)\,\cap\,\Gamma) & =
2^{-2|E|}2^{n(G)-r(G)}\sum_{z\in\mathcal{C}_4^\perp}\prod_{e\in E}\psi(z_e)\\
& = 2^{-2|E|}2^{n(G)-r(G)}4^{-k(G)}P(G;4).
\end{align*}
Dividing this last equation through by $\mathbb{P}(\Gamma)=2^{-2|E|}F(G;4)$ gives
$$\mathbb{P}(|A|\equiv|B|\,(\bmod
3)\,\mid\,\Gamma)-\mathbb{P}(|A|\equiv|B|\!+\!1\,(\bmod
3)\,\mid\,\Gamma)=2^{|E|-2|V|}P(G;4)/F(G;4),$$
i.e.,
$${\rm Bias}(|A|\equiv |B|\,(\bmod 3)\,\mid\,\Gamma)-{\rm
Bias}(|A|\equiv |B|\!+\!1\,(\bmod 3)\,\mid\,\Gamma)=2^{|E|-2|V|+1}P(G;4)/F(G;4)$$
By Lemma~\ref{bias S} with $q=3$ and
$\Sigma=\{A,B\subseteq E: |A|-|B|\in S\}$, taking the difference
between the cases $S=\{0\}$ and $S=\{1\}$ we obtain
$${\rm Bias}(|A|\!-\!|B|\equiv 0\,(\bmod\, 3)\,)-{\rm
Bias}(|A|\!-\!|B|\equiv 1\,(\bmod\, 3)\,)$$
\begin{align*} & =3^{-1}2^{1-|E|}\big[(1-\mbox{$\frac{1}{2}$})^{|E|}(1-e^{2\pi i/3})+(1-\mbox{$\frac{1}{2}$})^{|E|}(1-e^{4\pi i/3})\big]\\
& =2^{1-2|E|}\end{align*}
The second statement of the theorem now results.
\end{pf}
In particular, a graph $G$ with a cycle double cover by triangles has $P(G;4)\neq 0$ if and only if
$\mathbb{P}(|A|\equiv|B|\,(\mbox{\rm mod}\, 3)\,\mid\,
\Gamma)>\frac{1}{3}$, i.e., the event that $A,B$ form an eulerian cover
of $G$ is positively correlated with $|A|\equiv|B|\,(\bmod\, 3)$.
\section{Eulerian subdigraphs of a $4$-regular graph}\label{4 regular}
In this final section we use MacWilliams duality \eqref{McW cwe} for
compete weight enumerators and Lemma~\ref{cwe sum} to derive some
further evaluations of the Tutte polynomial on $H_3$ similar in form
to Theorem~\ref{triang}.
Take $q=3$ and $G$ a $4$-regular graph (such as the line graph of
a plane cubic graph), for which the space of $\mathbb{F}_3$-flows has a natural
identification with the set of eulerian partial orientations of $G$. In a partial
orientation of a graph some edges may not be directed; in an {\em
eulerian} partial orientation each vertex
has the same number of incoming and outgoing directed edges.
A reference orientation $\gamma$ of $G$ is fixed. A partial
orientation $\alpha$ is defined corresponding to a
vector $x\in\mathbb{F}_3^E$ by making $\alpha$ direct an
edge $e$ the same way as $\gamma$ if $x_e=+1$, making $\alpha$ reverse the
direction of $\gamma$ if $x_e=-1$, and leaving $e$ undirected if
$x_e=0$.
Given that $G$ is $4$-regular, if $x$ is a $\mathbb{F}_3$-flow
then it defines an eulerian partial orientation $\alpha$.
If further $x$ is nowhere-zero then $\alpha$ is an eulerian orientation of
$G$.
For orientations $\alpha,\beta$, define $\alpha+\beta$ to be
the partial orientation
whose directed edges are those sharing the same direction in
$\alpha$ and $\beta$. If $x,y\in\mathbb{F}_3^E$ define $\alpha,\beta$
relative to the base orientation $\gamma$ of $G$ then $\alpha+\beta$
is the partial orientation defined by $-(x+y)$.
Suppose an orientation $\alpha$ is chosen uniformly at random. Let $\Sigma$ be the event that $|\alpha+\gamma|\equiv 0\,(\mbox{\rm
mod}\, 2)$.
Clearly ${\rm Bias}(\Sigma)=0.$
Let $\Gamma$ be the event that $\alpha$ is an eulerian orientation.
Then $\mathbb{P}(\Gamma)=2^{-|E|}F(G;3)$, since for the $4$-regular
graph $G$ the number of eulerian orientations is the number of
nowhere-zero $3$-flows.
Finding ${\rm Bias}(\Sigma\,\mid\,\Gamma)$ is a bit more difficult and
in order to state a partial result on this we need some definitions.
The {\em line graph} $L(H)$ of a graph $H$ has vertices the edges of $H$ and
adjacent vertices $e,f$ when $e$ and $f$ are incident in $H$. If $H$
is embedded in an orientable surface, the {\em medial graph} $M(H)$ of
$H$ is the graph obtained by placing vertices at the edges
of $H$ and joining vertices $e,f$ of $M(H)$ by an edge if they lie on
incident edges $e,f$ of $H$ and it is possible to draw a line joining
$e$ and $f$ without crossing any edges of $H$. (If edges $e,f$ are incident with
a vertex of degree $2$ then they are joined by two edges, neither of
which can be continuously transformed to the other without crossing
an edge of $H$.)
The medial graph $M(H)$ is $4$-regular.
Suppose now that $H$ is an
orientably embedded cubic graph. Then $M(H)$
is an embedding of $L(H)$ in the same orientable surface as $H$. A
vertex of $H$ lies in the interior of a
triangle of edges in $M(H)$, which we shall call a {\em black triangle} of
$M(H)$ (on account of the standard white-black face colouring of the medial graph). When the edges of the black triangles of $M(H)$ are directed in a
clockwise sense on the surface on which $M(H)$ is embedded, the
resulting orientation of $M(H)$ is eulerian. (The clockwise direction
traced by the edges of
a black triangle of $M(H)$ corresponds to a clockwise orientation of the
three edges at a vertex of $H$, called a vertex rotation in the
embedding of $H$.
\begin{thm}\label{even odd eul suborns} Let $G$ be the medial graph of a plane cubic
graph and let $\gamma$ be the orientation directing edges of $G$
clockwise around black triangles. Then, choosing an orientation
$\alpha$ of $G$ uniformly at random, the event $\Sigma$ that
$\alpha$ agrees with $\gamma$ on an even number of edges and the
event $\Gamma$ that $\alpha$ is eulerian have correlation given by
$${\rm Bias}(\Sigma\,\mid\,\Gamma)=\frac{P(G;3)}{F(G;3)}.$$
\end{thm}
\begin{pf} Let $\mathcal{C}_3$ be the space of $\mathbb{F}_3$-flows of
$G$. Then
\begin{align*}\mathbb{P}(\Gamma){\rm Bias}(\Sigma\,\mid\,\Gamma) & =\mathbb{P}(\Sigma\cap\Gamma)-\mathbb{P}(\overline{\Sigma}\cap\Gamma)\\
& =2^{-|E|}{\rm cwe}(\mathcal{C}_3;1_1-1_{-1})\\
& = 2^{-|E|}3^{-r(G)}{\rm
cwe}(\mathcal{C}_3^\perp;(-3)^{\frac{1}{2}}(1_{-1}-1_{1})\\
& = (-1)^{|V|}2^{-|E|}3^{k(G)}{\rm cwe}(\mathcal{C}_3^\perp;1_1-1_{-1}),\end{align*}
using $|E|=2|V|$ for $4$-regular graph $G$. A result of Penrose\footnote{Penrose quoted this theorem (in a different
formulation) in \cite{P71}, in which he mentioned that his proof was too lengthy for inclusion. An
elegant short proof has been given by Kaufmann~\cite{K90}. See also
\cite[Theorem 3.1]{EG96} for a
generalisation and for further citations -
for example Scheim~\cite{S74} found the result
independently and was the first to
publish a proof.
It remains an open problem~\cite{EG96} to characterise those edge $3$-colourable cubic graphs $H$ for
which the line graph $L(H)$ with a fixed orientation of its edges has the property that the
number of nowhere-zero $\mathbb{F}_3$-tensions of $L(H)$ with an even number of
edges with value $-1$ differs from those with an odd number of edges
with value $-1$. The line graph $L(K_{3,3})$ does not have this
property. The theorem of Penrose and Scheim is that when $H$ is a planar cubic
graph nowhere-zero $\mathbb{F}_3$-tensions of $L(H)$ either all have an
even number of edges with value $-1$ or all an odd number of such
edges.} says that
for any nowhere-zero $\mathbb{F}_3$-tension $x$ of $G$ (corresponding to
an edge $3$-colouring of the plane cubic graph $H$) we have
$$\prod_{e\in E}(1_1-1_{-1})(x_e)=(-1)^{\#\{e\in E:x_e=-1\}}=(-1)^{|V|}$$
when $G$ has its fixed orientation $\gamma$ clockwise around black triangles (or any
other orientation $\beta$ with $|\beta+\gamma|$ even). With
$\mathbb{P}(\Gamma)=2^{-|E|}F(G;3)\neq 0$ the theorem is proved.
\end{pf}
We move on now to choosing pairs of orientations of a $4$-regular graph
and formulate an analogue of Proposition~\ref{squares with q=2} for
eulerian orientations rather than the eulerian subgraphs of that proposition.
Suppose $\alpha,\beta$ are orientations of $G$ chosen uniformly at random.
Let $\Sigma$ be the event that $|\alpha+\beta|$ is even.
Note that if $\beta$ is another orientation then
$|\alpha+\gamma|$ and $|\beta+\gamma|$ have the same parity if and
only if $|\alpha+\beta|$ is even. We have ${\rm Bias}(\Sigma)=0$ as before.
Let $\Gamma$ be the event that $\alpha+\beta$ is an eulerian partial
orientation of $G$, i.e., the consistently directed edges of $\alpha$
and $\beta$ form an eulerian partial orientation of $G$.
\begin{lem} \label{PGamma} {\rm (Cf. \cite[Corollary 5]{AJG05})}
The event $\Gamma$ that orientations $\alpha,\beta$ of a
$4$-regular graph agree on an eulerian partial orientation has
probability
$$\mathbb{P}(\Gamma)=4^{-|E|}T(G;2,4).$$
\end{lem}
\begin{pf} Let $\mathcal{C}_3$ be the space of $\mathbb{F}_3$-flows of $G$.
If orientations $\alpha,\beta$ are defined relative to the base
orientation $\gamma$ of $G$ by the nowhere-zero vectors
$x, -y\in\mathbb{F}_3^E$, then the event that $\alpha+\beta$ is eulerian
coincides with the event that $x-y\in\mathcal{C}_3$, i.e., $x,y$ belong
to the same coset of $\mathcal{C}_3$. Using Lemma~\ref{cwe sum}
\begin{align*}\mathbb{P}(\Gamma) &
=4^{-|E|}\sum_{\mathcal{C}_3+z\in\mathbb{F}_3^E/\mathcal{C}_3}{\rm cwe}(\mathcal{C}_3+z;1_{1,-1})^2\\
& = 4^{-|E|}3^{-r(G)}{\rm cwe}(\mathcal{C}_3^\perp;(21_0-1_{1,-1})^2)\\
& = 4^{-|E|}3^{-r(G)}{\rm hwe}(\mathcal{C}_3^\perp;4)\\
& =4^{-|E|}T(G;2,4).\end{align*}
\end{pf}
This lemma leads us to our promised theorem
\begin{thm}\label{eul orn correlation}
Let $G$ be a $4$-regular graph and $\alpha,\beta$ two orientations of
$G$ chosen uniformly at random. Then the event $\Sigma$ that
$\alpha$ agrees with $\beta$ on an even number of edges and the
event $\Gamma$ that $\alpha$ agrees with $\beta$ in an eulerian partial
orientation of $G$ have correlation given by
$${\rm Bias}(\Sigma\,\mid\,\Gamma)=\frac{3^{|E|-|V|}P(G;3)}{T(G;2,4)}.$$
\end{thm}
\begin{pf} Using Lemma~\ref{cwe sum}
\begin{align*}\mathbb{P}(\Gamma){\rm Bias}(\Sigma\,\mid\,\Gamma) & =
\mathbb{P}(\Sigma\cap\Gamma)-\mathbb{P}(\overline{\Sigma}\cap\Gamma)\\
& = 4^{-|E|}\sum_{\mathcal{C}_3+z\in\mathbb{F}_3^E/\mathcal{C}_3}|{\rm
cwe}(\mathcal{C}_3+z;1_1-1_{-1})|^2\\
& = 4^{-|E|}3^{-r(G)}{\rm
cwe}(\mathcal{C}_3^\perp;|(-3)^{\frac{1}{2}}(1_{-1}-1_1)|^2)\\
& =4^{-|E|}3^{-r(G)}{\rm
cwe}(\mathcal{C}_3^\perp;31_{1,-1})=4^{-|E|}3^{n(G)}(-1)^{r(G)}T(G;-2,0)\\
& = 4^{-|E|}3^{|E|-|V|}P(G;3).\end{align*}
Lemma~\ref{PGamma} now gives the result.
\end{pf}
{\small
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,044
|
Sitting in the heart of beautiful Palm Beach Gardens, the Financial Center at the Gardens is the premier office building in North Palm Beach County. This institutionally owned 10-story Class A office property features a soaring atrium, beautifully renovated lobbies and common areas, 100 percent structured parking, manned security and a new café and wine lounge. The Financial Center at the Gardens is home to some of the area's most prestigious tenants, including Morgan Stanley, UBS, Bankrate, and JP Morgan Chase.
The Financial Center at the Gardens is located within walking distance to numerous amenities at Downtown at the Gardens, a premier shopping and dining destination, and the Gardens Mall, a regionally renowned shopping experience.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,554
|
package com.vladmihalcea.book.hpjp.hibernate.flushing;
import com.vladmihalcea.book.hpjp.util.AbstractTest;
import com.vladmihalcea.book.hpjp.util.providers.Database;
import org.hibernate.Session;
import org.junit.Test;
import jakarta.persistence.*;
/**
* @author Vlad Mihalcea
*/
public class HibernateDeleteEntityTest extends AbstractTest {
@Override
protected Class<?>[] entities() {
return new Class<?>[] {
Post.class
};
}
@Override
protected boolean nativeHibernateSessionFactoryBootstrap() {
return true;
}
@Test
public void test() {
doInJPA(entityManager -> {
Post post = new Post();
post.setId(1L);
post.setTitle("High-Performance Java Persistence");
entityManager.persist(post);
});
doInJPA(entityManager -> {
Post post = new Post();
post.setId(1L);
entityManager.unwrap(Session.class).delete(post);
});
}
@Entity(name = "Post")
@Table(name = "post")
public static class Post {
@Id
private Long id;
private String title;
public Long getId() {
return id;
}
public void setId(Long id) {
this.id = id;
}
public String getTitle() {
return title;
}
public void setTitle(String title) {
this.title = title;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,606
|
Unlike the 6.0.1 release in December, this month's adjustments appear to be entirely focused on security fixes. Google has a bulletin describing the relevant security issues addressed with this release. There are also a couple of builds specific to the Pixel C, including one that appears quite large. Like some of the previous changelogs, this is merely because it technically starts from a base version (i.e. Android Wear 5.1.1) and incorporates all of the patches made since then. The other update merely increments a build number, which means the changes were probably made to the Pixel C's proprietary hardware binaries or to the other partitions included with its factory image.
As always, if there's anything interesting beyond the security fixes, or maybe something special about how something was fixed, drop a line in the comments.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,866
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{"url":"https:\/\/studyfinance.com\/weighted-average-cost-of-capital\/","text":"# Weighted Average Cost of Capital (WACC)\n\nThe weighted average cost of capital (WACC) is a calculation of a company or firm\u2019s cost of capital that weighs each category of capital (common stock, preferred stock, bonds, long-term debts, etc.). The ratio of debt to equity in a company is used to determine which source should be utilized to fund new purchases. An increase in a company\u2019s WACC signifies an increased risk and a decrease in valuation.\n\n## Weighted Average Cost of Capital Formula\n\n$WACC = \\bigg( \\dfrac{E}{V} \\times Re \\bigg) + \\bigg( \\dfrac{D}{V} \\times Rd \\times (1 - Tc) \\bigg)$\n\u2022 Re = Cost of equity\n\u2022 Rd = Cost of debt\n\u2022 E = Market value of the firm\u2019s equity\n\u2022 D = Market value of the firm\u2019s debt\n\u2022 V = E + D = Total market value of the firm\u2019s financing\n\u2022 E\/V = Percentage of financing that is equity\n\u2022 D\/V = Percentage of financing that is debt\n\u2022 Tc = Corporate tax rate\n\nIn the WACC calculation, the cost of each capital component is multiplied by its proportional weight. It is then multiplied by the corporate tax rate.\n\n## Weighted Average Cost of Capital Example\n\nUsing the following values, do a quick calculation of a fictional company\u2019s WACC:\n\n\u2022 Re = 5% or .05\n\u2022 Rd = 4% or .04\n\u2022 E = $5,500,000 \u2022 D =$1,400,000\n\u2022 Tc = 20% or .20\n\u2022 V = $6,900,000 (E+D_ $WACC = \\bigg( \\dfrac{\\5{,}500{,}000}{\\6{,}900{,}000} \\times 5\\% \\bigg) + \\bigg( \\dfrac{\\1{,}400{,}000}{\\6{,}900{,}000} \\times 4\\% \\times (1 - 20\\%) \\bigg) = 4.63\\%$ Using this example, one can determine that the company has a WACC of 4.63%, which is the amount of money the company needs to pay investors for each$1 of funding. In this case, it amounts to $.0463 per$1.\n\n## Analysis of Weighted Average Cost of Capital\n\nThe cost of equity (Re) can be difficult to determine, as each share of stock in a company doesn\u2019t have a specific value or price, and the value or price that investors are willing to pay is always fluctuating. Ultimately, the cost of equity is how much a company must spend to keep stock prices steady and meet its investors\u2019 required rate of return. The cost of equity can be found using the capital asset pricing model (CAPM), which uses a company\u2019s beta, expected return of the market, and the risk-free rate.\n\nThe cost of debt (Rd) is calculated using the actual interest rate or market interest rate that a company is paying. Interest is generally deductible, so the savings generated by these tax deductions is represented at the end of the formula (1-Tc).\n\nThe market value of a firm\u2019s equity (E) refers to the value of all outstanding shares, or those owned by insiders and shareholders investing in the company.\n\nThe market value of a firm\u2019s debt (D) refers to the value of all outstanding debt, which can be found on a company\u2019s balance sheet.\n\nThe corporate tax rate (Tc), or effective tax rate, can be determined by dividing the total tax paid by that company\u2019s taxable income.\n\nMany analysts and investors tend to come up with different WACC numbers for the same company due to the many different assumptions that need to take place when determining the cost of equity. As a result, WACC is often used as more of a speculation tool than a concrete number used in final investing decisions.\n\n## Weighted Average Cost of Capital Conclusion\n\n\u2022 The Weighted Average Cost of Capital is used to determine whether debt or equity should be used to finance a purchase.\n\u2022 WACC is not a concrete number, it is very assumption-based and subject to change.\n\u2022 WACC incorporates all aspects of a company\u2019s capital, including preferred stock, common stock, bonds, and other possible long-term debt.\n\u2022 Cost of equity can be found by utilizing the CAPM.\n\u2022 WACC is an essential part of any economic value added (EVA) calculations.\n\u2022 There are many values included in the calculation of WACC, namely the market value of a company\u2019s equity, the market value of a company\u2019s debt, the cost of equity and cost of debt for that company, the total market value of that company\u2019s financing, and the corporate tax rate.\n\n## Weighted Average Cost of Capital Calculator\n\nYou can use the WACC calculator below to quickly work out the weight average cost of capital by entering the required numbers.","date":"2021-01-21 04:33:02","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\": 2, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.49931350350379944, \"perplexity\": 2140.5240738945445}, \"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-2021-04\/segments\/1610703522242.73\/warc\/CC-MAIN-20210121035242-20210121065242-00552.warc.gz\"}"}
| null | null |
Q: Getting all my check-ins from a single category via Foursquare's API So here's what I'm trying to do: I have a space on my website that says X cups of coffee. Ideally the X would be replaced with the number of times I have checked into a venue on Foursquare that falls under the coffee shop category. Right now, I have to manually update the number from my site's backend. This is just too much work, and I'd rather be able to check-in and have the Foursquare API update the number automatically. Is this possible, and if so, what would be the easiest way to execute it?
Thanks so much in advance for your suggestions and help!
A: You can use the realtime push API to notify your server whenever you check in on Foursquare. You can then use the information in the check-in to update statistics on your website.
Here is the documentation for the push API: https://developer.foursquare.com/overview/realtime
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,960
|
package fixtures.report;
import java.util.ArrayList;
import java.util.List;
import java.util.Map;
import fixtures.report.implementation.AutoRestReportServiceImpl;
public final class CoverageReporter {
private static AutoRestReportService client = new AutoRestReportServiceImpl("http://localhost:3000");
private CoverageReporter() { }
public static void main(String[] args) throws Exception {
Map<String, Integer> report = client.getReport().getBody();
// Body cannot be null
report.put("putStringNull", 1);
report.put("OptionalIntegerParameter", 1);
report.put("OptionalStringParameter", 1);
report.put("OptionalClassParameter", 1);
report.put("OptionalArrayParameter", 1);
// Put must contain a body
report.put("OptionalImplicitBody", 1);
// OkHttp can actually overwrite header "Content-Type"
report.put("HeaderParameterProtectedKey", 1);
// Redirects not suppoted by OkHttp
report.put("HttpRedirect301Put", 1);
report.put("HttpRedirect302Patch", 1);
int total = report.size();
int hit = 0;
List<String> missing = new ArrayList<>();
for (Map.Entry<String, Integer> entry : report.entrySet()) {
if (entry.getValue() != 0) {
hit++;
} else {
missing.add(entry.getKey());
}
}
System.out.println(hit + " out of " + total + " tests hit. Missing tests:");
for (String scenario : missing) {
System.out.println(scenario);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,922
|
<?xml version="1.0" encoding="UTF-8"?>
<deployment id="org.openntf.red.third-party.feature">
<runtime id="RED Server"/>
</deployment>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,942
|
{"url":"https:\/\/mathoverflow.net\/questions\/327020\/find-probability-of-non-stationary-inputs-into-turing-machine","text":"Find probability of non-stationary inputs into Turing machine?\n\nConsider some finite string $$x=(x_1,x_2,...,x_{n-1},x_n)$$ that is drawn from a non-stationary process. Would it be possible to use the algorithmic probability formula, defined by Solomonoff as, $$P_M(x)=\\sum_{i=1}^{\\infty}2^{-|s_{i}(x)|}$$ to predict $$x_n$$, by computing the probability $$P_M(x_n | x_1,...x_{n-1})=P_M(x_1,x_2,...,x_{n-1},x_n)\/P_M(x_1,x_2,...,x_{n-1})$$? Or is it a necessity that the process that $$x$$ is drawn from be stationary?\n\nEdit: Here $$s_i(x)$$ refers to the $$i$$-th program which generated $$x=(x_1,x_2,...,x_{n-1})$$ and we want to predict the probability that $$x_n$$ is either a $$1$$ or a $$0$$.\n\n\u2022 Interesting, but I'm a bit confused. I mean, I can use any probability to predict anything. So are you asking whether it's a good idea or not? That depends where $x$ comes from. But this feels philosophical ... also, it's not computable, so in that sense, I think \"no, not possible\". \u2013\u00a0usul Apr 5 '19 at 12:52\n\u2022 @litmus I think your question shows some confusions. First, I believe you are using $s_i(x)$ to refer to the $i$-th program which generates $x$, which should be explained. Second, it is not $s_i(x)$ that has a universal distribution, rather the \"universal distribution\" is $P_M$ itself, as defined above. Lastly, it is not clear what you mean by a \"non-stationary binary string\". Usually a random process (i.e., a distribution over infinite strings) is said to be is stationary or not stationary, not a particular finite string. \u2013\u00a0Artemy Apr 6 '19 at 19:56\n\u2022 @Artemy Hello again Artemy! Thank you for the comments, I have updated the question. \u2013\u00a0litmus Apr 6 '19 at 20:19\n\u2022 That helps but I might still not be entirely understanding your question. The equation you have states exactly how to compute (or, more specifically, lower semicompute) $P_M(x)$ for any finite string $x$. Are you asking whether it is allowed for $x$ to be sampled from some non-stationary random process, say with distribution $Q$? If so, then yes -- the above probability measure is defined for any finite string whatsoever. \u2013\u00a0Artemy Apr 7 '19 at 0:06\n\u2022 @Artemy Yes, that is exactly what I wanted to know, thanks! If you want to elaborate your comment as an answer instead, I'd be very happy to accept it. Also, please feel free to edit my question if you think that there are some parts that could be more clear for posterity. \u2013\u00a0litmus Apr 8 '19 at 14:49\n\nThe equation you have for algorithmic probability states how to compute (or, more specifically, lower semicompute) $$P_M(x)$$ for any finite string $$x$$, so it can also be used to predict $$x_n$$ by evaluating $$P_M(x_n \\vert x_1, \\dots , x_{n-1})$$. It holds for any finite string $$x$$ -- it does not matter what distribution (if any) $$x$$ is drawn from.","date":"2020-05-29 12:19:19","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\": 17, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8754178881645203, \"perplexity\": 240.58965805178062}, \"config\": {\"markdown_headings\": false, \"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-2020-24\/segments\/1590347404857.23\/warc\/CC-MAIN-20200529121120-20200529151120-00306.warc.gz\"}"}
| null | null |
Гиљермо Себастијан Корија (; рођен 13. јануара 1982. године) је бивши аргентински тенисер. Најбољи пласман на АТП листи му је треће место.
Каријера
Гиљермо Корија је најбоље резултате постигао на шљаци. Свој први турнир освојио је 2001. године у Виња дел Мару, Чиле 2001. године. Највећи успех му је играње у финалу Ролан Гароса 2004. године. И поред предности од два сета у финалу је изгубио од сународника Гастона Гаудија са 6–0, 6–3, 4–6, 1–6, 6–8. Од већих резултата још треба издвојити освајање два турнира из Мастерс серије: Хамбург 2003. и Монте Карло 2004. године. Укупно је освојио девет АТП турнира у каријери. Повукао се 28. априла 2009. у двадесетседмој години.
Гренд слем финала
Појединачно: 1 (0—1)
АТП Мастерс финала
Појединачно: 7 (2–5)
Пласман на АТП листи на крају сезоне
Референце
Спољашње везе
АТП профил Гиљерма Корије
Гиљермо Корија на интернет страници ИТФ
Рођени 1982.
Аргентински тенисери
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,875
|
/**
* Provides implementation of the Classifier service.
*/
package org.onosproject.vtnrsc.classifier.impl;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,071
|
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout
xmlns:android="http://schemas.android.com/apk/res/android"
android:id="@+id/choosefile_layout"
android:layout_width="fill_parent"
android:layout_height="fill_parent"
android:orientation="vertical">
<ListView
android:id="@+id/list"
android:layout_width="fill_parent"
android:layout_height="fill_parent"
android:scrollbars="vertical"
android:layout_weight="1">
</ListView>
<LinearLayout
xmlns:android="http://schemas.android.com/apk/res/android"
android:id="@+id/choosefile_layout"
android:layout_width="wrap_content"
android:layout_height="fill_parent"
android:layout_gravity="center"
android:orientation="horizontal">
<Button
android:id="@+id/buttonfileConfirm"
android:layout_width="125px"
android:layout_height="wrap_content"
android:text="Confirm" />
<Button
android:id="@+id/buttonfileCancle"
android:layout_width="125px"
android:layout_height="wrap_content"
android:text="Cancle" />
</LinearLayout>
</LinearLayout>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,902
|
Q: Spinner value Changing based on parent spinner value changing hi everyone ,
I have three spinners .In that i want to change second spinner value change based on first spinner value and third spinner change based on second spinner value all values are fetched and displayed from database.I am using ArrayAdapter .I was searching for the last two weeks . But i dont get solution kindly help me i am running out of time pls.
A: you have to just do set adapter like:
package com.example.mapsdemo;
import java.util.ArrayList;
import android.app.Activity;
import android.os.Bundle;
import android.view.Menu;
import android.view.View;
import android.widget.AdapterView;
import android.widget.AdapterView.OnItemSelectedListener;
import android.widget.ArrayAdapter;
import android.widget.Spinner;
public class MainActivity extends Activity {
private Spinner spin1;
private Spinner spin2;
ArrayList<String> a = new ArrayList<String>();
ArrayList<String> b = new ArrayList<String>();
ArrayList<String> c = new ArrayList<String>();
// private ImageView imageView;
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_main);
spin1 = (Spinner) findViewById(R.id.spinner1);
spin2 = (Spinner) findViewById(R.id.spinner2);
fillarray();
spin1.setAdapter(new ArrayAdapter<String>(MainActivity.this,
android.R.layout.simple_dropdown_item_1line, a));
/*
* spin2.setAdapter(new ArrayAdapter<String>(MainActivity.this,
* android.R.layout.simple_dropdown_item_1line, b));
*/
spin1.setOnItemSelectedListener(new OnItemSelectedListener() {
@Override
public void onItemSelected(AdapterView<?> arg0, View arg1, int pos,
long arg3) {
// TODO Auto-generated method stub
if (pos == 0) {
spin2.setAdapter(new ArrayAdapter<String>(
MainActivity.this,
android.R.layout.simple_dropdown_item_1line, b));
} else {
spin2.setAdapter(new ArrayAdapter<String>(
MainActivity.this,
android.R.layout.simple_dropdown_item_1line, c));
}
}
@Override
public void onNothingSelected(AdapterView<?> arg0) {
// TODO Auto-generated method stub
}
});
}
private void fillarray() {
// TODO Auto-generated method stub
a.clear();
a.add("a");
a.add("b");
a.add("c");
a.add("d");
a.add("e");
b.clear();
b.add("1");
b.add("2");
b.add("3");
b.add("4");
c.clear();
c.add("Android");
c.add("ios");
}
}
your layout file like:
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="match_parent" >
<Spinner
android:id="@+id/spinner1"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:layout_alignParentLeft="true"
android:layout_alignParentTop="true"
android:layout_marginTop="40dp" />
<Spinner
android:id="@+id/spinner2"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:layout_alignParentLeft="true"
android:layout_below="@+id/spinner1"
android:layout_marginTop="80dp" />
</RelativeLayout>
A: may be this will help,
ArrayAdapter<String> secondspinnerAdapter;
ArrayAdapter<String> firstspinnerAdapter = new ArrayAdapter<String>(
MainActivity.this, android.R.layout.simple_spinner_item,
firstspinnervalue);
first_spinner.setAdapter(firstspinnerAdapter);
firstspinnerAdapter
.setDropDownViewResource(android.R.layout.simple_spinner_dropdown_item);
firstspinner.setOnItemSelectedListener(new OnItemSelectedListener() {
@Override
public void onItemSelected(AdapterView<?> arg0, View arg1,
int arg2, long arg3) {
// TODO Auto-generated method stub
// get data from database add to arraylist
secondspinnerAdapter = new
ArrayAdapter<String>( mainActivity.this,
android.R.layout.simple_spinner_item, arraylistvalue);
second_spinner.setAdapter(secondspinnerAdapter);
second_spinner.setSelection(arg2, false);
secondspinnerAdapter.setDropDownViewResource
(android.R.layout.simple_spinner_dropdown_item);
}
@Override
public void onNothingSelected(AdapterView<?> arg0) {
// TODO Auto-generated method stub
}
});
first spinner item is selected, then get data from database and fill arraylist and set secondspinner.
same way follow second spinner item selected.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,835
|
Q: scrapy on ubuntu server I have a problem I just cannot resolve.
After installing scrapy (with pip) I get and error when trying to make startup project:
File "/usr/local/bin/scrapy", line 5, in <module>
from pkg_resources import load_entry_point
File "/usr/lib/python2.7/dist-packages/pkg_resources.py", line 2749, in <module>
working_set = WorkingSet._build_master()
File "/usr/lib/python2.7/dist-packages/pkg_resources.py", line 446, in _build_master
return cls._build_from_requirements(__requires__)
File "/usr/lib/python2.7/dist-packages/pkg_resources.py", line 459, in _build_from_requirements
dists = ws.resolve(reqs, Environment())
File "/usr/lib/python2.7/dist-packages/pkg_resources.py", line 628, in resolve
raise DistributionNotFound(req)
pkg_resources.DistributionNotFound: Scrapy==1.0.3.post1-g83a06ed
Does anyone familiar with this? I tried a lot of things including reinstalling packages.
I`m using DigitalOcean server with ubuntu 14.04 and python 2.7.9
Thanks,
Aviad
A: Stumbled across this the other day:
Scrapy failing in terminal
You may need to upgrade Scrapy.
easy_install --upgrade scrapy
or
pip install --upgrade scrapy
If you don't have easy_install it can be installed with
sudo apt-get install python-setuptools
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,651
|
\section{Introduction}
A {\it unital $\ell$-group} $(G,u)$ is
an abelian group $G$ equipped with a translation invariant lattice-order
and with a distinguished {\it order unit}, i.e.
an element $0\leq u\in G$ whose positive integer multiples eventually
dominate every element of $G$.
A {\it unital $\ell$-homomorphism} between unital $\ell$-groups is
a group homomorphism that preserves the order unit and the lattice structure.
As a particular case of a general definition \cite[p. 286]{BS1981},
a unital $\ell$-group $(G,u)$ is said to be
{\it finitely presented} if there exists a finite set
$\{g_1,\ldots,g_n\}\subseteq G$ along with a finite set of equations
$s_1=t_1,\ldots,s_m=t_m$ in the language of
unital $\ell$-groups with $n$-variables such that
\begin{enumerate}
\item $s_i(g_1,\ldots,g_n)=t_i(g_1,\ldots,g_n)$ for each $i=1,\ldots,m$ and
\item if $(H,v)$ is a unital $\ell$-group and $h_1,\ldots,h_n\in H$ satisfy
$s_i(h_1,\ldots,h_n)=t_i(h_1,\ldots,h_n)$ for each $i=1,\ldots,m$,
then there exists a unique unital $\ell$-homomorphism
$h\colon (G,u)\rightarrow(H,v)$
such that $h(g_1)=h_1,\ldots, h(g_n)=h_n.$
\end{enumerate}
We denote by $\FP$
the category of finitely presented unital
$\ell$-groups with unital $\ell$-homomorphisms.
For $n=1,2,\ldots$ we let $\McNn$ denote the unital $\ell$-group of
all continuous functions $f\colon [0,1]^{n}\rightarrow\mathbb R$
having the following property:
there are linear polynomials $p_{1},\ldots,p_{m}$ with integer coefficients
such that for all $x\in [0,1]^{n}$ there is $i\in \{1,\ldots,m\}$
with $f(x)=p_{i}(x)$.
$\McNn$ is equipped with the pointwise operations $+,-,\max,\min$ of
$\mathbb R$,
and with the constant function $1$ as the distinguished order unit.
$\McNn$ is a free object in the category of unital $\ell$-groups,
in the following sense:
\begin{proposition}\label{proposition:free}{\rm (\cite[Corollary
4.16]{Mu1986})}
The coordinate maps $\xi_ {i} \colon [0,1]^{n}\to \mathbb R$ together with
the order unit $1$ form a generating set of $\McNn$.
%
For every unital $\ell$-group $(G,u)$ and $0\leq g_{1},\ldots,g_{n}\leq u$,
if the set $\{g_{1},\ldots,g_{n}, u\}$ generates $G$ then
there is a unique unital $\ell$-homomorphism $\psi$ of $\McNn$ onto $G$
such that $\psi(\xi_ {i})=g_{i}$ for each $i=1,\ldots,n.$
\end{proposition}
An {\it ideal} $\mathfrak i$ of a unital $\ell$-group $(G,u)$
is the kernel of a unital $\ell$-homomorphism of $(G,u)$,
(\cite[p.8 and 1.14]{Go1986}).
$\mathfrak i$ is {\it principal} if it is singly (=finitely) generated.
As a
consequence of Proposition \ref{proposition:free},
a unital $\ell$-group $(G,u)$ is finitely presented
iff for some $n=1,2,\ldots,$ $(G,u)$
is isomorphic to the quotient of $\McNn$ by
some principal ideal $\mathfrak j$,
in symbols, $(G,u)\cong \McNn/\mathfrak j$.
The characterization of finitely presented unital $\ell$-groups presented in
Proposition \ref{proposition:free} relies
on the free objects $\McNn$ and their universal property. In \cite{CM2011} an intrinsic characterization of finitely presented unital $\ell$-groups is
given in terms of special sets of generators, called bases. The notion of
{basis}
was introduced in \cite{MM2007} as a purely algebraic counterpart of
Schauder bases.
In \cite[Theorem 4.5]{MM2007}
it is proved that if a
unital $\ell$-group $(G,u)$ is isomorphic to an $\ell$-group of
real-valued functions defined on some set $X$ (that is,
$G$ is {\emph{Archimedean}}) then it is finitely presented
iff it has a basis. In \cite[Theorem 3.1]{CM2011} it is proved that the Archimedean assumption
can be dropped: thus, $(G,u)$ is finitely presented iff it has a basis.
In Section 2 we give a detailed account of
the main tools used in this paper, namely
the categorical duality
between finitely presented unital $\ell$-groups
and rational polyhedra, and the combinatorial
representation of rational polyhedra as weighted
abstract simplicial complexes.
Section 3 is devoted to the construction
of limits and co-limits in these two categories.
Finally, in Section 4, all the machinery of the earlier chapters
will be combined with the algebraic-topological analysis of projective
unital $\ell$-groups in \cite{CM20XX}
to give geometric and algebraic characterizations of finitely generated subalgebras of
the free unital $\ell$-groups $\McNn$.
\section{Preliminaries}
\subsection{Regular triangulations}
We refer to \cite{Ew1996}, \cite{Gl1970} and \cite{St1967} for
background in elementary polyhedral topology and simplicial complexes.
For any simplex $S$ we denote by $\ver(S)$ the set of its vertices.
For any $F\subseteq \ver(S)$, the convex hull $\conv(F)$ is called a
{\it face} of $S$.
A {\it polyhedron} $P$ in $\R^{n}$ is a finite union of (always
closed) simplexes $P=S_{1}\cup\cdots\cup S_{t}$ in $\R^n$.
A simplex $S$ is said to be {\it rational} if the coordinates of each
$v\in \ver(S)$ are rational numbers.
$P$ is said to be a {\it rational polyhedron} if there are rational
simplexes $T_{1},\ldots,T_{l}$
such that $P=T_{1}\cup\cdots\cup T_{l}$.
For every simplicial complex ${\Delta}$, its {\it support}
$|\Delta|$ is the pointset union of all simplexes of $\Delta$,
and $\ver(\Delta)$ is the set of its vertices, i.e.
the set of the
vertices of its simplexes.
We say that the simplicial complex $\Delta$ is {\it rational} if all
simplexes of $\Delta$ are rational.
Given a rational polyhedron $P$, a {\it triangulation} of
$P$ is a rational simplicial complex $\Delta$ such that $P=|\Delta|$.
In \cite[Theorem 1]{Be1977} it is proved that
$\Delta$ exists for every rational polyhedron $P$.
In the rest of this paper every simplex, polyhedron, and simplicial
complex will be rational.
Accordingly, the adjective ``rational'' will be omitted unless it is
strictly necessary.
For $v$ a rational point in $\R^{n}$ we let $\den(v)$ denote the
least common denominator of the coordinates of $v$.
The vector $\widetilde{v}=\den(v)(v,1)\in\Zed^{n+1}$ is called the
{\it homogeneous correspondent} of $v$.
A simplex $S$ is called {\it regular} if the set of homogeneous
correspondents of its vertices is part of a basis of the free abelian
group $\Zed^{n+1}.$
By a {\it regular triangulation} of a polyhedron $P$ we understand a
triangulation of $P$ consisting of regular simplexes.
The following proposition was proved in \cite[Theorem 1.2]{Mu1986}
under the assumption that $X\subseteq [0,1]^{n}$. However,
it is easy to see that the proof is the same for
all $X\subseteq \mathbb R^n$ (also see
\cite[Lemma 3.1]{MuXXXX}):
\begin{proposition}\label{proposition:poly}
For any
set $X\subseteq \R^{n}$ the following statements are equivalent:
\begin{enumerate}
\item $X$ coincides with the support of some regular complex $\Delta$;
\item $X$ is a rational polyhedron.
\end{enumerate}
\end{proposition}
\subsection{Farey subdivisions}
Given a polyhedron $P$ and triangulations $\Delta$ and $\Sigma$ of $P$
we say that $\Delta$ is a {\it subdivision} of $\Sigma$ if every
simplex of $\Delta$ is contained in a simplex of $\Sigma$.
For any point $p \in P$,
the {\it blow-up $\Delta_{(p)}$ of $\Delta$ at $p$} is the
subdivision of $\Delta$
given by replacing every simplex $S\in \Delta$ that contains $p$
by the set of all simplexes of the form $\conv(F\cup\{p\})$,
where $F$ is any face of $S$ that does not contain $p$
(see \cite[p. 376]{Wl1997} or \cite[III, Definition 2.1]{Ew1996}, where
blow-ups are called stellar subdivisions).
For any regular $m$-simplex $S =\conv(v_{0},\dots,v_m) \subseteq
\mathbb R^{n}$,
the {\it Farey mediant} of (the vertices of) $S$
is the rational point $v$ of $S$ whose homogeneous correspondent
$\tilde v$ equals $\widetilde{{v}_0}+\cdots+\widetilde{{v}_m}$.
If $S$ belongs to a triangulation $\Delta$ and $v$ is the Farey mediant of $S$
then the blow-up $\Delta_{(v)}$ is a regular triangulation iff
so is $\Delta\,\,\,$ (\cite[V, 6.2]{Ew1996}).
$\Delta_{(v)}$ will be called the {\it Farey blow-up} of $\Delta$ at $v$.
By a {\it (Farey) blow-down} we understand the inverse of
a (Farey) blow-up.
\smallskip
The proof of the ``weak Oda conjecture'' by
Morelli \cite{Mo1996}
and W\l odarczyk
\cite{Wl1997} yields:
\begin{lemma}\label{lemma:weakoda}
Let $P$ be a polyhedron.
%
Then any two regular triangulations of $P$ are connected by a
finite path of Farey blow-ups and Farey blow-downs.
\end{lemma}
For later use in this paper, we
recall here some properties of regular triangulations.
\begin{lemma}
\label{Lem_Triang-Subset}
Let $P\subseteq Q\subseteq\R^{n}$ be rational polyhedra and
$\Delta$ be a regular triangulation of $P$.
Then there exists a regular triangulation $\Delta_Q$ of $Q$ such that
the set
$\Delta_{P}=\{S\in\Delta_Q\mid S\subseteq P\}$ is a subdivision of
$\Delta$.
%
Moreover, $\Delta_Q$ can be so chosen that $\Delta_P$ is a
{ \rm full}
subcomplex of $\Delta_Q$, in the sense that
$\Delta_P=\{S\in\Delta_{Q}\mid \ver(S)\subseteq P\}$.
\end{lemma}
\begin{proof}
Let $\nabla_0$ be a rational triangulation of $Q$.
From \cite[Addendum 2.12]{RS1972} we obtain
%
a triangulation $\nabla_1$ of $Q$ which is
a subdivision of $\nabla_0$ and also satisfies
$S=\bigcup\{T\in \nabla_1 \mid T\subseteq S\}$,
for each $S\in\Delta$.
By \cite[Lemma 3.4]{RS1972}, there is a
subdivision $\nabla_2$ of $\nabla_1$ such
that $\{T\in \nabla_2\mid T\subset P\}$
is a full subcomplex of $\nabla_2$. By \cite[Corollary, p. 242]{Be1977},
there is no loss of generality
to assume that $\nabla_2$ is rational.
The desingularization process described in
\cite[Chapter 9]{CDM2000}
then yields
a regular triangulation $\nabla_3$ which is a subdivision of
$\nabla_2$. Since $\nabla_3$ is obtained form $\nabla_2$ by blow-ups,
then $\{T\in \nabla_3\mid T\subset P\}$ is a
regular triangulation of $P$ which is also
a subdivision of $\Delta$ and a
full subcomplex of $\nabla_3$.
\end{proof}
\begin{lemma}\label{Lem_Sub-Triang-Subset}
Let $P\subseteq Q\subseteq\R^{n}$ be rational polyhedra,
and $\Delta_P$ and $\Delta_Q$ be regular triangulations of $P$ and $Q$
such that $\Delta_P\subseteq \Delta_Q$.
%
If $\nabla_P$ is a regular subdivision of $\Delta_P$
then there exists a regular triangulation $\nabla_Q$ of $Q$ such that
$\nabla_P\subseteq \nabla_Q$
and $\nabla_Q$ is a subdivision of $\Delta_Q$.
\end{lemma}
\begin{proof}
Let $K=\{S\in \Delta_Q\mid \ver(S)\subseteq P\mbox{ and }S\nsubseteq P\}$ and $S$ be a maximal element in $K$.
Then the Farey blow up $(\Delta_Q)_{(v)}=\nabla_1$, where $v$ is the Farey mediant of $S$, is a regular triangulation of $Q$ such that $\Delta_P\subseteq \nabla_1$. By the maximality of $S$ in $K$, $K_1=\{S\in \nabla_1\mid \ver(S)\subseteq P\mbox{ and }S\nsubseteq P\}=K\setminus\{S\}$. Repeating this process, we obtain a sequence of regular complexes $\nabla_1, \ldots, \nabla_{r}$ and a sequence of sets $K=K_0\supseteq K_1\supseteq\cdots\supseteq K_{r}$ where each $\nabla_{k+1}$ is obtained by blowing-up $\nabla_{k}$ at the Farey mediant of some maximal $S$ in $K_{k}$ and $K_{k+1}=K_{k}\setminus S$.
Since $K$ is finite this process terminates at some $t$. By construction,
$\Delta_P$ is a full subcomplex of $\nabla_t$ and $\nabla_t$ is a subdivision of $\Delta_Q$.
For each $S\in \nabla_t$ we define:
$$
\ver_P(S)=\ver(S)\cap P\ \ \mbox{, }\ \ \ver_Q(S)=\ver(S)\setminus \ver_P(S),\mbox{ and}
$$
$$U_S=\{\conv(\ver_Q(S)\cup T)\mid T\in \nabla_P\mbox{ and }T\subseteq S\cap P\}.$$
Since $\Delta_P$ is a full subcomplex of $\nabla_t$ and $\nabla_Q$ is a subdivision of $\Delta_Q$, it follows that
$\nabla_0=\bigcup\{U_S\mid S\in \nabla_t\}$ is a simplicial complex and a triangulation of $Q$.
An application to $\nabla_0$ of the desingularization procedure
of \cite[1.2]{Mu1988} yields a regular triangulation
$\nabla_Q$. $\nabla_Q$ is obtained from $\nabla_0$ by a suitable sequence of blow ups at non-regular simplexes. Since $\nabla_P\subseteq \nabla_0$ is regular, none of its simplexes is modified by the application of this procedure. Therefore, $\nabla_P\subseteq\nabla_Q$ and $\nabla_Q$ is a subdivision of $\Delta_Q$, as desired.
\end{proof}
\subsection{The category of rational polyhedra}
\begin{definition}(\cite[Definition 3.1]{Mu2011})
Given a rational polyhedra $P\subseteq \R^{n}$ a map $\eta\colon P\rightarrow \R^{m}$ is called a {\it \Zed-map}
if there is a triangulation $\Delta$ of $P$
such that over every simplex $T$ of $\Delta$, $\eta$ coincides
with an affine linear map $\eta_{T}$ with integer coefficients.
\end{definition}
The following lemmas are easy consequences of the definition. For detailed proofs in the case of rational polyhedra contained in some $n$-cube $[0,1]^{n}$ see \cite[\S 3]{Mu2011}.
\begin{lemma}\label{Lem_ImagesZmorph0}
Suppose $P\subseteq \R^{n}$ and $\eta\colon P\rightarrow R^{m}$ is a \Zed-map.
Then $\eta(P)$ is a polyhedron in $\R^m$.
\end{lemma}
\begin{lemma}\label{Lem_StableTriangul}
Let $\eta\colon P\rightarrow Q$ be a \Zed-map and
$\Delta$ a regular triangulation of $P$.
%
Then there exists a regular triangulation $\nabla$ of $P$
such that $\nabla$ is a subdivision of $\Delta$ and $\eta$ is
linear over each simplex in $\nabla$.
\end{lemma}
\begin{lemma}\label{Lem_CarZed}
Given polyhedra $P\subseteq \R^{n}$, $Q\subseteq \R^{m}$ and
a map $\eta\colon P\rightarrow Q$, let
$\xi_1,\ldots,\xi_m\colon Q\rightarrow \R$ denote the coordinate maps.
Then
%
$\eta$ is a $\Zed$-map iff $\xi_i\circ\eta:P\rightarrow
\xi_i(P)$ is a \Zed-map for each $i=1,\ldots,m$.
\end{lemma}
\begin{lemma}\label{Lem_ZeroP}
Given a rational polyhedron $P\subseteq \R^{n}$. A subset $P\subseteq Q$ is a rational polyhedron
iff
there exists a \Zed-map $f\colon Q\rightarrow \R$ such
that $P=f^{-1}(0)$.
\end{lemma}
\begin{proof}
In \cite[Propositions 5.1 and 5.2]{MM2007} and \cite[Lemma 3.2]{MuXXXX}, the result was proven for the case $P\subseteq [0,1]^n$. The same argument replacing $[0,1]^n$ by an arbitrary rational polyhedron $Q$ proves the result of this lemma.
\end{proof}
We denote by $\QP$ the category
whose objects are rational polyhedra in
$\mathbb R^n$\,\,\,({\rm for}\,\, $n=1,2,\ldots),$
and whose arrows are $\Zed$-maps.
Following \cite[Definition 4.4]{MuXXXX} and \cite[Definition 3.9]{Mu2011}, a map $\eta\colon P\rightarrow Q$ is a {\it \Zed-homeomorphism} if
it is a one-one \Zed-map of $P$ onto $Q$ and its inverse
$\eta^{-1}$ is also a \Zed-map.
Thus {$\Zed$-homeomorphisms} are the same
as iso-arrows of the category $\QP$.
\medskip
A proof of the following result can be obtained from
\cite[\S 3]{Mu2011}:
\begin{theorem}[\bf Duality]\label{Theo_Baker-Beynon}
Let the functor $\McN\colon \QP\rightarrow \FP$ be defined by
$$
\begin{tabular}{ll}
Objects:& For $P\in \QP$ a polyhedron,\\& $\McN(P)$ is the set of
all \,\Zed-maps
from $P$ into $\R$.\\[0.2cm]
Arrows:& For $\eta\colon P\rightarrow Q$ a \Zed-map,\\&
$\McN(\eta)(f)=f\circ \eta$,
for each $f\in \McN(Q)$.
\end{tabular}
$$
Then $\McN$ yields a duality between the
categories $\QP$ and $\FP$.
Stated otherwise, $\McN$ is a categorical
equivalence between $\QP$ and the opposite category of $\FP$.
\end{theorem}
\begin{lemma}\label{Lem_ImagesZmorph}
Suppose $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$
are polyhedra and $\eta\colon P\rightarrow Q$ is a \Zed-map.
We then have $$\McN(Q)/{\rm ker}(\McN(\eta))\cong\McN(\eta(P)).$$
\end{lemma}
\begin{proof}
Observe that $f\in {\rm ker}(\McN(\eta))$ iff
$f\circ \eta (P)=f(\eta(P))=\{0\}$.
%
Therefore the kernel of the onto map $h\colon\McN(Q)\rightarrow
\McN(\eta(P))$
given by $f\mapsto f\upharpoonright \eta(P)$,
coincides with ${\rm ker}(\McN(\eta))$.
%
In conclusion,
$\McN(Q)/{\rm ker}(\McN(\eta))\cong\McN(\eta(P))$
\end{proof}
\subsection{Combinatorics of rational polyhedra}
Building on \cite{CM20XX} and \cite{MuXXXX},
in this section we introduce a functor
from the category of abstract simplicial complexes with weighted
vertices into the category of rational polyhedra.
This will be used
to construct
limits and co-limits of rational polyhedra. Using
the dual equivalence of Theorem \ref{Theo_Baker-Beynon} we will then characterize
finitely generated subalgebras of free unital $\ell$-groups.
\smallskip
Let us recall that a {\it (finite) abstract simplicial complex} is a
pair $({V},\Sigma)$,
where ${V}$ is a finite
set, whose elements are called the {\it vertices} of $({V},\Sigma)$,
and $\Sigma$ is a collection of subsets of ${\mathscr V}$ whose union
is ${V}$,
having the property that every subset of an element of $\Sigma$ is
again an element of $\Sigma$.
A {\it weighted abstract simplicial complex} is a triple $({
V},\Sigma, \omega)$
where $({ V},\Sigma)$ is an abstract simplicial complex
and $\omega$ is a map of ${ V}$ into the set $ \{1,2,3,\ldots\}.$
Given two weighted abstract simplicial complexes $ \mathfrak{W}=
( V,\Sigma,\omega)$ and $\mathfrak{W}' = (
V',\Sigma',\omega') $
a simplicial map $\gamma\colon V \rightarrow V',$
(that is, $\gamma(S)\in \Sigma'$ for each $S\in\Sigma$)
is a morphism from $\mathfrak{W}$ into $\mathfrak{W}'$,
if $\omega'(\gamma(v))$ divides $\omega(v)$ for all $v\in { V}$.
We denote $\wasc$ the category of weighted abstract simplicial complexes.
It is easy to see that two weighted abstract simplicial complexes
$ \mathfrak{W}= ( V,\Sigma,\omega)$ and $\mathfrak{W}' =
( V',\Sigma',\omega')$
are isomorphic in $\wasc$ iff
there is a one-one map $\gamma$
from $ V$ onto ${ V}'$
having the following properties:
\begin{itemize}
\item $\omega'(\gamma(v))=\omega(v)$ for all $v\in { V}$,
and
\item $\{w_{1},\ldots,w_{k}\}\in \Sigma$ iff
$\{\gamma(w_{1}),\ldots,\gamma(w_{k})\}\in \Sigma'$
for each $\{w_{1},\ldots,w_{k}\}\subseteq V$.
\end{itemize}
When this is the case
we say that $\mathfrak{W}$ and $\mathfrak{W}'$ are {\it
combinatorially isomorphic}.
\medskip
Let $\mathfrak{W}=({ V},\Sigma, \omega)$ be a weighted
abstract simplicial complex with vertex set ${
V}=\{v_{1},\ldots,v_{n}\}$.
Let
$e_{1},\ldots,e_{n}$ be the standard basis vectors of
$\mathbb R^{n}$.
We then use the notation $\Delta_{\mathfrak{W}}$
for the complex whose vertices are the following points
in $\mathbb R^n$
$$
v'_{1} = e_{1}/\omega(v_{1}),\ldots,v'_{n}=e_{n}/\omega(v_{n}),
$$
and whose $k$-simplexes ($k=0,\ldots,n$) are given by
$$
\conv(v'_{i(0)},\ldots, v'_{i(k)})\in \Delta_{\mathfrak{W}}\quad
\text{ iff }\quad
\{ v_{i(0)},\ldots, v_{i(k)}\}\in \Sigma.
$$
Trivially, $\Delta_{\mathfrak{W}}$ is a regular triangulation of the
polyhedron $|\Delta_{\mathfrak{W}}|\subseteq [0,1]^{n}$. The
polyhedron $|\Delta_{\mathfrak{W}}|$ is called the {\it geometric
realization} of $\mathfrak{W}$ and will be denoted
$\Pol(\mathfrak{W})$.
\medskip
In order to extend $\Pol$ to a functor from $\wasc$ into $\QP$
we prepare
\begin{lemma}\cite[Lemma 3.7]{Mu2011}\label{Lem-LinearMap}
Let $S=\conv(x_{1},\ldots,x_{k})\subseteq \R^{n}$ be a regular
${(k-1)}$-simplex, and
$\{y_{1},\ldots,y_{k}\}$ a set of rational points in $\R^{n}.$
%
Then the following conditions are equivalent:
\begin{itemize}
\item[(i)] For each $i=1,\ldots,k,$ $\den(y_{i})$ is a divisor
of $\den(x_{i})$. \smallskip
\item[(ii)] For some integer matrix $M\in\Zed^{n\times n}$ and
integer vector $b\in\Zed^{n}$, $M x_{i}+b=y_{i}.$
\end{itemize}
\end{lemma}
\begin{corollary}\label{Cor_Div_Denominators}
Let $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$ be polyhedra
and $\eta \colon P\rightarrow Q$ a $\Zed$-map.
%
Then for every rational point $x\in P,\,\,\,\,$
$
\den(\eta (x))\mbox{ divides }\den(x).
$
\end{corollary}
\begin{corollary}\label{Cor_ExtensionToZed}
Let $P\subseteq\R^{n}$ be a polyhedron, $\Delta$ a regular
triangulation of $P$ and
$f\colon
\ver(\Delta)\rightarrow \mathbb{Q}^{m}$ a map
such that $\den(f(v))$ divides $\den(v)$ for each $v\in\ver(\Delta)$.
%
Then there exists a unique $\Zed$-map $\eta\colon P\rightarrow
\R^{m}$ satisfying the following two conditions:
\begin{enumerate}
\item $\eta$ is linear on each simplex of $\Delta$;
\item $\eta\restrict{\ver(\Delta)}=f$.
\end{enumerate}
\end{corollary}
\bigskip
\noindent Let $\mathfrak{W}= ( V,\Sigma,\omega)$ and $\mathfrak{W}' =
( V',\Sigma',\omega')$ be weighted abstract simplicial
complexes, with
${ V}=\{v_{1},\ldots,v_{n}\}$ and ${
V'}=\{v'_{1},\ldots,v'_{m}\}$.
Let $\gamma\colon V \rightarrow V'$ be a morphism
of weighted abstract simplicial complexes.
Corollary \ref{Cor_ExtensionToZed} yields
a unique \Zed-map $\Pol(h)\colon
\Pol(W)\rightarrow \Pol(W')$ with the following properties:
\begin{itemize}
\item
$\Pol(h)$ is linear on each simplex $S$ of $\Delta_{W}$, and
\item
for each $i=1,\ldots,n,\,\,\,$
$\eta(e_{i}/\omega(v_{i}))=e_{j}/\omega'(v'_{j})$ whenever
$\gamma(v_i)=v'_{j}$.
\end{itemize}
As a consequence, $\Pol$ is a faithful functor from $\wasc$ into $\QP$.
\smallskip
For every regular complex $\Delta$, the {\it skeleton} of $\Delta$
is the weighted abstract simplicial complex
$\mathfrak{W}(\Delta)=(V,\Sigma,\omega)$ given by the following stipulations:
\begin{enumerate}
\item $V=\ver(\Delta)$.
\item For every $v\in\ver(\Delta)$,
$\omega(v)=\den(v).$
\item For every subset $W=\{w_{1},\ldots,w_{k}\}$
of $V$, $W\in \Sigma$ iff $\conv(w_{1},\ldots,w_{k})\in\Delta.$
\end{enumerate}
Let $\{v_1,\ldots,v_m\}$ be the vertices of
a regular triangulation $\Delta$ of a polyhedron $P$.
Let
\begin{equation}\label{Eq:IsoinCube}
\iota_{\Delta}\colon P\rightarrow \Pol(\mathfrak{W}(\Delta))
\end{equation}
be
the unique \Zed-map given by Corollary \ref{Cor_ExtensionToZed} which
is linear on each simplex of $\Delta$ and also satisfies
$\iota_\Delta(v_i)=e_i/\den(v_i)$. Then
$\iota_\Delta$ is a \Zed-homeomorphism.
Since $\Pol(\mathfrak{W}(\Delta))\subseteq [0,1]^m$, as a byproduct we obtain that each rational polyhedron is \Zed-homeomorphic to a polyhedron contained in some $m$-cube.
We have just proved
that for each object $P$ of $\QP$ there exists an object
$\mathfrak{W}$ of $\wasc$ such that $\Pol(\mathfrak{W})$ is
isomorphic to $P$ in $\QP$.
Yet, $\Pol$ does not define an equivalence between the
categories $\wasc$ and $\QP$,
because $\Pol$ is not full:
\begin{example}
Let the \Zed-map
$\eta\colon[1/4,1/3]\rightarrow[0,2/3]$ be defined by $\eta(x)=8x-2$.
Let the rational point $a\in [1/4,1/3]$ be
such that $[1/4,a]$ is a regular $1$-simplex.
Writing $a=k/l$ for $k, l\in \{1,2,\ldots\}$ with $\gcd(k,l)=1$, it
follows that $l=4k-1$, whence $\eta(a)=8(k/(4k-1))-2=2/(4k-1)$.
%
Thus $[0,2/(4k-1)]=\eta([1/4,a])$ is not regular.
%
As a consequence, there is no
regular $\Delta$ triangulation of $[1/4,1/3]$ such that $\eta(\Delta)$
is a regular triangulation of the simplex $[0,2/3]$.
Since for each morphism $\gamma\colon \mathfrak{W} \rightarrow
\mathfrak{W}'$ of weighted abstract simplicial complexes,
$\Pol(h)$ satisfies
$\Pol(h)(\Delta_{\mathfrak{W}})\subseteq\Delta_{\mathfrak{W}'}$,
we conclude that $\Pol$ is not full.
\end{example}
\section{Properties of the category of Rational Polyhedra}
In this section we will study some properties of the category $\QP$
of rational polyhedra,
and the dual properties of the category of finitely presented unital
$\ell$-groups.
\subsection{$\Zed$-maps and limits}
The category of unital $\ell$-groups is small complete and small co-complete.
This is a consequence of the categorical equivalence between unital
$\ell$-groups and the equational class of MV-algebras,
\cite[Theorem 3.9]{Mu1986}.\footnote{The small completeness (small co-completeness) for equational classes of algebras follows from Birkhoff Theorem (see \cite[Theorem 11.9]{BS1981}) and the construction of limits (co-limits) by products and equalizers (co-products and co-equalizers) (see \cite[\S 5.2. Theorem 1]{McL1969}).}
It follows that finitely presented unital $\ell$-groups are closed
under finite co-limits,
whence, by Theorem \ref{Theo_Baker-Beynon}, $\QP$ is closed under
finite limits.
We next
construct finite limits in $\QP$.
This will be the key tool
to describe monic and epic arrows in the category $\QP$
in Theorem \ref{Theo_monicepi}.
\begin{theorem}[\bf Limits]\label{Teo_QP_SmallComplete}
The category $\QP$ is closed under finite limits.
\end{theorem}
\begin{proof} In view of \cite[\S V.2. Corollary 2]{McL1969}, we
only need to prove that $\QP$ has finite products and equalizers.
\medskip
\noindent {\it Finite products}: It is easy to see that the set $\{1\}$ is the terminal object $\QP$.
Therefore, $\QP$ admits the empty product.
Suppose that $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$ are polyhedra.
%
The product of $P$ and $Q$ in $\Set$,
$$
P\times Q=\left\{ (x,y)\in\R^n\times\R^m\mid x\in P \text{ and } y\in Q \right\}
$$
is a rational polyhedron.
%
Using Lemma \ref{Lem_CarZed}, it is easy to see that the
projections $\pi_{P}$ and $\pi_{Q}$ are $\Zed$-maps.
Suppose that $R\subseteq \R^{l}$ is a polyhedron and $\eta\colon
R\rightarrow P$ and $\mu\colon R\rightarrow Q$ are \Zed-maps.
%
An application of Lemma \ref{Lem_CarZed} shows that
the unique map $(\eta,\mu)\colon R\rightarrow P\times Q$ such that
$\pi_P\circ (\eta,\mu)=\eta$ and $\pi_Q\circ (\eta,\mu)=\mu$ is a \Zed-map.
%
Thus the cone $\xymatrix{P&P\times
Q\ar[l]_{\pi_{P}}\ar[r]^{\pi_{Q}}& Q}$ is universal in $\QP$,
and $P\times Q$ is the product of $P$ and $Q$ in the category $\QP$.
\medskip
\noindent {\it Equalizers}: Let $P\subseteq \R^{n}$ and $Q\subseteq
\R^{m}$ be polyhedra and $\eta,\mu :P\rightarrow Q$ two
$\Zed$-maps. Let
$$
E=\left\{ x\in P\mid\mu(x)=\eta(x)\right\}
$$
be the equalizer of $\mu$ and $\eta$ in $\Set$.
%
To see that $E$ is a polyhedron, for each $i=1,\ldots,m,$
let $f_i\colon P\rightarrow \R$ be defined by
$f_i=|\xi_i\circ \mu-\xi_i\circ\eta|$, where $\xi_i\colon
Q\rightarrow \R$ are
the coordinate maps.
%
Writing $f=f_1+\cdots+f_m$ it follows that $E=f^{-1}(0)$.
%
By Lemma \ref{Lem_ZeroP}, $E$ is a rational polyhedron.
If $R\subseteq \R^{l}$ is a polyhedron and $\nu\colon
R\rightarrow P$ is a \Zed-map such that $\nu\circ \eta=\nu\circ\mu$
then $\nu(R)\subseteq E$.
%
Since inclusions are $\Zed$-maps, the proof is complete.
\end{proof}
\begin{theorem}\label{Theo_monicepi}
Let $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$ be polyhedra and
$\eta : P\rightarrow Q$ a $\Zed$-map.
%
Then
\begin{itemize}
\item[(i)] $\eta$ is monic in $\QP$ iff it is one-one.
\item[(ii)] $\eta$ is epic in $\QP$ iff it is onto.
\end{itemize}
\end{theorem}
\begin{proof}
(i) For the nontrivial direction, suppose
$\eta$ is a monic arrow in $\QP$.
%
If $\eta$ is not one-one (absurdum hypothesis) let
$$
E=\left\{(x,y)\in P\times P\mid\eta(x)=\eta(y)\right\}.
$$
The proof of Theorem \ref{Teo_QP_SmallComplete} shows that
%
$E$ is the equalizer of $\eta\circ\pi_{1}$ and $\eta\circ\pi_{2}$, where
$\pi_{1}$ and $\pi_{2}$ are the projections of $P\times P$ onto $P$.
%
By our assumption, there exists a rational point $(x,y)\in E$
such that $x,y\in P$ and $x\neq y$.
%
Let $d=\den(x)\cdot \den(y)$ and $S=\left\{ \frac{1}{d}
\right\}\subseteq \R$.
%
By Lemma \ref{Lem-LinearMap}, the maps $\mu_{1},\mu_{2}:S\rightarrow P$
defined by $\mu_{1}(\frac{1}{d})=x$ and $\mu_{2}(\frac{1}{d})=y$
are \Zed-maps.
%
Clearly, $\eta\circ\mu_{1}=\eta\circ\mu_{2}$ and $\mu_{1}\neq\mu_{2}$,
a contradiction with the assumption that $\eta$ is monic in $\QP$.
\medskip
(ii) For the nontrivial direction, suppose that $\eta$ is epic in
$\QP$ but is not onto $Q$ (absurdum hypothesis).
%
By Lemma \ref{Lem_ImagesZmorph}, $\eta(P)\subseteq Q$ is a polyhedron.
%
By Lemma \ref{Lem_Triang-Subset},
%
there exists a regular triangulation
$\Delta_{Q}$
of $Q$ such that $\{S\in\Delta_{Q}\mid \ver(S)\subseteq\eta(P)\}$
is a regular triangulation of $\eta(P)$.
%
Let $\mu_{1}\colon Q\rightarrow Q\times [0,1]$ be defined by
$$
\mu_{1}(v)=(v,0).
$$
%
Let $\mu_{2}$ be the unique $\Zed$-map of Corollary
\ref{Cor_ExtensionToZed},
satisfying the following conditions
for each $S\in\Delta_{Q}$ and every $v\in \ver(S)$:
$$
\mu_{2}(v)= \left\{\begin{tabular}{cl}
$(v,0)$ & if $v\in \eta(P),$ \\
$(v,1)$ & if $v\notin \eta(P),$
\end{tabular}\right.
$$
with $\mu_{2}$ being
linear on each simplex of $\Delta_Q$.
It is easy to see that $\mu_{1}\circ\eta=\mu_{2}\circ\eta$ but
$\mu_{1}\neq\mu_{2}$,
a contradiction with the assumption
that $\eta$ is epic. \end{proof}
The foregoing result
is well known to the specialist: in particular,
(ii) can also be derived from
Theorem \ref{Theo_Baker-Beynon}
in combination with \cite[Lemma 3.8]{Mu2011}.
We have given a proof for the
sake of completeness.
\subsection{Cantor-Bernstein-Schr\"oder theorem}\label{SecZedHomeo}
In the previous subsection we
have characterized monic and epic arrows in the category $\QP$.
In \cite[Proposition 3.15]{Mu2011} a characterization of iso-arrows
in $\QP$ is given in terms of preservation of denominators of rational points.
Using this result, in Corollary \ref{Cor_CBS}
we will prove a (dual) Cantor-Bernstein-Schr\"oder theorem for finitely
presented unital $\ell$-groups.
By Theorem \ref{Theo_Baker-Beynon} we immediately have
\begin{lemma}\label{Lem_ZedHomeo}
Let $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$ be polyhedra and
$\eta \colon P\rightarrow Q$ a \Zed-map.
%
Then $\eta$ is a \Zed-homeomorphism in $\QP$ iff
$\McN(\eta)\colon\McN(Q)\rightarrow \McN(P)$ is an isomorphism in
$\FP$.
\end{lemma}
\begin{theorem}\cite[Proposition 3.5]{Mu2011}\label{Theo_TriangZedHomeo}
Let $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$ be polyhedra and
$\eta \colon P\rightarrow Q$ a one-one \Zed-map of $P$ onto $Q$.
Then the following conditions are equivalent:
\begin{itemize}
\item[(i)] $\eta$ is a \Zed-homeomorphism.
\item[(ii)] $\den(\eta(x))=\den(x)$ for each rational point $x\in P$.
\item[(iii)] For each regular simplex $S\subseteq P$, $\eta(S)$ is
a regular simplex of $Q$ and $\den(\eta(x))=\den(x)$ for each $x\in
\ver(S)$.
\item[(iv)] For some (equivalently, for every) regular
triangulation $\Delta$ of $P$ such that $\eta$ is linear on each
simplex of $\Delta$, $\eta(\Delta)$ is a regular triangulation of $Q$
and $\den(\eta(x))=\den(x)$ for each $x\in \ver(\Delta)$.
\end{itemize}
\end{theorem}
\begin{definition}
Let $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$ be polyhedra and
$\eta \colon P\rightarrow Q$ a $\Zed$-map.
%
Then $\eta$ is a {\it strict $\Zed$-map} if it is a
$\Zed$-homeomorphism onto its range.
\end{definition}
From Theorem~\ref{Theo_TriangZedHomeo} we obtain:
\begin{corollary}\label{Cor-PreserDen}
Let $P\subseteq \R^{n}$ and $Q\subseteq\R^{m}$ be
polyhedra and $\eta\colon P\rightarrow Q$ a one-one $\Zed$-map.
%
Then the following conditions are equivalent:
\begin{itemize}
\item[(i)] $\eta$ is a strict \Zed-map.
\item[(ii)] $\den(\eta(x))=\den(x)$ for each rational point $x\in P$.
\item[(iii)] For each regular simplex $S\subseteq P$, $\eta(S)$
is a regular simplex of $Q$ and $\den(\eta(x))=\den(x)$ for each
$x\in \ver(S)$.
\item[(iv)] For some (equivalently, for every) regular
triangulation $\Delta$ of $P$ such that $\eta$ is linear
on each simplex of $\Delta$, $\eta(\Delta)$ is a regular
triangulation of $\eta(P)$ and $\den(\eta(x))=\den(x)$ for each
$x\in \ver(\Delta)$.
\end{itemize}
\end{corollary}
By Theorem \ref{Theo_monicepi}, every strict $\Zed$-map is a monic
$\Zed$-map, but the converse does not hold in general.
From Theorem \ref{Theo_Baker-Beynon}, monic \Zed-maps correspond to epi unital $\ell$-homomorphisms.
The following theorem
shows that strict \Zed-maps correspond to onto (or equivalently regular epi)
unital
$\ell$-homomorphisms:
\begin{theorem}\label{Theo_StricOnto}
Let $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$ be polyhedra and
$\eta\colon P\rightarrow Q$ be a \Zed-map.
%
Then $\eta$ is a strict $\Zed$-map iff
$\McN(\eta)\colon\McN(Q)\rightarrow \McN(P)$ is an onto map.
\end{theorem}
\begin{proof}
$(\Rightarrow)$
%
In order to prove that $\McN(\eta)$ is onto $\McN(P)$, let $f\in\McN(P)$.
%
By Proposition \ref{proposition:poly} and Lemma
\ref{Lem_StableTriangul}, there exists a regular triangulation
$\Delta_P$ of $P$ such that $\eta$ and $f$ are linear over each
simplex $S\in\Delta.$
%
Since $\eta$ is a strict $\Zed$-map, by Theorem \ref{Theo_TriangZedHomeo},
$\eta(\Delta_P)$ is a regular triangulation of $\eta(P)$.
%
By Lemma \ref{Lem_Triang-Subset} there exists a regular triangulation
$\Delta_Q$
of $Q$ such that the set $\nabla=\{S\in\Delta_Q\mid \ver(S)\subseteq
\eta(P)\}$ is a subdivision of $\eta(\Delta_P)$. Then
$\eta^{-1}\colon\eta(P)\rightarrow P$ is linear over each simplex of
$\nabla$. By Theorem \ref{Theo_TriangZedHomeo},
$\eta^{-1}(\nabla)$ is a regular triangulation of $P$ and is a
subdivision of $\Delta_P$.
Let $g\in\McN(Q)$
be
uniquely determined by the following conditions:
\begin{enumerate}
\item $g$ is linear over each simplex $S\in\Delta_Q,$
\item for every $v\in \ver(\Delta_Q)$,
$$
g(v)=\left\{\begin{tabular}{ll}
$f(\eta^{-1}(v))$ & if $v\in\eta(P),$ \\
$0$ & if $v\not\in\eta(P).$
\end{tabular}\right.
$$
\end{enumerate}
%
The existence and uniqueness of $g$
is ensured by Lemma~\ref{Lem-LinearMap}.
It follows that
$g\circ\eta$ is linear over each $S\in\eta^{-1}(\nabla)$.
%
For every $x\in \eta^{-1}(\nabla)$ we can write
$$
g\circ\eta(x)=g(\eta(x))=f(\eta^{-1}(\eta(x)))=f(x).
$$
Therefore, $g\circ\eta=h(g)=f$, and $\McN(\eta)$ is onto.
%
\medskip
$(\Leftarrow)$
Assume $\McN(\eta)$ is an onto map.
%
By Lemma \ref{Lem_ImagesZmorph}(ii),
$$
\McN(P)\cong\McN(Q)/\ker(\McN(\eta))\cong\McN(\eta(P)).
$$
By Lemma~\ref{Lem_ZedHomeo}, $\eta\colon P\rightarrow\eta(P)$ is a
$\Zed$-homeomorphism.
\end{proof}
\begin{theorem}
Let $P\subseteq \R^{n}$ be a polyhedron and $\eta\colon P\rightarrow P$
a one-one (equivalently, a monic) $\Zed$-map.
%
Then $\eta$ is a $\Zed$-homeomorphism.
\end{theorem}
\begin{proof}
By Theorem \ref{Theo_TriangZedHomeo}, it is enough to prove that
$\eta$ preserves denominators and is onto $P$.
%
For each $k=1,2,\ldots$, let $P_{k}=\{x\in P\cap \mathbb{Q}^{n}\mid
\den(x)=k\}$.
%
Since $P$ is a bounded set, each $P_{k}$ is a finite set.
\smallskip
\noindent {\it Claim}: $\eta$ preserves denominators.
Equivalently, $\eta(P_{k})=P_{k}$ for each $k=1,2,\ldots$.
The proof is by induction on $k$.
%
For the basis case, let $x\in P_{1}$.
%
Then $\den(\eta(x))$ divides $\den(x)=1$, i.e. $\eta(P_{1})\subseteq P_{1}$.
%
Since $\eta$ is one-one and $P_{1}$ is finite, $\eta(P_{1})=P_{1}.$
For the induction step, suppose that for every $j<k$, $\eta(P_{j})=P_{j}$.
%
Let $x\in P_{k}$. %
%
Since $\den(\eta(x))$ divides $k$, then $\den(\eta(x))\leq k$.
%
Assume $\eta(x)\not\in P_{k}$ (absurdum hypothesis).
%
Then $\eta(x)\in P_{k'}$ for some $k'<k$.
By hypothesis, there exists $y\in P_{k'}$ such that $\eta(y)=\eta(x)$.
%
Since $\den(x)=k'\neq k= \den(y)$, then
$x\neq y$, thus contradicting the fact that $\eta$.
Our claim is settled.
\smallskip
To see that $\eta$ is onto $P$, first let us
observe that since $P$ is a rational polyhedron,
$$
P={\rm cl}(\bigcup_{k\geq 1} P_{k}),
$$
where ${\rm cl}$ denotes topological closure.
%
The continuity of $\eta$ now yields
\begin{eqnarray}
\nonumber \eta(P) &=& \textstyle\eta({\rm cl}(\bigcup_{k\geq 1}
P_{k})) = {\rm cl}(\eta(\bigcup_{k\geq 1}P_{k}))\\
\nonumber &=& \textstyle{\rm cl}(\bigcup_{k\geq
1}\eta(P_{k})) = {\rm cl}(\bigcup_{k\geq 1} P_{k})\\
\nonumber &=& P,
\end{eqnarray}
%
i.e. $\eta$ is onto $P$. The proof is complete.
\end{proof}
\begin{theorem}
Let $P\subseteq \R^{n}$ and $Q\subseteq \R^{m}$ be polyhedra,
and $\eta
\colon P\rightarrow Q$ and $\mu\colon Q\rightarrow P$ be one-one
\Zed-maps.
%
Then $P$ is $\Zed$-homeomorphic to $Q$.
\end{theorem}
\begin{proof}
Since $\nu=\mu\circ\eta$ is a one-one $\Zed$-map from $P$ into
itself, by the previous theorem $\nu$ is a \Zed-homeomorphism.
We claim that $\nu^{-1}\circ\mu\colon Q\rightarrow P$ is the inverse of $\eta$.
%
Denoting by $Id_{P}$ and $Id_{Q}$
the identity maps over $P$ and $Q$, we get
$$
(\nu^{-1}\circ\mu)\circ\eta=\nu^{-1}\circ(\mu\circ\eta)=\nu^{-1}\circ(\nu)=Id_{P}.
$$
Symmetrically,
$
\mu\circ(\eta\circ\nu^{-1}\circ\mu)=\mu=\mu\circ Id_{Q}.
$
Since $\mu$ is a one-one $\Zed$-map, then
$\eta\circ(\nu^{-1}\circ\mu)=Id_{Q}.$
\end{proof}
As a corollary we obtain a (dual) Cantor-Bernstein-Schr\"oder
theorem for finitely presented unital $\ell$-groups:
\begin{corollary}\label{Cor_CBS} For any finitely presented unital $\ell$-groups
$(G_1,u_1)$ and $(G_2,u_2)$ the following conditions are equivalent:
\begin{enumerate}
\item $(G_1,u_1)$ and $(G_2,u_2)$ are isomorphic.
\item There are onto homomorphisms
$f\colon(G_1,u_1)\rightarrow(G_2,u_2)$ and
$g\colon(G_2,u_2)\rightarrow(G_1,u_1)$.
\item There are epic homomorphisms
$h\colon(G_1,u_1)\rightarrow(G_2,u_2)$ and
$l\colon(G_2,u_2)\rightarrow(G_1,u_1)$.
\end{enumerate}
\end{corollary}
It is easy to check
that if in item (ii) we replace ``onto'' by ``one-one'' the result
is no longer valid.
\subsection{Co-limits}
Having constructed finite limits in the category of rational polyhedra,
we devote this section to a special type of co-limits that will find
use in the rest of the paper.
{}From
Lemmas
\ref{Lem_Triang-Subset},
\ref{Lem_Sub-Triang-Subset},
\ref{Lem_StableTriangul}
and Theorem \ref{Theo_TriangZedHomeo} we obtain
\begin{lemma}\label{Lem_TrianofVstrictmap}
Let $P\subseteq \R^{n}$, $Q\subseteq \R^{m}$ and $D\subseteq
\R^{k}$ be polyhedra and
$\eta\colon P\rightarrow Q$ and $\mu\colon P\rightarrow D$ be
strict $\Zed$-maps. Then there exist
regular triangulations $\Delta_P$, $\Delta_Q$, $\Delta_R$ of $P$, $Q$,
$R$ satisfying the following conditions:
\begin{itemize}
\item[(i)] $\eta$ and $\mu$ are linear over each simplex of $\Delta_{P}$;
\item[(ii)] $\eta(\Delta_P)$ is a full subcomplex of $\Delta_Q$;
\item[(iii)] $\mu(\Delta_P)$ is a full subcomplex of $\Delta_R$.
\end{itemize}
Moreover, for any regular triangulations
$\nabla_P$, $\nabla_Q$, $\nabla_R$
of $P$, $Q$, $R$ we may insist that
$\Delta_P$, $\Delta_Q$, $\Delta_R$ subdivide
$\nabla_P$, $\nabla_Q$, $\nabla_R$, respectively.
\end{lemma}
\begin{theorem}[\bf Co-limits]\label{Theo-Colimits}
$\QP$ has finite co-products and pushouts of strict \Zed-maps.
\end{theorem}
\begin{proof}
{\it Finite co-products}:
Considered as a rational polyhedron,
the empty set is the intial object of $\QP$, and is also the empty
co-product.
Next, for any polyhedra
$P\subseteq \R^{m}$ and $Q\subseteq \R^{n}$ let
$k={\rm max}\{m,n\}$ and
$$
C=\left\{ (x,0_{m+1},\ldots,0_{k+1})\mid x\in P\right\}\cup
\left\{(y,1_{n+1},\ldots,1_{k+1})\mid y\in Q\right\}.
$$
%
Then
$C$ is the co-product of $P$ and $Q$ in $\Set$ and the inclusion maps
$$
\begin{tabular}{ccc}
$i_{P}\colon P\rightarrow C$ & and & $i_{Q}\colon Q\rightarrow C$ \\
$i_{P}(x)=(x,0_{m+1},\ldots,0_{k+1})$ & &
$i_{Q}(y)=(y,1_{n+1},\ldots,1_{k+1})$
\end{tabular}
$$
are $\Zed$-maps.
Let $R\subseteq \R^{l}$ be a polyhedron and $\eta\colon
P\rightarrow R$ and $\mu\colon Q\rightarrow R$ be \Zed-maps.
%
By
Lemma \ref{Lem_CarZed},
the unique map $\langle\eta,\mu\rangle\colon C\rightarrow R$
defined by
$\langle\eta,\mu\rangle\circ i_P=\eta$ and
$\langle\eta,\mu\rangle\circ i_Q=\mu$ is a \Zed-map.
%
Then $C$ is the co-product of $P$ and $Q$ in the category $\QP$.
%
\medskip
\noindent {\it Pushouts of strict \Zed-maps}:
Let $P\subseteq \R^{n}$, $Q\subseteq \R^{m}$ and $D\subseteq
\R^{k}$ be polyhedra and
$\eta\colon P\rightarrow Q$ and $\mu\colon P\rightarrow D$ be
strict $\Zed$-maps.
%
With reference to Lemma \ref{Lem_TrianofVstrictmap},
let $\Delta_{P}$, $\Delta_{Q}$ and $\Delta_{R}$ be regular
triangulations of $P$, $Q$ and $R$
such that $\eta$ and $\mu$ are linear over each simplex of $\Delta_{P}$,
$\eta(\Delta_P)$ is a full subcomplex of $\Delta_Q$,
and $\mu(\Delta_P)$ is a full subcomplex of $\Delta_R$.
%
Let $f=\eta\restrict{\ver(\Delta_P)}$ and $g=\mu\restrict{\ver(\Delta_P)}$.
%
By definition,
$f\colon \mathfrak{W}(\Delta_P)\rightarrow \mathfrak{W}(\Delta_Q)$
and $g\colon \mathfrak{W}(\Delta_P)\rightarrow
\mathfrak{W}(\Delta_R)$ are d-maps
and the following diagram commutes:
$$
\xymatrix{
{}&{Q\ar[rr]^{\iota_{\Delta_Q}}}&{}&{\Pol(\mathfrak{W}_{\Delta_Q})}\\
{P\ar[ur]^{\eta}\ar[dr]^{\mu}\ar[rr]^{\iota_{\Delta_P}}}&{}&{\Pol(\mathfrak{W}_{\Delta_P})\ar[ur]^{\Pol(f)}\ar[dr]^{\Pol(g)}}&{}\\
{}&{R\ar[rr]^{\iota_{\Delta_R}}}&{}&{\Pol(\mathfrak{W}_{\Delta_R})}\\
}
$$
Since $\eta$ and $\mu$ are one-one, then so are
$f$ and $g$, and we can write
\begin{center}
\begin{tabular}{lcl}
$\ver(\Delta_{P})$ & $=$ & $\{v_{1},\ldots,v_{r}\}$ \\
$\ver(\Delta_{Q})$ & $=$ &
$\{f(v_{1}),\ldots,f(v_m),w_{1},\ldots,w_{s}\}$ \\
$\ver(\Delta_{R})$ & $=$ &
$\{g(v_{1}),\ldots,g(v_m),z_{1},\ldots,z_{t}\}$
\end{tabular}
\end{center}
Let $W=\ver(\Delta_{Q})\setminus f(\ver(\Delta_{P}))$ and
$Z=\ver(\Delta_{R})\setminus g(\ver(\Delta_{P}))$.
%
Without loss of generality,
$W\cap Z=\emptyset$.
We now define the weighted abstract simplicial complex
$\mathfrak{W}=\langle V,\Sigma,\omega\rangle$ by the following stipulation:
\begin{itemize}
\item $V=\ver(\Delta_{P})\cup W\cup Z $;
\item $\omega(v)=\den(v)$ for each $v\in V$;
\item $X\in\Sigma$ if
\begin{itemize}
\item[(a)] either $X\subseteq \ver(\Delta_{P})\cup W$ and
$f(X\cap \ver(\Delta_{P}))\cup (X\cap W)\in\Sigma_{\Delta_{Q}}$, or
\item[(b)]$X\subseteq \ver(\Delta_{P})\cup Z$ and $g(X\cap
\ver(\Delta_{P}))\cup (X\cap Z)\in\Sigma_{\Delta_{R}}$.
\end{itemize}
\end{itemize}
Let $i_Q\colon \ver(\Delta_Q)\rightarrow V$ and
$i_R\colon \ver(\Delta_R)\rightarrow V$ be defined as follows:
$$
i_Q(v)=\left\{\begin{tabular}{ll}
$v$& if $v\in W$\\
$w$& if $w\in \ver(\Delta_{P})$ and $v=f(w)$,
\end{tabular}\right.
$$
$$
i_R(v)=\left\{\begin{tabular}{ll}
$v$& if $v\in Z$\\
$w$& if $w\in \ver(\Delta_{P})$ and $v=g(w)$.
\end{tabular}\right.
$$
By definition of $\Sigma$ and $\omega$, the
maps
$i_Q$ and $i_R$ preserve weights,
whence they are morphisms in $\wasc$.
%
By Corollary \ref{Cor-PreserDen},
$\Pol(i_Q)$ and $\Pol(i_R)$ are strict \Zed-maps and
we have a commutative diagram
$$
\xymatrix{
{}&{Q\ar[rr]^{\iota_{\Delta_Q}}}&{}&{\Pol(W_{\Delta_Q})\ar[dr]^{\Pol(i_Q)}}&{}\\
{P\ar[ur]^{\eta}\ar[dr]^{\mu}\ar[rr]^{\iota_{\Delta_P}}}&{}&{\Pol(W_{\Delta_P})\ar[ur]^{\Pol(f)}\ar[dr]^{\Pol(g)}}&{}&{\Pol(W)}\\
{}&{R\ar[rr]^{\iota_{\Delta_R}}}&{}&{\Pol(W_{\Delta_R})\ar[ur]^{\Pol(i_R)}}&{}\\
}
$$
\medskip
\medskip
Observe that $\rho_Q=\Pol(i_Q)\circ \iota_{\Delta_Q}$ and
$\rho_R=\Pol(i_R)\circ \iota_{\Delta_R}$ are strict \Zed-maps. Since
$i_Q(f(v))=i_R(g(v))=v$ for each $v\in \ver(\Delta_P)$, we have
$\rho_Q\circ \eta=\rho_R\circ \mu=\rho_P$.
\medskip
We {\it claim} that for all $x\in Q$ and $y\in R$ with $\rho_Q(x)=\rho_R(y)$,
there exists $z\in P$ such that $\eta(z)=x$ and $\mu(z)=y$.
As a matter of fact,
let $S\in \Sigma_{\Delta_{Q}}$
and $T\in \Sigma_{\Delta_{R}}$
satisfy $x\in S$
and $y\in T$.
Then $\rho_Q(x)=\rho_R(y)\in\rho_Q(S)\cap\rho_R(T)\in \Sigma$, whence
$\rho_Q(S)\cap\rho_R(T)=\conv(\rho_Q(\ver(S))\cap \rho_R(\ver(T)))$.
Our assumptions about
$i_Q$, $i_R$ and $W\cap Z=\emptyset$ are to the effect
that
$\rho_Q(\ver(S))\cap \rho_R(\ver(T))\subseteq\ver(\Delta_P)$.
Then there exists $z\in P$ such that $\rho_P(z)=
\rho_Q\circ \eta(z)=\rho_R\circ \mu(z)=\rho_Q(x)=\rho_R(y)$.
Since $\rho_Q$ and $\rho_R$ are one-one, $\eta(z)=x$ and $\mu(z)=y$, as claimed.
\medskip
We have proved:
\begin{equation}
\rho_Q(Q)\cap\rho_R(R)=\rho_Q\circ f(P)=\rho_R\circ
g(P)=\rho_P(P).\label{Eq_Amalgam}
\end{equation}
In order to prove that $\Pol(\mathfrak{W})$ is the pushout $Q\coprod_P R$ in $\wasc$,
let $U$ be a rational polyhedron, together with \Zed-maps
$\gamma_Q\colon Q\rightarrow U$ and $\gamma_R\colon R\rightarrow U$
such that $\gamma_Q\circ\eta=\gamma_R\circ\mu$.
%
Repeated applications
of Lemma \ref{Lem_StableTriangul} provide regular triangulations $\nabla_Q$
of $Q$ and
$\nabla_R$ of $R$ such that $\gamma_Q$ and $\rho_Q$ are linear on
each simplex of $\nabla_Q$, and $\gamma_R$ and $\rho_R$ are linear
on each simplex of $\nabla_R$.
%
Lemma \ref{Lem_TrianofVstrictmap} yields regular triangulations
$\Lambda_P$, $\Lambda_Q$, $\Lambda_R$ of $P$, $Q$, $R$
such that
\begin{itemize}
\item[(i)] $\eta$ and $\mu$ are linear on each simplex of $\Lambda_P$;
\item[(ii)] $\eta(\Lambda_P)\subseteq \Lambda_Q$ and
$\mu(\Lambda_P)\subseteq \Lambda_R$;
\item[(iii)] $\Lambda_Q$, $\Lambda_R$ are subdivisions of
$\nabla_Q$, $\nabla_R$.
\end{itemize}
Thus
$\gamma_Q$ and $\rho_Q$ are linear on each simplex of $\Lambda_Q$,
and $\gamma_R$ and $\rho_R$ are linear on each simplex of
$\Lambda_R$.
%
As a consequence,
$\rho_Q(\Lambda_Q)\cap\rho(\Lambda_R)=\rho_Q\circ\eta(\Lambda_P)=\rho_R\circ\mu(\Lambda_P)$
is a regular triangulation of
$\rho_Q(Q)\cap\rho_R(R)=\rho_Q\circ\eta(P)=\rho_R\circ\mu(P)$.
%
Thus, $\rho_Q(\Lambda_Q)\cup\rho(\Lambda_R)=\Lambda$ is a well
defined simplicial complex.
%
Moreover, $\Lambda$ is a regular triangulation of $\Pol(W)$.
Finally, let $\zeta\colon \Pol(W)\rightarrow U$ be the unique
\Zed-map given by Corollary \ref{Cor_ExtensionToZed} which is linear
over each simplex of $\Lambda$ and
also satisfies
$$
\zeta(v)=\left\{\begin{tabular}{rl}
$\gamma_Q(x)$& if $v=\rho_Q(x)$\\
$\gamma_R(y)$& if $v=\rho_R(y)$,
\end{tabular}\right.
$$
on each vertex $v$ of $\Lambda$.
Since $\rho_Q(\nabla_Q)\cap\rho(\nabla_R)=\rho_Q\circ\eta(\nabla_P)$
then $\zeta$ is well defined.
For any vertex
$x$ of $\nabla_Q$,
$\gamma_Q(x)=\zeta(\rho(x))$.
Since $\gamma$ is linear over each simplex of
$\nabla_Q$ then $\gamma_Q=\zeta\circ\rho_Q$.
Similarly, $\gamma_R=\zeta\circ\rho_R$,
which proves that $\Pol(W)$ is the pushout $Q\coprod_P R$ in $\wasc$.
\end{proof}
\begin{corollary}
\label{corollary:scazonte}
Let $(G_1,u_1)$, $(G_2,u_2)$, $(G_3,u_3)$ be
finitely presented unital $\ell$-groups
with onto homomorphisms
$f\colon G_1\rightarrow G_3$, $g\colon G_2\rightarrow G_3$.
Then the fiber product $G=\{(a,b)\in G_1\times G_2\mid f(a)=g(b)\}$
(with $(u_1,u_2)$ as the distinguished order unit)
is a finitely presented unital $\ell$-group.
\end{corollary}
\begin{proof}
The result
follows from Theorems \ref{Theo_Baker-Beynon}, \ref{Theo_StricOnto}
and \ref{Theo-Colimits},
upon observing that for all rational polyhedra $P,Q,R$ and strict
\Zed-maps $\eta\colon P\rightarrow Q$, $\mu\colon P\rightarrow R$ the
unital $\ell$-group $\McN(Q\coprod_P R)$ is isomorphic to the fiber
product $\{(f,g)\in \McN(Q)\times \McN(R)\mid f\circ\eta=g\circ\mu\}$.
\end{proof}
In Corollary \ref{Cor_FiberProj} we will prove that if in
Corollary \ref{corollary:scazonte}
we also assume
that $(G_1,u_1)$, $(G_2,u_2)$, and $(G_3,u_3)$ are projective, then so is
$G$.
\section{Exact unital $\ell$-groups}
Working in the framework of intuitionistic logic, in
his paper \cite{DJ1982}, de Jongh calls ``exact'' a formula
$\varphi$ such that the Heyting algebra
presented by $\varphi$ is embeddable into a free Heyting algebra
(see also Section \ref{Sec_admissible}).
Accordingly, in this paper we say that
a unital $\ell$-group $(G,u)$ is {\it exact} if it is finitely presented
and there exist a positive integer $n$ and a one-one unital
$\ell$-homomorphism $g $ of $(G,u)$ into the free
unital $\ell$-group $\McN([0,1]^{n})$. By Lemma \ref{Lem_ImagesZmorph0} and Theorem \ref{Theo_Baker-Beynon}, a unital $\ell$-group is exact iff it is isomorphic to a finitely generated unital $\ell$-subgroup of $\McNn$. This equivalent definition can also be obtained as an application of the equivalence between MV-algebras and unital $\ell$-groups and
\cite[Corollary 6.6]{Mu2011}.
In this section we will give a characterization of exact unital $\ell$-groups.
A unital $\ell$-group $(G,u)$ is {\it projective} if whenever
$\psi\colon (G_1,u_1)\rightarrow(G_2,u_2)$ is a unital
$\ell$-homomorphism onto $(G_2,u_2)$
and $\phi\colon (G,u)\to(G_2,u_2)$ is a unital $\ell$-homomorphism, there
is a unital $\ell$-homomorphism $\theta\colon (G,u)\to(G_1,u_1)$ such that
$\phi= \psi \circ \theta$.
A finitely generated unital $\ell$-group $(G,u)$ is projective iff it is
a {\it retraction} of the free unital $\ell$-group $\McNn$ for some
$n\in\{1,2,\ldots\}$.
In other words, there is a homomorphism $\iota\colon (G,u)\to \McNn$ and a
homomorphism $\sigma\colon \McNn \to (G,u)$ such that
$\sigma\circ\iota$ is the identity map $Id_{G}$
on $G$.
Lemma \ref{Lem_ImagesZmorph} yields the following inclusions
for unital $\ell$-groups:
\begin{equation}
\label{Eq_Inclusions}
\mbox{Finitely Presented }\supseteq\mbox{ Exact }\supseteq\mbox{
Finitely generated projective}.
\end{equation}
As shown in
\cite{CM20XX}, the class of finitely generated projective unital
$\ell$-groups (includes but)
does not coincide with the class of finitely presented unital $\ell$-groups.
An example of an exact unital
$\ell$-group which is not projective is as follows:
\begin{example}
Let $P=\{(x,y)\in[0,1]^{2}\mid x\in\{0,1\}\mbox{ or } y\in\{0,1\}\}$.
The map $\eta\colon [0,1]\rightarrow P$ defined by
$$
\eta(a)=\left\{\begin{tabular}{ll}
$(3x,0)$& if $0\leq a\leq 1/3$,\\
$(1,6x-2)$& if $1/3\leq a\leq {1/2}$,\\
$(4-6x,1)$& if $1/2\leq a\leq {2}/{3}$,\\
$(0,3-3x)$& if ${2}/{3}\leq a\leq 1$,
\end{tabular}\right.
$$
is a \Zed-map onto $P$. By Theorem \ref{Theo_monicepi},
$\McN(\eta)\colon\McN(P)\rightarrow \McN([0,1])$ is one-one.
Then $\McN(P)$ is exact. Since $P$ is not simply connected, by
\cite[Theorem 4.2]{CM20XX} $\McN(P)$ is not projective.
\end{example}
As is well known (see \cite{Be1977A}),
for $\ell$-groups the three notions in
(\ref{Eq_Inclusions}) coincide.
\subsection{Strongly regular triangulations}
The notion of {\it strongly regular triangulation} was introduced in \cite[Definition 3.1]{CM20XX} and it has a key role in our characterization of exact unital $\ell$-groups.
\begin{definition}\label{definition:strongly}
A simplex $S$ is said to be {\it strongly regular} if it is regular
and the greatest common divisor of the denominators of the vertices of $S$
is equal to
$1$.
%
A triangulation $\Delta$ of a polyhedron $P\subseteq \cube$ is
said to be {\it strongly regular}
if each maximal simplex of $\Delta$ is strongly regular.
\end{definition}
\begin{lemma}\label{Lem_subdiv preserves den}
Let $S$ be a regular k-simplex.
%
Then for every regular k-simplex $T$ such that $T\subseteq S$,
the greatest common divisor of the denominators of the vertices of $T$
is equal to the greatest common divisor of the denominators of the
vertices of
$S$.
\end{lemma}
\begin{proof}
In view of Lemma \ref{lemma:weakoda} it is no loss of
generality to assume that
$T$ is one of the maximal simplexes obtained by blowing up
$S$ at the Farey mediant $v$ of its vertices.
%
Let $v_1,\ldots,v_k$ be the vertices of $S$.
%
Since $\den(v)$ is equal to $\sum_{j=0}^k \den(v_j)$, then
for each $i=1,\ldots,k$
the
greatest common divisor of the integers
$\den(v_1),\ldots,\den(v_k)$ coincides with the greatest common
divisor of the set of integers
$$
\{ \den(v_0),\ldots,\den(v_{i-1}), \den(v),\den(v_{i+1}),\ldots,\den(v_k)\}
$$
Thus the
greatest common divisor of the denominators of the vertices $T$
is equal to the greatest common divisor of the denominators of the
vertices $S$.
\end{proof}
The following result was first proved in \cite[Lemma 3.2]{CM20XX}:
\begin{corollary}
\label{Cor-BlowupPreserGCD}
Let $\Delta$ and $\nabla$ be regular triangulations of a
polyhedron $P\subseteq\cube$.
%
Then $\Delta$ is strongly regular iff so is $\nabla$.
%
\end{corollary}
Since strong regularity does not depend on the regular triangulation
$\Delta$ of $P$,
without fear of ambiguity we may say that a polyhedron $P$
is {\it strongly regular} if some (equivalently, each) of its regular
triangulations
is strongly regular.
\begin{example}\label{Ex_CubeStronglyTriang}
The $n$-cube $\cube$ is strongly regular.
To see this, let us equip the set $\{0,1\}^{n}$ with the following
partial order $(a_1,\ldots,a_1)\leq (b_1,\ldots,b_n)$ iff
$a_i\leq b_i$ for each $i=1,\ldots,n$.
%
Let $\Delta$ be the triangulation of $\cube$ formed by the
simplexes $\conv(C)$ whenever $C$ is a chain in the poset
$(\{0,1\}^{n},\leq)$.
This is called the {\em standard triangulation} of the cube in
\cite[p. 60]{Se1982}.
Since the denominator of every vertex of $\Delta$ is $1$, it
follows that $\Delta$ is strongly regular. The desired conclusion now
follows from Corollary \ref{Cor-BlowupPreserGCD}.
\end{example}
For all
$v,w\in\R^{n}$, we let $\dist(v,w)$ denote
their Euclidean distance in $\R^{n}$.
For each $0\leq\delta\in\R$ and $v\in\R^{n}$, we use
the notation
$B(\delta,v)=\{w\in\R^{n}\mid \dist(v,w)<\delta\}$.
The
dimension
of the ambient space will always be clear from the context.
\medskip
{}From
Lemma \ref{Lem_subdiv preserves den}
we obtain the following characterization of strong regularity:
\begin{corollary}\label{Cor_SRandCoprvectors}
Let $P\subseteq\R^{n}$ be a polyhedron. Then the following
conditions are equivalent:
\begin{itemize}
\item[(i)] $P$ is strongly regular.
\item[(ii)] For each $v\in P$ and each $0<\delta\in\R$, there exists
$w\in P$ such that $\dist(v,w)<\delta,$ and $\den(v)$ and $\den(w)$ are
{\em coprime}, in the sense that ${\rm gcd}(\den(v),\den(w))=1$.
\end{itemize}
\end{corollary}
For any set $T\subseteq\R^{n}$, we let
${\rm aff}(T)$ denote the {\it affine hull} of $T$, i.e.
$${\rm
aff}(T)=\left\{\sum_{i=0}^{m}\lambda_i v_i\mid \mbox{for some }v_i\subseteq T, \lambda_i\in\R, \sum_{i=0}^{m}\lambda_i=1\mbox{ and }m=1,2,\ldots
\right\}.
$$
Further, ${\rm relint}(T)$ denotes the
relative interior of $T$, that is, the interior of $T$ in the relative
topology of ${\rm aff}(T)$.
\bigskip
For later use in the proofs of
Theorems \ref{Theo_StrongPreserved}
and \ref{Thm_AnchvsStReg}, we
record here the following elementary
characterization:
\begin{lemma}\label{Lem_MaxSimpTriang}
Let $\Delta$ be a triangulation of a polyhedron $P\subseteq \R^{n}$
and $T\in \Delta$. Then the following conditions are equivalent:
\begin{itemize}
\item[(i)] $T$ is maximal in $\Delta$.
\item[(ii)] Whenever $w\in{\rm relint}(T)$ and
$v\in\R^{n}$ does not lie in the affine hull
of $T$, then $\conv(v,w)$ is not contained in $P$.
\item[(iii)] For every $v\in{\rm relint}(T)$ there exists
$0<\delta\in\R$ such that $B(\delta,v)\cap P\subseteq T$.
\end{itemize}
\end{lemma}
\begin{theorem}\label{Theo_StrongPreserved}
Let $P$ and $Q$ be polyhedra and $\eta\colon P\rightarrow Q$
be a \Zed-map onto $Q$.
%
If $P$ has a strongly regular triangulation then $Q$ has a strongly
regular triangulation.
\end{theorem}
\begin{proof}
Let $\Delta$ be a regular triangulation of $Q$ and $S$ a maximal
simplex of $\Delta$.
%
Let $d$ denote the greatest common divisor of the denominators of the vertices of $S$.
%
Let $v\in {\rm relint}(S)$. By
Lemma \ref{Lem_MaxSimpTriang} there exists $0<\epsilon\in\R$ such
that $B(\epsilon,v)\cap Q\subseteq S$.
%
Since $\eta$ is a continuous onto map,
there exist $w\in P$ and $0<\delta\in\R$ such that
$\eta(B(\delta,w)\cap P)\subseteq B(\epsilon,v)$.
Since $P$ is strongly regular,
Corollary \ref{Cor_SRandCoprvectors}
yields $z\in B(\delta,w)\cap P$ such that $\den(w)$ and $\den(z)$
are coprime. Then $\eta(w)=v,\eta(z)\in S$,
and $d$ is a common divisor of
$\den(\eta(w))$ and $\den(\eta(z))$.
%
By Corollary \ref{Cor_Div_Denominators}, $d$ is a
common divisor of
$\den(w)$ and $\den(z)$.
%
We conclude that $d=1$.
\end{proof}
\subsection{Finitely generated projective unital $\ell$-groups}
We now
collect some definitions and results from \cite{CM20XX} that will be
necessary for the geometrical description of exact unital
$\ell$-groups in Theorem \ref{Theo_WeakProj}.
In
Theorem \ref{Thm_AmalProj} it is also proved that
projectiveness is preserved under fiber products of onto homomorphisms
of unital $\ell$-groups.
A $\Zed$-map $\sigma\colon P\rightarrow P$ is a
{\it \Zed-retraction of P} if $\sigma\circ\sigma = \sigma$.
The rational polyhedron $R=\sigma(P)$ is said to be a
\Zed{\it-retract of $P$}.
A rational polyhedron $Q$ is said to be a {\it \Zed-retract} if it is
a \Zed-retract of $[0,1]^{n}$ for some $n\in\{1,2,\ldots\}$.
The following is a consequence of Theorem \ref{Theo_Baker-Beynon}
(see \cite[Theorem 1.2]{CM2009} for details):
\begin{theorem}
A unital $\ell$-group $(G,u)$ is finitely generated projective iff it is isomorphic to $\McN(P)$ for some \Zed-retract $P$.
\end{theorem}
A simplex $T\in\Sigma$
of
an abstract simplicial complex $\langle V,\Sigma\rangle$
is said to have a {\it free face} $F$
if
\begin{itemize}
\item $\emptyset\neq F\subseteq T$ is a {\it facet} (=maximal proper subset)
of $T$, and
\item whenever $F\subseteq S\in\Sigma$ then $S=F$ or $S=T$.
\end{itemize}
It follows that $T$ is a maximal simplex of $\Sigma$, and the removal
from $\Sigma$ of both $T$ and $F$ results in the subcomplex $\langle
V',\Sigma'=\Sigma\setminus\{T,F\}\rangle$ of $\langle
V,\Sigma\rangle$, where $V'=V\setminus F$ if $F$ is a singleton and
otherwise $V'=V$.
The transition from $\langle V,\Sigma\rangle$ to $\langle
V',\Sigma'\rangle$ is called an {\it (abstract) elementary collapse}.
If a simplicial complex $\langle W,\Gamma\rangle$ can be obtained
from $\langle V,\Sigma\rangle$ by a sequence of elementary collapses
we say that $\langle V,\Sigma\rangle$ {\it collapses to} $\langle
W,\Gamma\rangle$.
We say that
the simplicial complex
$\langle V,\Sigma\rangle$ is {\it collapsible} if it
collapses to the abstract simplicial complex consisting of one of its
vertices (equivalently,
any of its vertices \cite[p.248]{Wh1939}).
See \cite[\S III, Definition 7.2]{Ew1996}, \cite[p.247]{Wh1939} for the geometrical
counterpart of collapsibility.
For the purposes of this paper it is enough to observe that a regular triangulation
$\Delta$ is collapsible iff its skeleton
$\mathfrak{W}(\Delta)$ is collapsible.
\begin{theorem}\cite[Theorem 6.1]{CM20XX}\label{Thm_collapsible}
Let $P\subseteq [0,1]^{n}$ be a polyhedron. Suppose
\begin{itemize}
\item[(i)] $P$ has a collapsible triangulation $\nabla$;
\item[(ii)] $P$ contains a vertex $v$ of $\cube$;
\item[(iii)] $P$ is strongly regular.
\end{itemize}
Then $P$ is a $\Zed$-retract of $[0,1]^{n}$.
\end{theorem}
The following result states
that \Zed-retracts are preserved under pushouts of strict \Zed-maps.
\begin{theorem}\label{Thm_AmalProj}
Let $P\subseteq [0,1]^{n}$, $Q\subseteq [0,1]^{m}$ and $R\subseteq
[0,1]^{k}$ be
\Zed-retracts and
$\eta\colon P\rightarrow Q$ and $\mu\colon P\rightarrow D$ be
strict $\Zed$-maps. Then the pushout $Q\coprod_P R$ (whose existence
is ensured by Theorem \ref{Theo-Colimits}) is a \Zed-retract.
\end{theorem}
\begin{proof}
With the notation of the proof of Theorem \ref{Theo-Colimits}, the
pushout $Q\coprod_P R$ was realized therein as the rational
polyhedron $\Pol(\mathfrak{W})\subseteq[0,1]^{r+s+t}$ of a
certain weighted abstract simplicial complex $\mathfrak{W}$.
The embeddings of $P$, $Q$, $R$ into $Q\coprod_P R$,
were denoted $\rho_P$,
$\rho_Q$, and $\rho_R$.
Letting $A=[0,1]^{r+s}\times\{0\}^{t}$ and
$B=[0,1]^{r}\times\{0\}^{s}\times[0,1]^{t}$,
it was shown:
\begin{itemize}
\item[(i)]
$\rho_Q(Q)\subseteq A$;
\smallskip
\item[(ii)]
$\rho_R(R)\subseteq A$;
\smallskip
\item[(iii)] $\rho_Q(Q)\cup\rho_R(R)=Q\coprod_P R$;
\smallskip
\item[(iv)] $\rho_Q(Q)\cap\rho_R(R)=\rho_P(P)$.
\end{itemize}
\medskip
\noindent
Since $P$, $Q$, and $R$ are \Zed-retracts, by \cite[Lemma
4.2]{CM20XX} $\rho_P(P)$, $\rho_Q(Q)$, and $\rho_R(R)$ are
\Zed-retracts. We let $\gamma_P\colon[0,1]^{r+s+t}\rightarrow
\rho_P(P)$, $\gamma_Q\colon[0,1]^{r+s+t}\rightarrow \rho_Q(Q)$ and
$\gamma_R\colon[0,1]^{r+s+t}\rightarrow \rho_R(R)$ denote the
corresponding \Zed-retractions.
\medskip
The \Zed-retraction for $Q\coprod_P R$ will be constructed in three steps.
\medskip
\noindent{\it Step 1:}
If $x\in A\cup B$, then $\conv({\bf 0},x)\subseteq A\cup B$, where
${\bf 0}$ denotes the origin of $\R^{r+s+t}$. Therefore, by
\cite[Theorem 1.4]{CM2009}, there exists a \Zed-retraction
$\gamma_1\colon[0,1]^{r+s+t}\rightarrow A\cup B$ onto $A\cup B$.
\medskip
\noindent{\it Step 2:}
Combining Proposition \ref{proposition:poly} and Lemma
\ref{Lem_Triang-Subset}, there is a regular triangulation $\Delta$ of
$A\cup B$, such that $\Delta_{Q\coprod_P R}=\{S\in \Delta\mid
S\subseteq Q\coprod_P R\}$ is a full subcomplex of $\Delta$. By Lemma
\ref{Cor_ExtensionToZed} there exists a unique \Zed-map
$\gamma_2\colon A\cup B\rightarrow A\cup B$ satisfying
$$
\gamma_2(v)=\left\{\begin{tabular}{ll}
$v$& if $v\in Q\coprod_P R$,\\
$\gamma_P(v)$& otherwise,
\end{tabular}\right.
$$
with $\gamma_2 $ linear over each simplex $S\in \Delta$.
By construction, $\gamma_2$ has the following properties:
\begin{itemize}
\item[(a)] if $v\in Q\coprod_P R$, then $\gamma_2(v)=v$;
\item[(b)] if $\gamma_2(A)\cap \gamma_2(B)=\rho_P(P)$
\end{itemize}
\medskip
\noindent{\it Step 3:}
By (a), we have $Q\coprod_P R\subseteq \gamma_2(A\cup B)$. Again by Proposition
\ref{proposition:poly} and Lemma \ref{Lem_Triang-Subset}, there is a
regular triangulation $\Lambda$ of $\gamma_2(A\cup B)$, such that
$\Lambda_{Q\coprod_P R}=\{S\in \Lambda\mid S\subseteq Q\coprod_P R\}$
is a full subcomplex of $\Lambda$. Let $\gamma_3\colon \gamma_2(A\cup
B)\rightarrow Q\coprod_P R$ be defined by:
$$
\gamma_3(v)=\left\{\begin{tabular}{ll}
$\gamma_Q(v)$& if $v\in \gamma_2(A)$,\\
$\gamma_R(v)$& if $v\in \gamma_2(B)$.
\end{tabular}\right.
$$
By (b) and (iv), if $v\in \gamma_2(A)\cap\gamma_2(B)$, $v\in
\rho_P(P)=\rho_Q(Q)\cap\rho_R(R)$, which implies that $
\gamma_Q(v)=\gamma_R(v)=v$.
This shows that $\gamma_3$ is well defined.
\bigskip
We {\it claim} that the map
$\gamma=\gamma_3\circ\gamma_2\circ\gamma_1\colon[0,1]^{r+s+t}\rightarrow
Q\coprod_P R$ is a \Zed-retraction.
If $v\in Q\coprod_P R$, by definition of $\gamma_1$, $\gamma_2$ and
$\gamma_3$ it follows that
$\gamma_1(v)=\gamma_2(v)=\gamma_3(v)=v$.
If $v\notin Q\coprod_P R$, then $\gamma_1(v)\in A\cup B$. Assume
first $\gamma_1(v)\in A$. Then $\gamma_2\circ\gamma_1(v)\in\gamma_2(A)$. Further,
$\gamma(v)=\gamma_3(\gamma_2\circ\gamma_1(v))=\gamma_Q(v)\in\rho_Q(Q)\subseteq
Q\coprod_P R$. Similarly, if $\gamma_1(v)\in B$ then
$\gamma(v)\in Q\coprod_P R$. Therefore, $\gamma[0,1]^{r+s+t}\rightarrow Q\coprod_P R$ is a
\Zed-retraction onto $Q\coprod_P R$, as claimed.
\medskip
The proof is complete
\end{proof}
Combining Corollary \ref{corollary:scazonte}
with the foregoing theorem
we get:
\begin{corollary}\label{Cor_FiberProj}
Let $(G_1,u_1)$, $(G_2,u_2)$, $(G_3,u_3)$ be finitely
generated projective unital $\ell$-groups
and $f\colon G_1\rightarrow G_3$ and $g\colon G_2\rightarrow
G_3$ onto homomorphisms.
Then the fiber product $G=\{(a,b)\in G_1\times G_2\mid f(a)=g(b)\}$
(with $(u_1,u_2)$ as the distinguished order unit)
is a finitely generated projective unital $\ell$-group.
\end{corollary}
\subsection{Geometric realization of exact unital $\ell$-groups}
The rest of the section is devoted to proving
\begin{theorem}\label{Theo_WeakProj}
A unital $\ell$-group $(G,u)$ is exact iff there exists
a polyhedron $P\subseteq\R^{n}$ satisfying the following conditions:
\begin{enumerate}
\item $(G,u)\cong\McN(P)$;
\item $P$ is connected;
\item $P\cap\Zed^{n}\neq\emptyset$;
\item $P$ is strongly regular.
\end{enumerate}
\end{theorem}
For the proof we prepare:
\begin{lemma}\label{Lem_CarWeakProj}
A unital $\ell$-group $(G,u)$ is exact iff there exist integers
$m,n\geq 0$, and a \Zed-map $\eta\colon [0,1]^{n}\rightarrow
\R^{m}$ such that $(G,u)\cong \McN(\eta([0,1]^{n}))$.
\end{lemma}
\begin{proof}
$(\Rightarrow)$
For some $m,n\in\{1,2,\ldots\}$ there
exist unital $\ell$-ho\-mo\-mor\-phisms
$$f\colon\McN([0,1]^{m})\rightarrow (G,u)
\,\,\,\mbox{and}\,\,\,
g\colon(G,u)\rightarrow \McN([0,1]^{n})$$
with
$f$ onto $(G,u)$ and $g$ one-one.
%
Then $$(G,u)\cong\McN([0,1]^{m})/{\rm ker}f= \McN([0,1]^{m})/{\rm
ker}(f\circ g).$$
Theorem \ref{Theo_Baker-Beynon} yields
a \Zed-map $\eta\colon[0,1]^{n}\rightarrow [0,1]^{m}$ such that
$g\circ f=\McN(\eta)$.
%
Therefore, $h\in{\rm ker}(f\circ g)$ iff $h\circ\eta=0$ iff
$h(\eta([0,1]^{n}))=\{0\}$,
%
whence $\McN([0,1]^{m})/{\rm ker}(f\circ g)=\McN([0,1]^{m})/{\rm
ker}(\McN(\eta))=\McN(\eta([0,1]^{n}))$.
\smallskip
$(\Leftarrow)$
As observed in Section 2,
(see \eqref{Eq:IsoinCube} in particular)
every polyhedron is \Zed-homeomorphic to a
polyhedron contained in some unit cube $[0,1]^{m}$.
Thus without loss of generality we can
assume $\eta([0,1]^{n})\subseteq [0,1]^{m}$. Let $\eta\colon
[0,1]^{n}\rightarrow [0,1]^{m}$ be a \Zed-map such that $(G,u)\cong
\McN(\eta([0,1]^{n}))$. Let further
$\mu\colon\eta([0,1]^{n})\rightarrow [0,1]^{m}$ and $\nu\colon
[0,1]^{n}\rightarrow \eta([0,1]^{n}) $ respectively be a strict and an
onto \Zed-map such that $\eta=\mu\circ\nu$.
%
By Theorems \ref{Theo_monicepi} and \ref{Theo_StricOnto},
$\McN(\mu)\colon \McN([0,1]^{m})\rightarrow \McN(\eta([0,1]^{n})) $ is
an onto unital $\ell$-homomorphism and $\McN(\nu)\colon
\McN(\eta([0,1]^{n}))\rightarrow \McN([0,1]^{n})$ is a one-one unital
$\ell$-homomorphism.
%
Since $\McN(\eta([0,1]^{n}))\cong(G,u)$, $(G,u)$ is finitely
generated and is
isomorphic to a subalgebra of the free unital $\ell$-group
$\McN([0,1]^{n})$.
\end{proof}
\begin{lemma}\label{Lem_CarZimages}
Let $P\subseteq\R^{n}$ be a polyhedron.
%
Then for some $l=1,2,\ldots$ there is a \Zed-map $\eta$
of $[0,1]^{l}$ onto $P$
iff $P$ satisfies the following three conditions:
\begin{enumerate}
\item $P$ is connected;
\item $P\cap\Zed^{n}\neq\emptyset$;
\item $P$ is strongly regular.
\end{enumerate}
\end{lemma}
\begin{proof}
$(\Rightarrow)$ If $\eta\colon [0,1]^{l}\rightarrow P$ is an onto
\Zed-map, then $P$ is connected because $\eta$ is continuous.
Combining Example \ref{Ex_CubeStronglyTriang} and Theorem
\ref{Theo_StrongPreserved}, it follows that $P$ is strongly regular.
%
By Corollary \ref{Cor_Div_Denominators}, $\den(\eta(0,\ldots,0))$
is a divisor of $\den(0,\ldots,0)$, that is,
$\den(\eta(0,\ldots,0))=1$. Then $\eta(0,\ldots,0)\in\Zed^{n}$.
%
\medskip
$(\Leftarrow)$
For some suitable strongly regular collapsible triangulation $\nabla$,
we will define onto \Zed-maps $\eta_1\colon\cube \rightarrow |\nabla|$,
and $\eta_2\colon |\nabla|\rightarrow P$ providing the required $\eta.$
\medskip
\noindent{\it Construction of $\nabla$:}
Let $\Delta$ be a regular triangulation of $P$.
$\nabla$ will be defined as the geometric realization of
a
weighted abstract simplicial complex $\mathfrak{W}$ arising
from $\Delta$.
Since $P$ is connected, the simple graph $H$ given by the
$1$-simplexes of $\Delta$ is connected.
As is well known, a {\em spanning tree}
of $H$ is a tree
$\mathcal{T}\subseteq\Delta$ such that
$\ver(\mathcal{T})=\ver(\Delta)=\{v_1,\ldots,v_n\}$.
%
By (ii), there is no loss of generality to assume
\begin{equation}\label{Eq:AsV1}
\den(v_1)=1.
\end{equation}
\bigskip
\noindent
{\it Vertices of $\mathfrak{W}$:}
%
Let us set
$$J=\{(i,j)\in\{1,\ldots,n\}^{2}\mid i\neq j \mbox{ and
}\conv(v_i,v_j)\in\Delta\}.$$
%
For each $i\neq j$ such that $\conv(v_i,v_j)\in\mathcal{T}$ let
$S_{i,j}$ be a maximal simplex in $\Delta$ such that
$\conv(v_i,v_j)\subseteq S_{i,j}$.
%
Let $K=\{(i,j,k)\in\{1,\ldots,n\}^{3}\mid i\neq j, j\neq k,i\neq k
\mbox{ and }\conv(v_i,v_j,v_k)\subseteq S_{i,j}\}$.
%
Then
$$
V=\{i,\ldots,n\}\cup J\cup K.
$$
\bigskip
\noindent
{\it Simplexes of $\mathfrak{W}$:} For each $i\in\{1,\ldots,n\}$,
let the set $\mathcal{F}_i\subseteq\mathcal{P}(V)$ be defined by:
%
$X\in\mathcal{F}_i$ if there are $j_1,\ldots,j_m\in\{1,\ldots,n\}$
such that $\conv(v_i,v_{j_1},\ldots, v_{j_m})\in\Delta$ and
$X\subseteq \{i,(i,j_i),\ldots,(i,j_m)\}$.
For each $i,j\in\{1,\ldots,n\}$ such that $i\neq j$ and
$\conv(v_i,v_j)\in\mathcal{T}$,
we further define $\mathcal{B}_{i,j}\subseteq\mathcal{P}(V)$ as follows:
%
$X\in \mathcal{B}_{i,j}$ if
$\conv(v_i,v_j,v_{k_1}\ldots, v_{k_m})=S_{i,j}$ and
$X\subseteq\{i,j,(i,j,k_1),\ldots,(i,j,k_m)\}$.
%
We next let
$$\Sigma=\bigcup\mathcal{F}_i\cup\bigcup \mathcal{B}_{i,j}$$
By definition, if $X\subseteq Y$ and $Y\in\Sigma$ then $X\in\Sigma$.
%
Moreover, for all $x\in V$ there exists $X\in\Sigma$ such that $x\in X$.
%
Therefore, $\langle V,\Sigma\rangle$ is an abstract simplicial complex.
\bigskip
\noindent
{\it Weights:} Finally we define $w\colon V\rightarrow
\{1,2,\ldots\}$ as follows:
\begin{equation}\label{Eq_DefWeight}
\begin{tabular}{lcl}
$w(i)$ & $=$ & $\den(v_i)$,\\
$w(i,j)$ & $=$ & $\den(v_j)$, \\
$w(i,j,k)$ & $=$ & $\den(v_k)$.
\end{tabular}
\end{equation}
\medskip
\noindent{\it Claim 1:} For every maximal simplex $X$ in $\Sigma$,
the greatest common divisor of the denominators of the vertices of $X$
is $1$.
By definition,
we either have
$$X=\{i\}\cup\{(i,j_1),\ldots,(i,j_m)\}
\,\,\,\mbox{ and $\conv(v_i,v_{j_1},\ldots, v_{j_m})$ is maximal in $\Delta$},$$
or
$$X=\{i,j,(i,j,k_1),\ldots,(i,j,k_m)\}\,\,\,
\mbox{and
$\conv(v_i,v_j,v_{k_1}\ldots, v_{k_m})=S_{i,j}$}.$$
%
\bigskip
\noindent
In either case
the claim follows by definition of $w$,
because $\Delta$ is strongly regular.
%
\bigskip
\noindent{\it Claim 2:} $\mathfrak{W}$ is collapsible.
By definition of $\mathfrak{W}$, we have:
\begin{itemize}
\item[(a)] $\mathcal{F}_i\cap \mathcal{F}_j=\{\emptyset\}$ whenever
$i\neq j$;
\item[(b)] $\mathcal{F}_{i}\cap \mathcal{B}_{j,k}=\{\emptyset\}$
whenever $i\neq j$ and $i\neq k$;
\item[(c)] $\mathcal{F}_{i}\cap
\mathcal{B}_{j,k}=\{\emptyset,\{i\}\}$ whenever $i= j$ or $i=k$;
\item[(d)] $\mathcal{B}_{i,j}\cap
\mathcal{B}_{k,l}=\{\emptyset,\{i\}\}$ whenever $i=k$ or $i=l$;
\item[(e)] $\mathcal{B}_{i,j}\cap \mathcal{B}_{k,l}=\{\emptyset\}$
whenever $\{i,j\}\cap\{k,l\}=\emptyset$.
\end{itemize}
The claim is now proved in 3 steps as follows:
\medskip
{\it Step 1 ($\mathcal{F}_{i}$):}
For each $i\in\{1,\ldots,n\}$,
$\langle\ver(\mathcal{F}_i),\mathcal{F}_i\rangle$ is
combinatorially isomorphic to
the closed star of $v_i$ (see \cite[\S III Definition 1.11]{Ew1996}), and
therefore, it is
a collapsible abstract simplicial complex.
%
Then $\langle\ver(\mathcal{F}_i),\mathcal{F}_i\rangle$ collapses
to the vertex $i$.
%
By (a-e), $\mathfrak{W}$ collapses to $\langle
V,\bigcup\mathcal{B}_{i,j}\rangle$.
\medskip
{\it Step 2 ($\mathcal{B}_{i,j}$):}
For each $i,j\in\{1,\ldots,2\}$
such that $i\neq j$ and $\conv(v_i,v_j)\subseteq \mathcal{T}$, the
abstract simplicial complex
$(\ver(\mathcal{B}_{i,j}),\mathcal{B}_{i,j})$ is combinatorially
isomorphic to the skeleton of the complex given by the simplex
$S_{i,j}$ and its faces.
%
Therefore,
$\langle \ver(\mathcal{B}_{i,j}),\mathcal{B}_{i,j}\rangle$ can be
collapsed to any of its faces.
In particular, it can be collapsed to
$\langle\{i,j\},\{\emptyset,\{i\},\{j\},\{i,j\}\}\rangle$.
%
Using (d) we see that
$(V,\bigcup\mathcal{B}_{i,j})$ collapses to the abstract simplicial
complex $(V,\Sigma')$ where $X\in\Sigma'$ iff
$X\subseteq\{i,j\}$ and $\conv(v_i,v_j)\subseteq\mathcal{T}$.
\medskip
{\it Step 3:}
The sequence of collapses defined in Steps
1 and 2 leads to an abstract simplicial complex $(V,\Sigma')$ which
is combinatorially isomorphic to the skeleton of $\mathcal{T}$.
%
Since $\mathcal{T}$ is a tree, it is collapsible and therefore
$(V,\Sigma')$ is collapsible, too.
\bigskip
Thus
$\mathfrak{W}$ is collapsible, and Claim 2 is settled.
%
\medskip
By (\ref{Eq_DefWeight}) and (\ref{Eq:AsV1}), $w(1)=\den(v_1)=1$.
{}From Claims 1 and 2 it follows that
$\Pol(\mathfrak{W})$ satisfies the hypotheses of Theorem
\ref{Thm_collapsible}.
%
Therefore, for some integer $l>0$
there is an onto \Zed-map $\eta\colon
[0,1]^{l}\rightarrow\Pol(\mathfrak{W})$.
%
\medskip
Finally let $f\colon V\rightarrow \ver(\Delta)$ be defined as follows:
$$
\begin{tabular}{lcl}
$f(i)$ & $=$ & $v_i$,\\
$f(i,j)$ & $=$ & $v_j$, \\
$f(i,j,k)$ & $=$ & $v_k$.
\end{tabular}
$$
By (\ref{Eq_DefWeight}), $\den(f(x))=w(x)$ for each $x\in V$. By
definition of $\mathcal{F}_i$ and $\mathcal{B}_{i,j}$, $\,\,\,f$
is a morphism from $\mathfrak{W}$ into the skeleton
$\mathfrak{W}(\Delta)$ of $\Delta$.
%
Then $\Pol(f)\colon\Pol(\mathfrak{W})\rightarrow
\Pol(\mathfrak{W}(\Delta))$ is a \Zed-map.
%
Since for every $m$-simplex $S=\conv(v_{i_0},\ldots,v_{i_m})
\in \Delta$ there is $X\in\Sigma$ (specifically,
$X\in \mathcal{F}_{i_0}$)
such that $f(X)=\{v_{i_0},\ldots,v_{i_m}\}$, it follows that
$\Pol(f)$ is onto $\Pol(W_\Delta)$.
\medskip
In conclusion,
$\iota_{\Delta}^{-1}\circ\Pol(f)\circ\eta\colon[0,1]^{m}\rightarrow
P$ is the desired \Zed-map onto $P$.
\end{proof}
\subsubsection*{Proof of Theorem \ref{Theo_WeakProj}}
This immediately follows from
Lemmas \ref{Lem_CarWeakProj} and \ref{Lem_CarZimages}.
\subsection{Intrinsic characterization of exact unital $\ell$-groups}
In \cite[Definition 2.1]{MMM2006}, {\it abstract Schauder basis} were defined
for abelian $\ell$-groups as isomorphic copies of Schauder basis.
In \cite[Theorem 3.1]{MMM2006} a characterization of abstract Schauder bases is presented.
Using this characterization, in \cite[Definition 4.3]{MM2007},
the notion of abstract Schauder basis was extended to unital $\ell$-groups and called {\it basis}.
In \cite[Theorem 4.5]{MM2007}
it is proved that an {\it archimedean} unital $\ell$-group $(G,u)$
is finitely presented iff it has a basis.
In \cite[Theorem 3.1]{CM2011},
the archimedean assumption was shown to be unnecessary.
Using this latter result in combination with
Theorem \ref{Theo_WeakProj},
will
provide in Theorem \ref{Theo-BasisExact} an algebraic description of exact unital $\ell$-groups.
We first need to recall some definitions.
We denote by ${\rm maxspec}(G,u)$ the set of
maximal ideals of $(G,u)$ equipped with the {\it spectral} topology:
a basis of closed sets for ${\rm maxspec}(G,u)$
is given by sets of the form
$ \{\mathfrak p \in {\rm maxspec}(G,u) \mid g\in \mathfrak p\},$
where $g$ ranges over all elements of $G$ (see \cite[\S 10]{BKW1977}).
As is well known,
${\rm maxspec} (G,u)$ is a nonempty compact Hausdorff space, \cite[Theorem 10.2.2]{BKW1977}.
\begin{definition}\label{def:basis}\cite[Definition 4.3]{MM2007}
Let $(G,u)$ be a unital $\ell$-group.
A {\em basis} of $(G,u)$ is a finite set $\mathcal B = \{b_{1},\ldots,b_{n}\}$
of elements $\not=0$ of the positive cone $G^{+}=\{g\in G \mid g\geq 0\}$ such that
\begin{itemize}
\item[(i)] $\,\mathcal B$ generates $G$ using the
group and lattice operations;
\item[(ii)] for each $k=1,2,\ldots$ and
$k$-element subset $C$ of $\mathcal B$
with $0\not= \bigwedge\{b\mid b\in C\}$,
the set $\{\mathfrak{m}\in{\rm maxspec(G,u)}\mid\mathfrak{m}\supseteq\mathcal{B}\setminus C \}$
is homeomorphic to a $(k-1)$-simplex;
\item[(iii)] there are integers $1 \leq m_{1},\ldots,m_{n}$ such that
$\sum_{i=1}^{n}m_{i}b_{i} = u$.
\end{itemize}
\end{definition}
\begin{theorem}\cite[Theorem 3.1]{CM2011}
Let $(G,u)$ be unital $\ell$-group. Then the following are equivalent:
\begin{itemize}
\item[(i)] $(G,u)$ is finitely presented;
\item[(ii)] $(G,u)$ has a basis.
\end{itemize}
\end{theorem}
Given a unital $\ell$-group and a basis
$\mathcal{B}=\{b_1,\ldots,b_m\}$
of $(G,u)$, let $\mathfrak{W}_{\mathcal{B}}=\{\mathcal{B},\Sigma_{\mathcal{B}},\omega_{\mathcal{B}}\}$ be the weighted abstract simplicial complex given by the following stipulations:
\begin{itemize}
\item[--]
$
S\in\Sigma_{\mathcal{B}} \mbox{ iff }\bigwedge S\neq 0$
\item[--]
$
\omega_{\mathcal{B}}(b_i)=m_i.$
\end{itemize}
\begin{proposition}\label{Prop:PolBasis}
Let $(G,u)$ be unital $\ell$-group and $\mathcal B$ be a basis for $(G,u)$.
Then $$(G,u)\cong \McN(\Pol(\mathfrak{W}_{\mathcal{B}})).$$
Recall that $\McN(\Pol(\mathfrak{W}_{\mathcal{B}}))$ is the unital $\ell$-group of \Zed-maps from the canonical realization of $\mathfrak{W}_{\mathcal{B}}$ into $\R$.
\end{proposition}
\begin{proof}
This is essentially the content of the proof of
\cite[Theorem 3.1]{CM2011}.
\end{proof}
\begin{theorem}\label{Theo-BasisExact}
Let $(G,u)$ be a unital $\ell$-group.
Then $(G,u)$ is exact iff it has a basis $\mathcal{B}=\{b_1,\ldots,b_n\}$
satisfying the following conditions:
\begin{itemize}
\item[(i)] There is an element $b_i\in\mathfrak{B}$, such that $m_i=1$;
\item[(ii)] For each maximal $S\in \mathfrak{W}_{\mathcal{B}}$
the greatest common divisor of $\{m_j\mid b_j\in S\}$ is $1$;
\item[(iii)] For each $b_i,b_j\in \mathcal{B}$
there exist a sequence $b_i=b_{k_1},b_{k_2},\ldots,b_{k_m}=b_j$,
such that $b_{k_l}\wedge b_{k_{l+1}}\neq 0$ for each $l\in\{1,\ldots, m-1\}$.
\end{itemize}
\end{theorem}
\begin{proof}
Immediate
from Theorem \ref{Theo_WeakProj}, Proposition \ref{Prop:PolBasis},
upon noting that
\begin{itemize}
\item[--] Condition
(i) is equivalent to $\Pol(\mathfrak{W}_{\mathcal{B}})\cap\mathbb{Z}^{n}\neq\emptyset$;
\item[--]
Condition
(ii) is equivalent to $\Pol(\mathfrak{W}_{\mathcal{B}})$ being strongly regular;
\item[--] Condition
(iii) is equivalent to $\Pol(\mathfrak{W}_{\mathcal{B}})$ being connected.
\end{itemize}
\end{proof}
\begin{remark}
In \cite[Definition p.3]{Ma20XX}, working in the
framework of Abelian $\ell$-groups,
the author introduced the notion of a
{\em regular} set of positive elements.
This definition only depends
on the algebraic/combinatorial notions of {\em starrable set} and {\em $1$-regularity}.
In \cite[Lemmas 2.1 and 2.6]{Ma20XX}
it is proved that, for Abelian $\ell$-groups,
regular set of positive generators
coincide with abstract Schauder bases.
Using this result one can prove that a subset
$\mathcal B$ of a unital $\ell$-group is a basis
iff it is a regular set of positive generators satisfying
condition (iii) in Definition~\ref{def:basis}.
This leads to a reformulation of Theorem~\ref{Theo-BasisExact}
where the exactness of a
unital $\ell$-group is characterized only in terms
of algebraic-combinatorial notions.
\end{remark}
\subsection{Admissible rules in \L ukasiewicz infinite-valued
calculus}\label{Sec_admissible}
Throughout this paper we have been going back and forth from
unital $\ell$-groups, rational polyhedra and weighted abstract simplicial
complexes.
Using the
categorical equivalence $\Gamma$ between unital $\ell$-groups and
MV-algebras,
the span of our paper can be further extended the algebraic counterparts of
\L ukasiewicz infinite-valued calculus \L$_\infty$.
This gives us an opportunity to discuss the underlying
algorithmic and proof-theoretic aspects of the theory developed so far.
We refer to \cite{CDM2000} and \cite{Mu2011} for background on \L$_\infty$
and MV-algebras, and to \cite{Ry1997} for background on admissible rules.
\medskip
For any set $X$, we will denote ${\sf FORM}_X$ to the algebra of formulas
in the language $\{\top,\neg,\oplus\}$ where $\top$ is a constant
$\neg$ is a unary connective and $\oplus$ is a binary connective and
whose variables are in $X$.
By definition,
a {\it substitution} $\sigma\colon{\sf FORM}_X\rightarrow \mathsf{FORM}_Y $
is
a homomorphism of the algebra ${\sf FORM}_X$ into $\mathsf{FORM}_Y$.
Two formulas $\psi,\varphi$ are said to be {\it equivalent in}
\L$_\infty$ (in symbols, $\psi\cong_{\text{\L}_\infty}\varphi$) if
the equation $\psi\approx\varphi$ is valid in every MV-algebra.
The algebra ${\mathsf{Free}MV}_X={\sf
FORM}_X/\cong_{\text{\L}_\infty}$ is the free algebra on $X$
generators in the variety of MV-algebras.
Let
$\Gamma$ be the categorical equivalence of \cite[\S 3]{Mu1986}
between MV-algebras and unital $\ell$-groups. Then for
any $n$-element set $X$, ${\mathsf{Free}MV}_X\cong \Gamma(\McN([0,1]^{n}))$.
A {\it rule} in \L$_\infty$ is a pair $(\Theta,\Sigma)$ where
$\Theta\cup \Sigma$ is a finite subset of ${\sf FORM}_X$ for some $X$.
A rule $(\Theta,\{\varphi\})$ is {\it derivable} in \L$_\infty$ if the
quasi-equation
$\bigwedge\{\psi\approx\top\mid\psi\in\Theta\})\rightarrow
\varphi\approx \top$ is valid in every MV-algebra. A formula
$\varphi$ is a {\it theorem} of \L$_\infty$ if
$(\emptyset,\{\varphi\})$ is derivable in \L$_\infty$.
A rule $(\Theta,\Sigma)$ is said to be {\it admissible} in \L$_\infty$
if for every substitution $\sigma$ such that $\sigma(\psi)$ is a
theorem of \L$_\infty$ for each $\psi\in\Theta$, then there is
$\varphi\in\Sigma$ such that $\sigma(\varphi)$ is a theorem of
\L$_\infty$.
In \cite{Je2010}, the author provides a basis for the
admissible rules
of \L$_\infty$.
To this purpose, he introduced the notion of admissibly saturated formula.
An equivalent reformulation is as follows:
\begin{definition}\label{Def:AdmSat}\cite[Definition 3.1]{Je2010} A formula $\varphi$ is {\em
admissibly saturated in \L$_\infty$} if for every finite set $\Sigma$
of formulas the following conditions are equivalent:
\begin{itemize}
\item[(i)] the rule $(\{\varphi\},\Sigma)$ is admissible in \L$_\infty$;
\item[(ii)] there exists $\psi\in\Sigma$ such that $(\{\varphi\},\{\psi\})$ is derivable in
\L$_\infty$.
\end{itemize}
\end{definition}
A formula $\varphi$ whose set of variables is $X$ is said to be {\it exact} in \L$_\infty$ if there exists a
substitution $\sigma\colon{\sf FORM}_X\rightarrow \mathsf{FORM}_Y $
such that $\sigma(\psi)$ is a theorem of \L$_\infty$ if
$(\{\varphi\},\{\psi\})$ is derivable in \L$_\infty$.
Equivalently, $\varphi$ is exact iff there exists $Y$ such
that
${\mathsf{Free}MV}_X/\theta([\varphi]_{\cong_{\text{\L}_\infty}},[\top]_{\cong_{\text{\L}_\infty}})$
is isomorphic to a subalgebra of ${\mathsf{Free}MV}_Y$, where
$\theta([\varphi]_{\cong_{\text{\L}_\infty}},[\top]_{\cong_{\text{\L}_\infty}})$
denotes the principal congruence generated by
$([\varphi]_{\cong_{\text{\L}_\infty}},[\top]_{\cong_{\text{\L}_\infty}})$.
Using the categorical equivalence
$\Gamma$
between MV-algebras
and unital $\ell$-groups, it follows that $\varphi$ is exact iff
${\mathsf{Free}MV}_X/\theta([\varphi]_{\cong_{\text{\L}_\infty}},[\top]_{\cong_{\text{\L}_\infty}})$ is isomorphic to $\Gamma(G,u)$ for some exact unital $\ell$-group.
As is well known,
{\it exact formulas are admissibly
saturated.} To help the reader,
we supply a short proof for the
special case of \L$_\infty$.
Let $\varphi $ be an
exact formula whose set of variables is $X$.
Let $(\{\varphi\},\Sigma)$ be an admissible rule in
\L$_\infty$. By Defintion \ref{Def:AdmSat} we need to prove that there exists
$\psi\in \Sigma$ such that $(\{\varphi\},\{\psi\})$ is derivable in \L$_\infty$.
Since $\varphi$ is exact, there exist
a substitution $\sigma\colon{\sf FORM}_X\rightarrow \mathsf{FORM}_Y $
such that $\sigma(\psi)$ is a theorem of \L$_\infty$ and
$(\{\varphi\},\{\psi\})$ is derivable in \L$_\infty$ iff $\sigma(\psi)$ is a theorem of \L$_\infty$.
Since $\sigma(\varphi)$ is a theorem of \L$_\infty$ and $(\{\varphi\},\Sigma)$
is an admissible rule, there exists
$\psi\in \Sigma$ such that $\sigma(\psi)$ is a
theorem of \L$_\infty$, i.e. $(\{\varphi\},\{\psi\})$ is derivable in \L$_\infty$.
This proves that $\varphi$ is admissibly saturated.
\medskip
In Theorem \ref{Thm_combination}
we will prove that exact
and admissibly saturated formulas coincide in \L$_\infty$.
\medskip
The following notion was introduced in \cite[Definition 4.5]{Je2009} to study the decidability of admissible rules in \L ukasiewicz infinite-valued calculus, and used in \cite{Je2010} to characterize admissibly saturated formulas:
A set $X\subseteq [0,1]^{n}$ is called a {\it anchored}
if for some $v_1,\ldots,v_m\in[0,1]^{n}$, $X=\conv(v_1,\ldots,v_m)$ and the affine hull of $X$ intersects
$\Zed^{n}$.
\begin{lemma}\label{Lem_SimplexAnchvsStReg}
Let $S\subseteq\R^{n}$ be a regular $t$-simplex,
$t=0,1,\ldots, n$. Then the following conditions are equivalent:
\begin{itemize}
\item[(i)] $S$ is strongly regular;
\item[(ii)] $S$ is anchored;
\item[(iii)] $S$ is union of finitely many anchored sets.
\end{itemize}
\end{lemma}
\begin{proof} The equivalence
(ii)$\Leftrightarrow$(iii)
immediately follows by definition.
\smallskip
To prove (i)$\Leftrightarrow$(ii),
let $\{v_0,\ldots,v_t\}$ be the vertices of $S$.
$(\Rightarrow)$ There exist integers $m_0,\ldots,m_t$ such that
$\sum_{i=0}^{t}m_i\den(v_i)=1$. Since $\den(v_i)v_i\in\Zed^{n}$,
the affine linear combination $\sum_{i=0}^{t}m_i\den(v_i)v_i$
lies in $\Zed^n$, whence $S$ is anchored.
$(\Leftarrow)$ By hypothesis,
there are $\lambda_0,\ldots,\lambda_t\in \R$ such that
$v=\sum_{i=0}^{t}\lambda_i v_i\in\Zed^{n}$ and
$\sum_{i=0}^{t}\lambda_i =1$. Then
\begin{eqnarray}
\nonumber \tilde{v}&=&\textstyle (\sum_{i=0}^{t}\lambda_i
v_i,1)=(\sum_{i=0}^{t}\lambda_i v_i,\sum_{i=0}^{t}\lambda_i)\\
\nonumber &=&\textstyle
\sum_{i=0}^{t}\lambda_i(v_i,1)=\sum_{i=0}^{t}\frac{\lambda_i}{\den(v_i)}(\den(v_i)(v_i,1))\\
\nonumber &=&\textstyle \sum_{i=0}^{t}\frac{\lambda_i}{\den(v_i)}\tilde{v_i}.
\end{eqnarray}
Since $S$ regular, $\frac{\lambda_i}{\den(v_i)}m_i\in\Zed$.
Therefore, $\sum_{i=0}^{t}m_i\den(v_i)=\sum_{i=0}^{t}\lambda_i=1$,
whence
the greatest common divisor
of $\den(v_0),\ldots,\den(v_t)$ is $1$.
\end{proof}
\begin{theorem}\label{Thm_AnchvsStReg}
A rational polyhedron $P\subseteq\R^{n}$ is strongly regular iff it is
a finite union of anchored sets.
\end{theorem}
\begin{proof}
$(\Rightarrow)$ Immediate from
Lemma \ref{Lem_SimplexAnchvsStReg}.
$(\Leftarrow)$
Suppose that $P$ is such that $P=S_1\cup\cdots\cup S_m$ for some anchored sets $S_1,\ldots,S_m\subseteq\R^{n}$
%
Let $\Delta$ be a regular triangulation of $P$, and
$T$ a maximal simplex in $\Delta$, with the intent of proving that
$T$ is strongly regular.
Let $v\in{\rm relint}(T)$.
%
Since $T\subseteq P \subseteq S_1\cup\cdots\cup S_m$ there exists
$S_i$ such that $v\in S_i$.
%
Lemma \ref{Lem_MaxSimpTriang} yields
$0<\delta_1\in\R$ such that $B(\delta_1,v)\cap P\subseteq T$.
%
Since $v\in S_i$, there exists a point
$w$ in $B(\delta_1,v)\cap{\rm relint}(S_i)$,
whence $w\in {\rm relint}(T)\cap{\rm relint}(S_i)$.
Again by Lemma \ref{Lem_MaxSimpTriang},
there exists $0<\delta_2\in\R$ such that $B(\delta_2,w)\cap P\subseteq T$.
%
By definition of the relative interior of $S_i$, there exists
$0<\delta_3\in\R$
such that $B(\delta_1�,w)\cap {\rm aff}(S_i)\subseteq S_i$.
%
Letting $\delta={\rm min}\{\delta_2,\delta_3\}$ we obtain
%
\begin{eqnarray}
\nonumber B(\delta,w)\cap{\rm aff}(S_i)&\subseteq& B(\delta,w)\cap S_i\\
\nonumber &\subseteq& B(\delta,w)\cap P\\
\nonumber &\subseteq& T.
\end{eqnarray}
Therefore, ${\rm aff}(S_i)\subseteq {\rm aff}(T)$, and $T$ is anchored.
%
Since $T$ is a regular simplex,
by Lemma \ref{Lem_SimplexAnchvsStReg},
$T$ is strongly regular, whence so is
$P$.
\end{proof}
\begin{theorem}\label{Thm_combination}
Let $\varphi$ be a formula in the language of MV-algebras. Then the
following are equivalent
\begin{itemize}
\item[(i)] $\varphi$ is admissibly saturated;
\item[(ii)] $\varphi$ is exact.
\end{itemize}
\end{theorem}
\begin{proof}
{}A combination of Theorems \ref{Theo_WeakProj} and
\ref{Thm_AnchvsStReg} with
\cite[Theorem 3.5]{Je2010}, again using
the categorical equivalence $\Gamma$ of \cite[Theorem 3.9]{Mu1986}.
\end{proof}
\subsection*{Acknowledgements}
\smallskip
The author would like to thank Professor Daniele Mundici for his
helpful comments and suggestions on previous drafts of this paper.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,341
|
package com.whinc.widget.lsystem.display;
import android.graphics.Canvas;
import android.graphics.Paint;
import java.io.IOException;
import java.io.ObjectInputStream;
import java.io.ObjectOutputStream;
import java.util.LinkedList;
import java.util.List;
/**
* Created by Administrator on 2015/11/20.
*/
public abstract class AbsDisplay implements Display {
private Generator mGenerator;
private float mDirection = -90.0f;
private float mAngle = 45.0f;
private float mFractionPosX = 0.5f;
private float mFractionPosY = 0.5f;
private float mStep = 10.0f;
private int mIterations = 1;
/* Don't serialize these fields */
private transient List<Float> mDirectionStack;
private transient List<Float> mFractionPosXStack;
private transient List<Float> mFractionPosYStack;
private transient List mState;
private transient Paint mPaint;
public AbsDisplay(String axiom, String delimiter, String... rules) {
mGenerator = new GeneratorImpl(axiom, delimiter, rules);
init();
}
private void init() {
mDirectionStack = new LinkedList<>();
mFractionPosXStack = new LinkedList<>();
mFractionPosYStack = new LinkedList<>();
mState = new LinkedList();
mPaint = new Paint(Paint.ANTI_ALIAS_FLAG);
}
private void writeObject(ObjectOutputStream out) throws IOException {
out.defaultWriteObject();
out.writeInt(getColor());
}
private void readObject(ObjectInputStream in) throws IOException, ClassNotFoundException {
in.defaultReadObject();
init();
setColor(in.readInt());
}
/**
* <P>Save state</P>
*/
private void storeState() {
mState.clear();
mState.add(mDirection);
mState.add(mAngle);
mState.add(mFractionPosX);
mState.add(mFractionPosY);
}
/**
* <P>restore state</P>
*/
private void restoreState() {
mDirection = (float) mState.remove(0);
mAngle = (float) mState.remove(0);
mFractionPosX = (float) mState.remove(0);
mFractionPosY = (float) mState.remove(0);
}
@Override
public Generator getGenerator() {
return mGenerator;
}
@Override
public void setGenerator(Generator generator) {
mGenerator = generator;
}
@Override
public String getPattern() {
if (mGenerator == null) {
return "";
}
return mGenerator.generate(mIterations);
}
@Override
public void setColor(int color) {
mPaint.setColor(color);
}
@Override
public int getColor() {
return mPaint.getColor();
}
@Override
public float getAngle() {
return mAngle;
}
@Override
public void setAngle(float angle) {
mAngle = angle;
}
@Override
public void setPercentX(float fraction) {
mFractionPosX = fraction;
}
@Override
public float getPercentX() {
return mFractionPosX;
}
@Override
public void setPercentY(float fraction) {
mFractionPosY = fraction;
}
@Override
public float getPercentY() {
return mFractionPosY;
}
@Override
public float getDirection() {
return mDirection;
}
@Override
public void setDirection(float degree) {
this.mDirection = degree;
}
@Override
public void rotateDirection(float deltaDegree) {
mDirection += deltaDegree;
}
@Override
public void saveDirection() {
mDirectionStack.add(mDirection);
}
@Override
public void restoreDirection() {
if (mDirectionStack.isEmpty()) {
throw new IllegalStateException("stack is empty! push and pop operation don't match");
}
mDirection = mDirectionStack.remove(mDirectionStack.size() - 1);
}
@Override
public void savePos() {
mFractionPosXStack.add(mFractionPosX);
mFractionPosYStack.add(mFractionPosY);
}
public void restorePos() {
if (mFractionPosXStack.isEmpty() || mFractionPosYStack.isEmpty()) {
throw new IllegalStateException("stack is empty! push and pop operation don't match");
}
mFractionPosX = mFractionPosXStack.remove(mFractionPosXStack.size() - 1);
mFractionPosY = mFractionPosYStack.remove(mFractionPosYStack.size() - 1);
}
@Override
public float getStep() {
return mStep;
}
@Override
public void setStep(float length) {
mStep = Math.max(0.0f, length);
}
@Override
public int getIterations() {
return mIterations;
}
@Override
public void setIterations(int iterations) {
mIterations = Math.max(0, iterations);
}
public void draw(Canvas canvas) {
storeState();
drawContent(canvas, mPaint);
restoreState();
}
public abstract void drawContent(Canvas canvas, Paint paint);
@Override
public String toString() {
return String.format("direction:%f, angle:%f, pos:(%f, %f), step:%f, iterations:%d",
mDirection, mAngle, mFractionPosX, mFractionPosY, mStep, mIterations);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,372
|
Alton Sports is committed and dedicated to ensuring that your privacy is protected, Alton Sports will only use your information to assist them in providing you with a better service. Whilst browsing the site such as registration, ordering products or entering competitions you may be asked for certain personal information by filling out and submitting one of the dedicated online forms.
It is at your discretion to whether you share any information; however, Alton Sports may require certain personal and demographic information such as your name, mailing address, email address, security data, and personal identification information to process orders placed.
When placing an order with Alton Sport you may be asked for credit details to process payment. Your details will only be used in accordance with your requests. If you change your mind and decide you no longer wish to receive information then email 'unsubscribe' to callam.altonsports@gmail.com with your name, billing address, email address and the details which you know longer wish to receive. Alton Sports will not share any information with third parties without your permission, unless it will assist us in providing you with the level of service and commitment you expect from us.
Alton Sports may periodically send promotional material about new products, special offers or other information we feel would benefit you as a consumer.
From time to time we may contact you for market research purposes via email in order to improve our services.
When you fill in any forms. For example, if an accident happens in store.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,791
|
El Markaz Shabab Al-Am'ari es club de fútbol palestino que milita en la Cisjordania Premier League, la liga de fútbol más importante de Palestina. Juega como local en Jerusalén.
Fue fundado en el año 1953 en la ciudad de Al-Ram y cuenta con 2 títulos de liga y un subtítulo de copa.
A nivel internacional ha participado en 1 torneo continental, en la Copa Presidente de la AFC 2012, donde perdió la final ante el Istiqlol de Tayikistán.
Estadio
Palmarés
Cisjordania Premier League: 2
1997, 2011
Sub-Campeón: 1
2008-09
Copa Yasser Arafat: 0
Sub-Campeón: 1
2009-10
Participación en competiciones de la AFC
Copa Presidente de la AFC: 1 aparición
2012: Finalista
Socios Deportivos
Persijap
Kelantan FC
Equipo 2012
Enlaces externos
Cisjordania Premier League
soccerway.com
Temporada de Archivo
Equipos de fútbol de Palestina
Equipos de fútbol fundados en 1953
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,617
|
{"url":"https:\/\/mywebsiteontheinternet.com\/blog\/hey-google-track-a-bunch-of-baby-stuff-for-me\/","text":"Hey Google, track a bunch of baby stuff for me!\n2020.04.18\n\nMy daughter was born last september. It\u2019s a big life event with many changes, but one (of many) things that caught me by surprise was the amount of data collection that came along with a newborn. Besides the doctors and nurses measuring her weight and length over those first months, there\u2019s also an expectation that the parents track all kinds of things: how often is the baby feeding and for how long, how long is the baby sleeping and when, how many times is the baby pooping and peeing\u2026\n\nAll these new data collection tasks come right when you\u2019re at your least capable of taking on a new project like that. You\u2019re overwhelmed with new skills or concerns, and braindead from sleep deprivation.\n\nTo make things as easy as possible for us, I decided I wanted it so that we could just tell Google Home that, \u201cThe baby fell asleep,\u201d and that information and a timestamp would be logged in a spreadsheet for later analysis.\n\nUsing IFTTT, it was pretty easy, but there were a couple little \u201cgotchas\u201d \u2013 if you happen to be a sleep addled new parent reading this, I\u2019ve decided to document my baby-related IFTTT recipes to make your life as easy as possible. Really, they could be adapted to anything you wanted to log.\n\nIFTTT (\u201cIf THIS then THAT\u201d) is kind of like digital glue. It lets you stick services to other services. Companies pay to have their services integrated, and then capable of talking to (and being talked to by) a wide range of other services. In my case, I wanted Google Assistant to trigger IFTTT to enter some information into a particular Google Sheet. Crucially, unlike glue, IFTTT is a one-way proposition. Information flows through it, but never back out. So it can write a new row on a spreadsheet, but will not be able to read you back the most recent row that was written. This presented a problem for tracking how long since the last breastfeed, which I got around with a fairly hacky solution.\n\n# The baby is asleep \/ the baby is awake\n\n## The baby is asleep\n\nThe Google Assistant side of this recipe is pretty straightforward. When we say, \u201cOK Google, baby fell asleep _______\u201d the IFTTT script is triggered. The blank is so that we can add some context, like, \u201cbaby fell asleep in her crib\u201d or \u201cbaby fell asleep after crying for 37 hours straight\u201d.\n\nThe Google Sheets side of things is slightly more complicated. The Sheets integration with IFTTT lets you add a new row to a google sheet, and lets you access two variables Created At and TextField. TextField is the \u201cblank\u201d (represented with a $ in IFTTT) from before. CreatedAt is the date, in long form, like \u201cJanuary 28, 2020 at 06:05PM\u201d. Let me break down what I\u2019m doing with this: =REGEXREPLACE(\"{{CreatedAt}}\";\" at .*$\"; \" \")|||=REGEXREPLACE(\"{{CreatedAt}}\";\"^.* at\";\" \")|||ASLEEP|||{{TextField}}\nEach ||| separates two cells in the row we\u2019re adding to the spreadsheet. So here we\u2019re adding a row with the first four cells filled in.\n\nThe first two cells are reformatting the CreatedAt variable from long form into a date (in the first cell) and a time (in the second). =REGEXREPLACE() is a google sheets function that does regular expression replacement. Basically, we\u2019re saying in the first cell, remove everything in CreatedAt after the date. And in the second cell, remove everything before the time.\n\nThe third cell simply says that the baby is ASLEEP at this time. This is useful because we also have an IFTTT set up for adding to this same spreadsheet when the baby wakes up.\n\nLastly, the fourth and final field includes whatever comment the speaker appended to their command. It\u2019s a comments column, essentially.\n\n## The baby is awake\n\nIt is also good to know when the baby wakes up and, crucially, how long the baby actually slept for. The Google Assistant side of this recipe is even simpler than the \u201casleep\u201d one, because we don\u2019t have any text field for additional comments:\n\nThe spreadsheet side of things is a little bit more complicated, however, because I wanted it to automatically calculate the duration of the sleep.\n\nLet\u2019s break this one down cell by cell: =REGEXREPLACE(\"{{CreatedAt}}\";\" at .*\\$\"; \" \") is just extracting the date, as before. Similarly, =REGEXREPLACE(\"{{CreatedAt}}\";\"^.* at\";\" \") is pulling out the time, and AWAKE is just telling us that the baby is awake.\n\nThen there\u2019s an empty cell for the column where comments for the falling asleep IFTTT are placed. Finally, we have this mess, which is just calculating how long the baby slept for:\n=IF(INDIRECT(ADDRESS(ROW()-1,COLUMN()-2))=\"ASLEEP\", TEXT(TIMEVALUE(INDIRECT(ADDRESS(ROW(),COLUMN()-3)))-TIMEVALUE(INDIRECT(ADDRESS(ROW()-1,COLUMN()-3))), \"hh:mm\"), \"\")\nWhat it does is check if the cell one row above and two columns over says \u201cASLEEP\u201d (to make sure the last entry was an \u201cASLEEP\u201d and not an \u201cAWAKE\u201d\u2026 sometimes you forget!), and if it is, it calculates the difference between the time of the previous row and the current time. Because IFTTT doesn\u2019t necessarily know where in the spreadsheet it is, you have to use these awkward INDIRECT(ADDRESS(row, column)) statements to select other cells in a relative position.\n\nWhat I get in the end looks something like this:\n\nEverything works pretty well \u2026except for Google\u2019s voice recognition. I assure you, I never let my enemies hold our baby.\n\n# Breastfeeding\n\nAnother thing you\u2019re often asked about, especially in the first months, is the frequency of feeding. Additionally, (and I realize this differs from region to region) we were told that we should be offering a newborn some milk every three hours. Just logging when the baby is fed is straightforward enough, and I made a script very much like the previous two:\n\n\u2026but what about when you\u2019re a tired parent and you can\u2019t remember when you last fed the baby, and want to know if it\u2019s been three hours or not? Well, it turns out that this is where things get tricky, because IFTTT travels in one direction. You can add rows to a spreadsheet just fine, and you can even calculate elapsed time (like the previous example), but there\u2019s no way to have IFTTT read that back to you.\n\nUltimately, I landed on a pretty hack-y solution, but it works. The Google Home app (that links with a Google Home device) allows you to make things called routines. These are basically meta-commands that you can define that act like you\u2019ve said a multiple things to Google Home. Google Home also happens to have a stopwatch function which we don\u2019t otherwise use.\n\nSo I defined two routines, one that looks like this:\n\nWhen\nI say \u201cfeeding baby\u201d\nAssistant will\nfeed baby routine\nclear and start stopwatch\n\nAnd another:\n\nWhen\nI say \u201clast feed\u201d\nAssistant will\ntime on stopwatch\n\nThe first routine will log the feed in the spreadsheet using the IFTTT recipe above, and then start the in-built stopwatch. Meanwhile, the second routine is just an alias for asking how much time on the stopwatch.\n\nIt\u2019s imperfect and a bit goofy, but it works. Unfortunately, the stopwatch is per-device, so if you have multiple homes or home minis they are unaware of the status of each other\u2019s stopwatches. Also annoying is that the assistant implementation on the phone doesn\u2019t seem to be capable of running a stopwatch for some reason. All the other commands work equally well on a phone as on the Google Home itself.\n\nSo there you have it. How to log baby things and whatever really using Google Home + IFTTT. Track to your heart\u2019s content!\n\ntags: \u00a0automation \u00a0baby \u00a0google \u00a0voice assistant \u00a0sleep deprivation \u00a0pee and poop","date":"2021-05-09 07:52:01","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\": 0.19248418509960175, \"perplexity\": 1779.0970562859218}, \"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-2021-21\/segments\/1620243988961.17\/warc\/CC-MAIN-20210509062621-20210509092621-00224.warc.gz\"}"}
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{"url":"https:\/\/stacks.math.columbia.edu\/tag\/006Y","text":"Example 6.7.6. Let $X$ be a topological space. Suppose for each $x\\in X$ we are given an abelian group $M_ x$. Consider the presheaf $\\mathcal{F} : U \\mapsto \\bigoplus _{x \\in U} M_ x$ defined in Example 6.4.5. This is not a sheaf in general. For example, if $X$ is an infinite set with the discrete topology, then the sheaf condition would imply that $\\mathcal{F}(X) = \\prod _{x\\in X} \\mathcal{F}(\\{ x\\} )$ but by definition we have $\\mathcal{F}(X) = \\bigoplus _{x \\in X} M_ x = \\bigoplus _{x \\in X} \\mathcal{F}(\\{ x\\} )$. And an infinite direct sum is in general different from an infinite direct product.\n\nHowever, if $X$ is a topological space such that every open of $X$ is quasi-compact, then $\\mathcal{F}$ is a sheaf. This is left as an exercise to the reader.\n\nThere are also:\n\n\u2022 3 comment(s) on Section 6.7: Sheaves\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).","date":"2022-07-01 01:40:56","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\": 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\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.980731725692749, \"perplexity\": 108.85346091795813}, \"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-2022-27\/segments\/1656103917192.48\/warc\/CC-MAIN-20220701004112-20220701034112-00714.warc.gz\"}"}
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Q: Max value of Y capacitor What is the Max value of Y cap (or what is max leakage current) should be select for EMI power Line Filter. understand increasing Y cap value will help to reduce Common mode noise but it will also increase the leakage current.
My product is Industrial grade, CISPR 11 CLASS A.
Thanks
A: First you have to know the leakage current limit for your class of equipment. For IEC60950 IT equipment it is 250uA. Then the leakage current will be given by: Cy = (Il/ ωV) – Ciw where Cy is the Y-capacitor value, Ciw is the interwinding capacitance of your transformer (specified or measured) ω is the line frequency and V is the line voltage.
Of course you need to build in margin for all the tolerances involved (check temperature variation as well) and you need to measure the leakage current to be sure the calculation is correct.
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(-) Remove Austria (32) filter Austria (32)
Project acronym AQUAMS
Project Analysis of quantum many-body systems
Researcher (PI) Robert Seiringer
Host Institution (HI) INSTITUTE OF SCIENCE AND TECHNOLOGYAUSTRIA
Summary The main focus of this project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g., Bose–Einstein condensation and superfluidity, which makes them interesting both from a physical and a mathematical point of view. The goal of this project is the development of new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and thus increase our understanding of physical systems. From the point of view of mathematical physics, there has been significant progress in the last few years in understanding the interesting phenomena occurring in quantum gases, and the goal of this project is to investigate some of the key issues that remain unsolved. Due to the complex nature of the problems, new mathematical ideas and methods will have to be developed for this purpose. One of the main question addressed in this proposal is the validity of the Bogoliubov approximation for the excitation spectrum of many-body quantum systems. While its accuracy has been successfully shown for the ground state energy of various models, its predictions concerning the excitation spectrum have so far only been verified in the Hartree limit, an extreme form of a mean-field limit where the interaction among the particles is very weak and ranges over the whole system. The central part of this project is concerned with the extension of these results to the case of short-range interactions. Apart from being mathematically much more challenging, the short-range case is the one most relevant for the description of actual physical systems. Hence progress along these lines can be expected to yield valuable insight into the complex behavior of these many-body quantum systems.
The main focus of this project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g., Bose–Einstein condensation and superfluidity, which makes them interesting both from a physical and a mathematical point of view. The goal of this project is the development of new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and thus increase our understanding of physical systems. From the point of view of mathematical physics, there has been significant progress in the last few years in understanding the interesting phenomena occurring in quantum gases, and the goal of this project is to investigate some of the key issues that remain unsolved. Due to the complex nature of the problems, new mathematical ideas and methods will have to be developed for this purpose. One of the main question addressed in this proposal is the validity of the Bogoliubov approximation for the excitation spectrum of many-body quantum systems. While its accuracy has been successfully shown for the ground state energy of various models, its predictions concerning the excitation spectrum have so far only been verified in the Hartree limit, an extreme form of a mean-field limit where the interaction among the particles is very weak and ranges over the whole system. The central part of this project is concerned with the extension of these results to the case of short-range interactions. Apart from being mathematically much more challenging, the short-range case is the one most relevant for the description of actual physical systems. Hence progress along these lines can be expected to yield valuable insight into the complex behavior of these many-body quantum systems.
Project acronym ARIPHYHIMO
Project Arithmetic and physics of Higgs moduli spaces
Researcher (PI) Tamas Hausel
Summary The proposal studies problems concerning the geometry and topology of moduli spaces of Higgs bundles on a Riemann surface motivated by parallel considerations in number theory and mathematical physics. In this way the proposal bridges various duality theories in string theory with the Langlands program in number theory. The heart of the proposal is a circle of precise conjectures relating to the topology of the moduli space of Higgs bundles. The formulation and motivations of the conjectures make direct contact with the Langlands program in number theory, various duality conjectures in string theory, algebraic combinatorics, knot theory and low dimensional topology and representation theory of quivers, finite groups and algebras of Lie type and Cherednik algebras.
The proposal studies problems concerning the geometry and topology of moduli spaces of Higgs bundles on a Riemann surface motivated by parallel considerations in number theory and mathematical physics. In this way the proposal bridges various duality theories in string theory with the Langlands program in number theory. The heart of the proposal is a circle of precise conjectures relating to the topology of the moduli space of Higgs bundles. The formulation and motivations of the conjectures make direct contact with the Langlands program in number theory, various duality conjectures in string theory, algebraic combinatorics, knot theory and low dimensional topology and representation theory of quivers, finite groups and algebras of Lie type and Cherednik algebras.
Project acronym CRYTERION
Project Cryogenic Traps for Entanglement Research with Ions
Researcher (PI) Rainer Blatt
Host Institution (HI) UNIVERSITAET INNSBRUCK
Summary Quantum computers offer a fundamentally new way of information processing. Within the scope of this proposal, quantum information processing with an ion trap quantum computer will be investigated. With the new combination of cryogenic technology and ion traps for quantum computing we intend to build a quantum information processor with strings of up to 50 ions and with two-dimensional ion arrays for an investigation of deterministic many-particle entanglement. The cryogenic traps will be applied for quantum simulations, for fundamental investigations concerning large-scale entanglement and for precision measurements enhanced by quantum metrology techniques employing entangled particles.
Quantum computers offer a fundamentally new way of information processing. Within the scope of this proposal, quantum information processing with an ion trap quantum computer will be investigated. With the new combination of cryogenic technology and ion traps for quantum computing we intend to build a quantum information processor with strings of up to 50 ions and with two-dimensional ion arrays for an investigation of deterministic many-particle entanglement. The cryogenic traps will be applied for quantum simulations, for fundamental investigations concerning large-scale entanglement and for precision measurements enhanced by quantum metrology techniques employing entangled particles.
Project acronym CSI.interface
Project A molecular interface science approach: Decoding single molecular reactions and interactions at dynamic solid/liquid interfaces
Researcher (PI) Markus Valtiner
Summary After decades of truly transformative advancements in single molecule (bio)physics and surface science, it is still no more than a vision to predict and control macroscopic phenomena such as adhesion or electrochemical reaction rates at solid/liquid interfaces based on well-characterized single molecular interactions. How exactly do inherently dynamic and simultaneous interactions of a countless number of interacting "crowded" molecules lead to a concerted outcome/property on a macroscopic scale? Here, I propose a unique approach that will allow us to unravel the scaling of single molecule interactions towards macroscopic properties at adhesive and redox-active solid/liquid interfaces. Combining Atomic Force Microscopy (AFM) based single molecule force spectroscopy and macroscopic Surface Forces Apparatus (SFA) experiments CSI.interface will (1) derive rules for describing nonlinearities observed in complex, crowded (water and ions) and chemically diverse adhesive solid/liquid interfaces; (2) uniquely characterize all relevant kinetic parameters (interaction free energy and transition states) of electrochemical and adhesive reactions/interactions of single molecules at chemically defined surfaces as well as electrified single crystal facets and step edges. Complementary, (3) my team and I will build a novel molecular force apparatus in order to measure single-molecule steady-state dynamics of both redox cycles as well as binding unbinding cycles of specific interactions, and how these react to environmental triggers. CSI.interface goes well beyond present applications of AFM and SFA and has the long-term potential to revolutionize our understanding of interfacial interaction under steady state, responsive and dynamic conditions. This work will pave the road for knowledge based designing of next-generation technologies in gluing, coating, bio-adhesion, materials design and much beyond.
After decades of truly transformative advancements in single molecule (bio)physics and surface science, it is still no more than a vision to predict and control macroscopic phenomena such as adhesion or electrochemical reaction rates at solid/liquid interfaces based on well-characterized single molecular interactions. How exactly do inherently dynamic and simultaneous interactions of a countless number of interacting "crowded" molecules lead to a concerted outcome/property on a macroscopic scale? Here, I propose a unique approach that will allow us to unravel the scaling of single molecule interactions towards macroscopic properties at adhesive and redox-active solid/liquid interfaces. Combining Atomic Force Microscopy (AFM) based single molecule force spectroscopy and macroscopic Surface Forces Apparatus (SFA) experiments CSI.interface will (1) derive rules for describing nonlinearities observed in complex, crowded (water and ions) and chemically diverse adhesive solid/liquid interfaces; (2) uniquely characterize all relevant kinetic parameters (interaction free energy and transition states) of electrochemical and adhesive reactions/interactions of single molecules at chemically defined surfaces as well as electrified single crystal facets and step edges. Complementary, (3) my team and I will build a novel molecular force apparatus in order to measure single-molecule steady-state dynamics of both redox cycles as well as binding unbinding cycles of specific interactions, and how these react to environmental triggers. CSI.interface goes well beyond present applications of AFM and SFA and has the long-term potential to revolutionize our understanding of interfacial interaction under steady state, responsive and dynamic conditions. This work will pave the road for knowledge based designing of next-generation technologies in gluing, coating, bio-adhesion, materials design and much beyond.
Project acronym CYFI
Project Cycle-Sculpted Strong Field Optics
Researcher (PI) Andrius Baltuska
Summary The past decade saw a remarkable progress in the development of attosecond technologies based on the use of intense few-cycle optical pulses. The control over the underlying single-cycle phenomena, such as the higher-order harmonic generation by an ionized and subsequently re-scattered electronic wave packet, has become routine once the carrier-envelope phase (CEP) of an amplified laser pulse was stabilized, opening the way to maintain the shot-to-shot reproducible pulse electric field. Drawing on a mix of several laser technologies and phase-control concepts, this proposal aims to take strong-field optical tools to a conceptually new level: from adjusting the intensity and timing of a principal half-cycle to achieving a full-fledged multicolor Fourier synthesis of the optical cycle dynamics by controlling a multi-dimensional space of carrier frequencies, relative, and absolute phases. The applicant and his team, through their unique expertise in the CEP control and optical amplification methods, are currently best positioned to pioneer the development of an optical programmable "attosecond optical shaper" and attain the relevant multicolor pulse intensity levels of PW/cm2. This will enable an immediate pursuit of several exciting strong-field applications that can be jump-started by the emergence of a technique for the fully-controlled cycle sculpting and would rely on the relevant experimental capabilities already established in the applicant's emerging group. We show that even the simplest form of an incommensurate-frequency synthesizer can potentially solve the long-standing debate on the mechanism of strong-field rectification. More advanced waveforms will be employed to dramatically enhance coherent X ray yield, trace the time profile of attosecond ionization in transparent bulk solids, and potentially control the result of molecular dissociation by influencing electronic coherences in polyatomic molecules.
The past decade saw a remarkable progress in the development of attosecond technologies based on the use of intense few-cycle optical pulses. The control over the underlying single-cycle phenomena, such as the higher-order harmonic generation by an ionized and subsequently re-scattered electronic wave packet, has become routine once the carrier-envelope phase (CEP) of an amplified laser pulse was stabilized, opening the way to maintain the shot-to-shot reproducible pulse electric field. Drawing on a mix of several laser technologies and phase-control concepts, this proposal aims to take strong-field optical tools to a conceptually new level: from adjusting the intensity and timing of a principal half-cycle to achieving a full-fledged multicolor Fourier synthesis of the optical cycle dynamics by controlling a multi-dimensional space of carrier frequencies, relative, and absolute phases. The applicant and his team, through their unique expertise in the CEP control and optical amplification methods, are currently best positioned to pioneer the development of an optical programmable "attosecond optical shaper" and attain the relevant multicolor pulse intensity levels of PW/cm2. This will enable an immediate pursuit of several exciting strong-field applications that can be jump-started by the emergence of a technique for the fully-controlled cycle sculpting and would rely on the relevant experimental capabilities already established in the applicant's emerging group. We show that even the simplest form of an incommensurate-frequency synthesizer can potentially solve the long-standing debate on the mechanism of strong-field rectification. More advanced waveforms will be employed to dramatically enhance coherent X ray yield, trace the time profile of attosecond ionization in transparent bulk solids, and potentially control the result of molecular dissociation by influencing electronic coherences in polyatomic molecules.
Project acronym Feel your Reach
Project Non-invasive decoding of cortical patterns induced by goal directed movement intentions and artificial sensory feedback in humans
Researcher (PI) Gernot Rudolf Mueller-Putz
Host Institution (HI) TECHNISCHE UNIVERSITAET GRAZ
Summary In Europe estimated 300.000 people are suffering from a spinal cord injury (SCI) with 11.000 new injuries per year. The consequences of spinal cord injury are tremendous for these individuals. The loss of motor functions especially of the arm and grasping function – 40% are tetraplegics – leads to a life-long dependency on care givers and therefore to a dramatic decrease in quality of life in these often young individuals. With the help of neuroprostheses, grasp and elbow function can be substantially improved. However, remaining body movements often do not provide enough degrees of freedom to control the neuroprosthesis. The ideal solution for voluntary control of an upper extremity neuroprosthesis would be to directly record motor commands from the corresponding cortical areas and convert them into control signals. This would realize a technical bypass around the interrupted nerve fiber tracts in the spinal cord. A Brain-Computer Interface (BCI) transform mentally induced changes of brain signals into control signals and serve as an alternative human-machine interface. We showed first results in EEG-based control of a neuroprosthesis in several persons with SCI in the last decade, however, the control is still unnatural and cumbersome. The objective of FEEL YOUR REACH is to develop a novel control framework that incorporates goal directed movement intention, movement decoding, error processing, processing of sensory feedback to allow a more natural control of a neuroprosthesis. To achieve this aim a goal directed movement decoder will be realized, and continuous error potential decoding will be included. Both will be finally joined together with an artificial kinesthetic sensory feedback display attached to the user. We hypothesize that with these mechanisms a user will be able to naturally control an neuroprosthesis with his/ her mind only.
In Europe estimated 300.000 people are suffering from a spinal cord injury (SCI) with 11.000 new injuries per year. The consequences of spinal cord injury are tremendous for these individuals. The loss of motor functions especially of the arm and grasping function – 40% are tetraplegics – leads to a life-long dependency on care givers and therefore to a dramatic decrease in quality of life in these often young individuals. With the help of neuroprostheses, grasp and elbow function can be substantially improved. However, remaining body movements often do not provide enough degrees of freedom to control the neuroprosthesis. The ideal solution for voluntary control of an upper extremity neuroprosthesis would be to directly record motor commands from the corresponding cortical areas and convert them into control signals. This would realize a technical bypass around the interrupted nerve fiber tracts in the spinal cord. A Brain-Computer Interface (BCI) transform mentally induced changes of brain signals into control signals and serve as an alternative human-machine interface. We showed first results in EEG-based control of a neuroprosthesis in several persons with SCI in the last decade, however, the control is still unnatural and cumbersome. The objective of FEEL YOUR REACH is to develop a novel control framework that incorporates goal directed movement intention, movement decoding, error processing, processing of sensory feedback to allow a more natural control of a neuroprosthesis. To achieve this aim a goal directed movement decoder will be realized, and continuous error potential decoding will be included. Both will be finally joined together with an artificial kinesthetic sensory feedback display attached to the user. We hypothesize that with these mechanisms a user will be able to naturally control an neuroprosthesis with his/ her mind only.
Project acronym FLATOUT
Project From Flat to Chiral: A unified approach to converting achiral aromatic compounds to optically active valuable building blocks
Researcher (PI) Nuno Xavier Dias Maulide
Summary "The stereoselective preparation of enantioenriched organic compounds of high structural complexity and synthetic value, in an economically viable and expeditious manner, is one of the most important goals in contemporary Organic Synthesis. In this proposal, I present a unified and conceptually novel approach for the conversion of flat, aromatic heterocycles into highly valuable compounds for a variety of applications. This approach hinges upon a synergistic combination of the dramatic power of organic photochemical transformations combined with the exceedingly high selectivity and atom-economy of efficient catalytic processes. Indeed, the use of probably the cheapest reagent (light) combined with a catalytic transformation ensures near perfect atom-economy in this journey from flat and inexpensive substructures to chiral added-value products. Conceptually, the photochemical operation is envisaged as a energy-loading step whereas the catalytic transformation functions as an energy-release where asymmetric information is inscribed into the products. The chemistry proposed herein will open up new vistas in enantioselective synthesis. Furthermore, groundbreaking and unprecedented methodology in the field of catalytic allylic alkylation is proposed that significantly expands (and goes beyond) the currently accepted "dogmas" for these textbook reactions. These include (but are not limited to) systematic violations of well-established rules "by design", new contexts for application, new activation modes and innovative leaving groups. Finally, the comprehensive body of synthetic technology presented will be applied to pressing target-oriented problems in Organic Synthesis. It shall result in a landmark, highly efficient total synthesis of Tamiflu, as well as in application to an environmentally important target (Fomannosin), allowing the easy production of analogues for biological testing."
"The stereoselective preparation of enantioenriched organic compounds of high structural complexity and synthetic value, in an economically viable and expeditious manner, is one of the most important goals in contemporary Organic Synthesis. In this proposal, I present a unified and conceptually novel approach for the conversion of flat, aromatic heterocycles into highly valuable compounds for a variety of applications. This approach hinges upon a synergistic combination of the dramatic power of organic photochemical transformations combined with the exceedingly high selectivity and atom-economy of efficient catalytic processes. Indeed, the use of probably the cheapest reagent (light) combined with a catalytic transformation ensures near perfect atom-economy in this journey from flat and inexpensive substructures to chiral added-value products. Conceptually, the photochemical operation is envisaged as a energy-loading step whereas the catalytic transformation functions as an energy-release where asymmetric information is inscribed into the products. The chemistry proposed herein will open up new vistas in enantioselective synthesis. Furthermore, groundbreaking and unprecedented methodology in the field of catalytic allylic alkylation is proposed that significantly expands (and goes beyond) the currently accepted "dogmas" for these textbook reactions. These include (but are not limited to) systematic violations of well-established rules "by design", new contexts for application, new activation modes and innovative leaving groups. Finally, the comprehensive body of synthetic technology presented will be applied to pressing target-oriented problems in Organic Synthesis. It shall result in a landmark, highly efficient total synthesis of Tamiflu, as well as in application to an environmentally important target (Fomannosin), allowing the easy production of analogues for biological testing."
Project acronym FLOODCHANGE
Project Deciphering River Flood Change
Researcher (PI) Guenter Bloeschl
Call Details Advanced Grant (AdG), PE10, ERC-2011-ADG_20110209
Summary Many major and devastating floods have occurred around the world recently. Their number and magnitude seems to have increased but such changes are not clear. More surprisingly, the exact causes of changes remain a mystery. Although, drivers such as climate and land use change are known to play a critical role, their complex interactions in flood generation have not been disentangled. The main objectives of this project are to understand how changes in land use and climate translate into changes in river floods, what are the factors controlling this relationship and what are the uncertainties involved. We decipher the relationship between changes in floods and their drivers by analysing the processes separately for different flood types such as flash floods, rain-on-snow floods and large scale synoptic floods. We then use data from catchments in transects across Europe to build a probabilistic flood-change model that explicitly describes the change mechanisms. The model is unconventional as it does not take a reductionist approach but conceptualises the dominant flood change processes at the catchment scale. We test the model on long high-quality flood data series. We use the model as well as the temporal and spatial data variability to quantify the sensitivity of floods to climate and land use change and estimate the uncertainties involved. The data are already available to me or will be made available through my excellent contacts in Europe. For the first time, it will be possible to systematise the effects of land use and climate on floods which will provide a vital step towards predicting how floods will change in the future.
Many major and devastating floods have occurred around the world recently. Their number and magnitude seems to have increased but such changes are not clear. More surprisingly, the exact causes of changes remain a mystery. Although, drivers such as climate and land use change are known to play a critical role, their complex interactions in flood generation have not been disentangled. The main objectives of this project are to understand how changes in land use and climate translate into changes in river floods, what are the factors controlling this relationship and what are the uncertainties involved. We decipher the relationship between changes in floods and their drivers by analysing the processes separately for different flood types such as flash floods, rain-on-snow floods and large scale synoptic floods. We then use data from catchments in transects across Europe to build a probabilistic flood-change model that explicitly describes the change mechanisms. The model is unconventional as it does not take a reductionist approach but conceptualises the dominant flood change processes at the catchment scale. We test the model on long high-quality flood data series. We use the model as well as the temporal and spatial data variability to quantify the sensitivity of floods to climate and land use change and estimate the uncertainties involved. The data are already available to me or will be made available through my excellent contacts in Europe. For the first time, it will be possible to systematise the effects of land use and climate on floods which will provide a vital step towards predicting how floods will change in the future.
Project acronym GEMIS
Project Generalized Homological Mirror Symmetry and Applications
Researcher (PI) Ludmil Katzarkov
Summary Mirror symmetry arose originally in physics, as a duality between $N = 2$ superconformal field theories. Witten formulated a more mathematically accessible version, in terms of topological field theories. Both conformal and topological field theories can be defined axiomatically, but more interestingly, there are several geometric ways of constructing them. A priori, the mirror correspondence is not unique, and it does not necessarily remain within a single class of geometric models. The classical case relates $\sigma$-models, but in a more modern formulation, one has mirror dualities between different Landau-Ginzburg models, as well as between such models and $\sigma$-models; orbifolds should also be included in this. The simplest example would be the function $W: \C \rightarrow \C$, $W(x) = x^{n+1}$, which is self-mirror (up to dividing by the $\bZ/n+1$ symmetry group, in an orbifold sense). While the mathematics of the $\sigma$-model mirror correspondence is familiar by now, generalizations to Landau-Ginzburg theories are only beginning to be understood. Today it is clear that Homologcal Mirror Symmetry (HMS) as a categorical correspondence works and it is time for developing direct geometric applications to classical problems - rationality of algebraic varieties and Hodge conjecture. This the main goal of the proposal. But in order to attack the above problems we need to generalize HMS and explore its connection to new developments in modern Hodge theory. In order to carry the above program we plan to further already working team Vienna, Paris, Moscow, MIT.
Mirror symmetry arose originally in physics, as a duality between $N = 2$ superconformal field theories. Witten formulated a more mathematically accessible version, in terms of topological field theories. Both conformal and topological field theories can be defined axiomatically, but more interestingly, there are several geometric ways of constructing them. A priori, the mirror correspondence is not unique, and it does not necessarily remain within a single class of geometric models. The classical case relates $\sigma$-models, but in a more modern formulation, one has mirror dualities between different Landau-Ginzburg models, as well as between such models and $\sigma$-models; orbifolds should also be included in this. The simplest example would be the function $W: \C \rightarrow \C$, $W(x) = x^{n+1}$, which is self-mirror (up to dividing by the $\bZ/n+1$ symmetry group, in an orbifold sense). While the mathematics of the $\sigma$-model mirror correspondence is familiar by now, generalizations to Landau-Ginzburg theories are only beginning to be understood. Today it is clear that Homologcal Mirror Symmetry (HMS) as a categorical correspondence works and it is time for developing direct geometric applications to classical problems - rationality of algebraic varieties and Hodge conjecture. This the main goal of the proposal. But in order to attack the above problems we need to generalize HMS and explore its connection to new developments in modern Hodge theory. In order to carry the above program we plan to further already working team Vienna, Paris, Moscow, MIT.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: Slim v4 URL decode returns 404 i am upgrading my application from slim v2 to v4
i have one route function menitoned below
$group->get('/get-data/{url}', '\V2:get_data');
when i pass the url variable a
api.app.com/get-data/xxxx%2fyyyyy
the route gives 404 ,
i tried with accessing the url in function with args but
function get_data($request,$response,$args){
$slug =$args['url'];
print_r($slug);die;
}
but it doesn't even entering into the function
can anyone help with how we can pass the dynamic para with %2f in slim v4
A: Slim uses fastroute as router. This is just a known limitation with FastRoute.
The reason is, %2f is internally converted to /, and this is also a path segment delimiter. You may also notice a similar behavior with %2e which is just a . dot. But don't ask me why this happens with the dot.
URL path parameters should be more "simple", for example just a numeric value or just simple strings. That's why many use just the 62 alphanumeric characters (i.e. A–Z, a–z, 0–9). *
For more complex queries, a typical query string might be more appropriate. This would then not affect the routing path.
If you want to use the Symfony router, check out this sample repo:
https://github.com/l0gicgate/slim4-symfony-router-exp
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,080
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\section{Introduction}
The chromatic number of a graph is the minimum number of different colors required to color the vertices so that
no edge connects vertices of the same color.
In \cite{galliardwolf, AHKS06, cnmsw, SS12} the concept of the quantum chromatic number $\chi_{q}(G)$
of a graph $G$ was developed and inequalities for estimating this parameter,
as well as methods for its computation, were presented.
In \cite{pt_chrom} several new variants of the quantum chromatic number, especially,
$\chi_{qc}(G)$ and $\chi_{qa}(G)$, were introduced.
The motivation behind these new chromatic numbers came from the
conjectures of Tsirelson and Connes and the fact that the set of correlations of
quantum experiments may possibly depend on which set of quantum mechanical axioms one
chooses to employ.
Given a graph $G$, the aforementioned chromatic numbers satisfy the inequalities
\[
\chi_{qc}(G) \le \chi_{qa}(G) \le \chi_{q}(G) \le \chi(G),
\]
where $\chi(G)$ denotes the classical chromatic number of the graph $G$.
If the strong form of Tsirelson's conjecture is true, then $\chi_{qc}(G) = \chi_{q}(G)$ for every graph $G$, while if
Connes' Embedding
Conjecture is true,
then $\chi_{qc}(G) = \chi_{qa}(G)$ for every graph $G$.
Thus, computing these invariants gives a means to test the corresponding conjectures.
The fractional chromatic number $\chi_{f}(G)$ of a graph $G$ is an important lower bound on $\chi(G).$
D.~Roberson and L.~Man\v{c}inska~\cite{arxiv:1212.1724} introduced a non-commutative analogue of the fractional
chromatic number, which they called the
{\it projective rank} of $G$, denoted $\xi_{f}(G)$
and proved that $\xi_{f}(G)$ is a lower bound for $\chi_{q}(G)$.
However, it is still not known if $\xi_{f}(G)$ is a lower bound for
the variants of the quantum chromatic number studied in \cite{pt_chrom}.
The paper \cite{PSSTW} introduced a new C*-algebra built from a graph and using traces on these algebras introduced a parameter $\xi_{tr}(G)$ which they called the
\emph{tracial rank} and showed that it is a lower bound for $\chi_{qc}(G)$.
They also gave a new interpretation of the projective rank and fractional chromatic number,
by proving that if one restricted the C*-algebras in the definition of the tracial rank to be
either finite dimensional C*-algebras or abelian, then one obtained the projective rank and fractional chromatic number,
respectively.
In this paper, we prove that if Connes' embedding conjecture is true, then the tracial and projective ranks
are equal for all graphs.
A key result of \cite{PSSTW}, that allowed the introduction of traces, was a correspondence between
certain quantum correlations, called ``synchronous correlations'' and
traces on a
particular
C*-algebra.
In this paper we further develop the theory of synchronous correlations.
Using the equivalence of the microstates conjecture with Connes' embedding conjecture, we are able to prove that
Connes' embedding conjecture is equivalent to equality of two families of sets of synchronous quantum correlations.
In \cite{PSSTW}, it was noted that the graph theoretic parameters that they were studying all belonged to a family of two person games that they called ``synchronous games''. We further develop the connection between traces and synchronous games. In particular, we introduce the ``synchronous value'' of a two person game and show that it is equal to the supremum of
the values of
all tracial states on a fixed element of a particular C*-algebra.
\section{Quantum correlation matrices and Non-Local Games}
\label{sec:Qmats}
Imagine that two non-communicating players, Alice and Bob, receive inputs from some finite set $X$ of cardinality $n$ and produce
outputs belonging to some finite set $O$ of cardinality $m.$
The game $\cl G$ has ``rules'' given by a function
\[ \lambda: X \times X \times O \times O \to \{0,1 \} \]
where $\lambda(x,y,a,b) =0$ means that if Alice and Bob receive inputs $x,y,$ respectively,
then producing respective outputs $a,b$ is ``disallowed''.
The game is {\bf synchronous} if whenever they receive the same input,
they must produce the same output, i.e., $\lambda (x,x, a,b) = 0, \, \forall a \ne b.$
A {\bf quantum strategy} for a game $\cl G$ means that Alice and Bob have finite
dimensional Hilbert spaces $H_A$, $H_B$ and for each input $x \in X$
Alice has a projective measurement $\{ E_{x,a} \}_{a \in O}$ on $H_A$, i.e., for each $x \in X$ Alice has a set of projections satisfying,
$\sum_{a\in O}E_{x,a}=I$,
and, similarly, for each input $y \in X$
Bob has a projective measurement $\{ F_{y,b} \}_{b \in O}$ on $H_B$;
moreover,
they share a state $\psi \in H_A \otimes H_B,$ i.e., a unit vector. In this case
\[ p(a,b|x,y) := \langle E_{x,a} \otimes F_{y,b} \psi, \psi \rangle \]
is the probability of getting outcomes $a,b$ given inputs $x,y.$
The set of $n \times m$ matrices of the form $\big( p(a,b|x,y) \big)$,
arising from all choices of finite dimensional Hilbert spaces $H_A$ and $H_B$, all projective measurements,
and all unit vectors,
is called the set of {\bf quantum correlation matrices}
and is, usually, denoted $Q(n,m).$
For
a slight improvement of
notation
from \cite{PSSTW} we set $C_q(n,m):= Q(n,m)$
and similarly for related sets of correlation matrices, as described below.
A quantum strategy is called
a {\bf winning quantum strategy} for the game if the probability of it ever producing a
disallowed output is 0. Thus, a winning quantum strategy
is
$\big( p(a,b|x,y) \big) \in C_q(n,m)$
such that
\[ \lambda(x,y,a,b) =0 \implies p(a,b|x,y) =0.\]
If the game is synchronous, then a winning quantum strategy must satisfy $p(a,b|x,x) =0$ for all $x$ and for all $a \ne b,$ and, consequently,
we call such a correlation tuple {\bf synchronous}.
We let $C^s_q(n,m)$ denote the set of all synchronous quantum correlation matrices.
By a {\bf commuting quantum strategy} we mean that there is a single
(possibly infinite dimensional)
Hilbert space $H$,
and for each $x \in X$ Alice has a projective measurement $\{ E_{x,a} \}_{a \in O}$ on $H$,
and for each $y \in X$ Bob has a projective measurement $\{F_{y,b} \}_{b \in O} $ on $H$,
satisfying $E_{x,a}F_{y,b} = F_{y,b}E_{a,x}, \, \forall x,y,a,b$ and they share a state $\psi \in H.$
In this case the probabilities are given by
\[ p(a,b|x,y) = \langle E_{x,a}F_{y,b} \psi, \psi \rangle.\]
We let $C_{qc}(n,m)$ denote the set of {\bf commuting quantum correlation matrices},
namely, those $n \times m$ matrices $(p(a,b,|x,y))$ arising as described above,
and
we let
$C^s_{qc}(n,m)$ denote the subset of synchronous
commuting quantum correlation
matrices.
Clearly, $C_q(n,m)\subseteq C_{qc}(n,m)$.
The {\bf strong Tsirelson conjecture} is the conjecture that $C_q(n,m) = C_{qc}(n,m)$ for all $n,m.$
It is known that the set $C_{qc}(n,m)$ is closed but it is still not known if $C_q(n,m)$ is closed. So we set
$C_{qa}(n,m)$ equal to the closure of $C_q(n,m)$ and let $C^s_{qa}(n,m)$ denote the synchronous elements
of $C_{qa}(n,m).$
The {\bf weak Tsirelson conjecture} is the conjecture that $C_{qa}(n,m) = C_{qc}(n,m)$ for all $n,m.$
Junge, Navascues, Palazuelos, Perez-Garcia, Scholtz and Werner \cite{jnppsw} proved that if Connes' embedding conjecture is true, then the
weak Tsirelson conjecture is true. Recently, Ozawa \cite{oz} proved the converse, so we now know that the weak
Tsirelson conjecture and Connes' embedding conjecture are equivalent.
A {\bf classical strategy} or {\bf local strategy} is any commuting quantum strategy for which
all the measurement operators commute. The set of these correlation matrices is generally denoted $LOC(n,m)$, but for consistency
of notation we denote these by $C_{loc}(n,m)$ and we let $C^s_{loc}(n,m)$ denote the synchronous matrices in this set.
In summary, we have four types of correlation matrices
\[ C_{loc}(n,m) \subseteq C_q(n,m) \subseteq C_{qa}(n,m) \subseteq C_{qc}(n,m),\]
together with their synchronous subsets
\[ C_{loc}^s(n,m) \subseteq C_q^s(n,m) \subseteq C_{qa}^s(n,m) \subseteq C_{qc}^s(n,m).\]
One important example of a synchronous game is the {\bf graph coloring game}. Given a graph $G=(V,E)$ on $n$
vertices where $V$ denotes the vertex set and $E \subseteq V \times V$ denotes the set of edges, we write $v \sim w$ whenever
$(v,w) \in E.$ In the graph coloring game the inputs are the vertices, i.e., $X=V$ and the outputs are a set of $c$ colors,
so without loss of generality, $O = \{ 1, \ldots, c \}.$ The rules are that whenever $v=w$ then Alice and Bob must both output the
same color, so the game is synchronous, and whenever $v \sim w$ then they must output different colors.
The quantum coloring number, $\chi_x(G)$ for $x= loc, q, qa, qc$ is the least integer $c$ for which there exists a winning strategy
corresponding to a correlation matrix in $C^s_x(n,c).$ Interestingly, the classical chromatic number $\chi(G)$ is equal to
$ \chi_{loc}(G)$~\cite{pt_chrom}.
In \cite[Theorem~5.4]{PSSTW} an important connection was made between synchronous correlation matrices and traces on C*-algebras.
Recall that a positive linear functional $\tau: \cl A \to \bb C$ on a unital C*-algebra $\cl A$ is called a {\bf tracial state} provided that $\tau(1) =1$ and $\tau(uv) = \tau (vu)$ for all $u,v \in \cl A.$
We summarize their result below.
\begin{thm}[\cite{PSSTW}]\label{synctr} $\big( p(a,b|x,y) \big) \in C^s_{qc}(n,m)$ if and only if there exists a unital C*-algebra $\cl A$ generated by projections
$\{ e_{x,a} \}_{1 \le x \le n, 1 \le a \le m}$ satisfying $\sum_{a=1}^m e_{x,a} = 1, \, \forall x$ and a tracial state $\tau: \cl A \to \bb C$ such that
\[ p(a,b|x,y) = \tau(e_{x,a}e_{y,b}), \forall x,y,a,b.\]
Moreover, $\big( p(a,b|x,y) \big) \in C^s_q(n,m)$ if and only if the C*-algebra $\cl A$ can be taken to be finite dimensional,
and $\big( p(a,b|x,y) \big) \in C^s_{loc}(n,m)$ if and only if the C*-algebra $\cl A$ can be taken to be abelian.
\end{thm}
\medskip
In order to bound quantum chromatic numbers,
the paper
\cite{PSSTW} introduced
some parameters
of a graph, denoted $\xi_x,$ for $x \in \{ loc, q, qa, qc \}$ that had many nice properties,
including the fact that
they are
multiplicative for strong graph product.
Given a correlation matrix $\big( p(a,b|v,w) \big)$, the {\bf marginal probability} that Alice produces output $a$ given input $v$ is
\[ p_A(a,v) = \langle E_{v,a} \psi, \psi \rangle . \]
Note that
\[ p_A(a,v) = \sum_{b=1}^m p(a,b|v,w) ,\]
where the sum is independent of $w.$ Similarly, the marginal probability that Bob produces output $b$ given input $w$ is
\[p_B(b|w) = \langle F_{w,b} \psi, \psi \rangle = \sum_{a=1}^m p(a,b|v,w).\]
When $\big( p(a,b|v,w) \big)$ is synchronous, it follows from Theorem~\ref{synctr} that $p_A(a|v) = p_B(a|v).$
\begin{defn}\label{def:xiX}
Let $G$ be a graph on $n$ vertices.
For $x \in \{loc, q, qa, qc \}$, let $\xi_{x}(G)$ be the infimum of the
positive real numbers
$t$ such that there exists $\big(p(a,b|v,w) \big)_{v,a,w,b} \in C^s_{x}(n,2)$
satisfying
\begin{gather*}
p_A(1|v)= p_B(1,v)= t^{-1}, \, \forall v, \\
v \sim w \implies p(1,1 | v, w)= 0.
\end{gather*}
\end{defn}
We have the following summary of the results in \cite{PSSTW}
(see \cite[Definition~5.9, Proposition~5.10, Theorem~6.8, Theorem~6.11]{PSSTW}).
\begin{thm}[\cite{PSSTW}] \label{xichar} Let $G$ be a graph on $n$ vertices. Then
$\xi_{qc}(G)$ is the reciprocal of the supremum of the set of all real numbers $\lambda$ for which there exists a unital C*-algebra
$\cl A$ generated by projections $\{ e_v \}_{v \in V}$ and a tracial state $\tau: \cl A \to \bb C$ satisfying:
\begin{align*}
\tau(e_v) \ge \lambda, &\, \forall v,\\
v \sim w \implies & e_ve_w =0.
\end{align*}
Moreover, if we require, in addition, that $\cl A$ be finite dimensional, then we obtain $\xi_{q}(G)$, which is equal to the Mancinska-Roberson
projective rank $\xi_f(G).$ If we require, in addition, that $\cl A$ be abelian, then we obtain $\xi_{loc}(G)$, which
is equal to the fractional chromatic number $\chi_f(G).$
\end{thm}
In the next section, we prove that if Connes' embedding is true, then $\xi_{qc}(G) = \xi_f(G)$ for every graph $G.$
\section{Microstates, correlation matrices and Connes' embedding conjecture}
\label{sec:CEC}
Connes' embedding conjecture is one of the most important outstanding problems in operator algebra theory.
Connes asked~\cite{C76} whether every II$_1$-factor having separable predual is embeddable in an ultrapower $R^\omega$
of the hyperfinite II$_1$-factor $R$.
This is, in essence, a question about approximation (in terms of moments) of elements of II$_1$ factors by matrices.
To use a term introduced by Voiculescu, if $(\Mcal,\tau)$ is a tracial von Neumann algebra
(namely, a von Neumann algebra $\Mcal$ with normal, faithful tracial state $\tau$)
and if $x_1,\ldots,x_n$ are self-adjoint elements of $\Mcal$,
we say that the tuple $(x_1,\ldots,x_n)$ {\em has matricial microstates} if for every $\eps>0$ and every
integer $N\ge1$, there is $k$ and there are $k\times k$ self-adjoint matrices $A_1,\ldots,A_n$ such that for all $p\le N$
and every $i_1,\ldots,i_p\in\{1,\ldots,n\}$, we have
\[
\big|\tr_k(A_{i_1}\cdots A_{i_p})-\tau(x_{i_1}\cdots x_{i_p})\big|<\eps.
\]
The set of matrices, $A_1, \ldots, A_n$ is called a $(N, \epsilon)$-matricial microstate for $x_1, \ldots, x_n.$
The following result is well known and follows quite easily from the definition of $R^\omega$; see, for example, \cite{V02}, p.\ 264, Remark~(d).
\begin{thm}\label{thm:embeddable}
For a countably generated, tracial von Neumann algebra $(\Mcal,\tau)$, the following are equivalent:
\begin{enumerate}[(i)]
\item $\Mcal$ is embeddable in $R^\omega$
\item all finite families of self-adjoint elements of $\Mcal$ have matricial microstates.
\end{enumerate}
Furthermore, assuming $\Mcal$ is finitely generated, the above conditions are also equivalent to the following condition:
\begin{enumerate}[(i)]
\setcounter{enumi}{2}
\item some finite generating set of $\Mcal$ (consisting of self-adjoint elements) has matricial microstates.
\end{enumerate}
\end{thm}
\begin{prop}\label{closureconnes}
If $C_{qc}^s(n,m)$ equals the closure of $C_q^s(n,m)$ for all $n,m,$ then Connes' embedding conjecture is true.
\end{prop}
\begin{proof}
By a result of Kirchberg~\cite{Ki93},
the truth of Connes' embedding conjecture is equivalent
to the ``unitary moments'' assertion,
which states that whenever $n\in\Nats$ and $u_1,\ldots,u_n$ are unitary elements of a II$_1$-factor $\Mcal$, (with tracial state denoted $\tau$)
and whenever $\eps>0$, there is $p\in\Nats$ and there are unitary matrices $U_1,\ldots,U_n\in M_p(\Cpx)$ such that
$|\tau(u_ku_\ell^*)-\tr_p(U_kU_\ell^*)|<\eps$ for all $k,\ell\in\{1,\ldots,n\}$.
(See~\cite{DJ11} for discussion of this slight modification of Kirchberg's formulation.)
We will observe that the ``unitary moments'' assertion follows if we assume $\overline{C_q^s(n,m)}=C_{qc}^s(n,m)$.
Let $u_1,\ldots,u_n$ be as above and let $\eps>0$.
Take an integer $m>6\pi/\eps$.
Let
\[
\ut_k=\sum_{j=1}^m\omega^je_{k,j}
\]
where $\omega=\exp(\frac{2\pi\sqrt{-1}}m)$ and
where $e_{k,j}$ is the spectral projection of the unitary $u_k$ for the arc
\[
\bigg\{\exp(2\pi t\sqrt{-1})\;\bigg|\;\frac{j-1}m\le t<\frac jm\bigg\}.
\]
Then
$\|\ut_k-u_k\|\le|1-\omega|<\eps/3$.
By hypothesis, there exist $p\in\Nats$ and projections $E_{k,j}$ in $M_p(\Cpx)$ such that
\[
\sum_{j=1}^mE_{k,j}=1\qquad(1\le k\le n)
\]
and
\[
\big|\tr_p(E_{k,i}E_{\ell,j})-\tau(e_{k,i}e_{\ell.j})\big|<\frac\eps{3m^2},\qquad(1\le k,\ell\le n,\;1\le i,j\le m).
\]
Let
\[
U_k=\sum_{j=1}^m\omega^jE_{k,j}.
\]
Then $U_k\in M_p(\Cpx)$ is unitary and $|\tr_p(U_kU_\ell^*)-\tau(\ut_k\ut_\ell^*)|<\eps/3$ for all $1\le k,\ell\le n$.
This implies $|\tr_p(U_kU_\ell^*)-\tau(u_ku_\ell^*)|<\eps$.
\end{proof}
As usual, we will denote by $\|\cdot\|_2$ the $2$-norm on $M_k(\bb C)$, given by $\|x\|_2=\tr_k(x^*x)^{1/2}$.
\begin{lemma}\label{lem:hdelta}
Let $\tau$ be a tracial state on a finite dimensional abelian C$^*$-algebra $A=\bb C^m$ and let $h\in A$
be a self-adjoint generator of $A$.
Let $\delta>0$.
Then there exists a positive integer $N$ and $\eps>0$ such that for all sufficiently large positive integers $k$,
if $a\in M_k(\bb C)$
is an $(N,\eps)$-microstate for $h$, then there is a unital $*$--representation $\pi:A\to M_k(\bb C)$
so that $\|\pi(h)-a\|_2<\delta$.
\end{lemma}
\begin{proof}
By Lemma~4.3 of~\cite{V94}, there is an integer $N>0$ and there is $\eps>0$ such that whenever
$a,b\in M_k(\bb C)$ are $(N,\eps)$-microstates for $h$, then there is a unitary $u\in M_k(\Cpx)$ such that
$\|a-ubu^*\|_2<\delta$.
For all $k$ sufficiently large, there is a unital $*$-representation $\pi:A\to M_k(\Cpx)$
so that $\pi(h)$ is an $(N,\eps)$-microstate for $h$.
Be Voiculescu's result, we can find unitary $u$ so that $\|a-u\pi(h)u^*\|<\delta$,
and replacing $\pi$ by $u\pi(\cdot)u^*$,
we are done.
\end{proof}
Let $\bb F(n,m)$ denote the free product of $n$ copies of the cyclic group of order $m,$ $\bb Z_m$ and let $\CFnm$ be the full group C$^*$-algebra.
Then $\CFnm$ is the universal unital free product C$^*$-algebra
\begin{equation}\label{eq:A}
\CFnm=*_1^n\Cpx^m
\end{equation}
of $n$ copies of the $m$--dimensional abelian C$^*$-algebra $\Cpx^m$.
\begin{defn}
Fix a set $H$ of self-adjoint elements of $\Cpx^m$, each of norm $\le1$, that generates $\Cpx^m$ as a unital algebra.
For every $j\in\{1,\ldots,n\}$,
let $H_j\subset\CFnm$ be the copy of $H$ in the $j$-th copy of $\Cpx^m$ in $\CFnm$, so that $\CFnm$ is generated by
$H_1\cup\cdots\cup H_n$.
For $\tau$ and $\sigma$ tracial states on $\CFnm$ and for $N\in\Nats$ and $\eps>0$, we will say that
{\em $\sigma$ approximates $\tau$ with tolerance $(N,\eps)$ for the generating set $H$}
if for every $p\in\{1,\ldots,N\}$ and $x_1,\dots,x_p\in H_1\cup\cdots\cup H_n$, we have
\[
\left|\tau(x_1\cdots x_p)-\sigma(x_1\cdots x_p)\right|<\eps.
\]
\end{defn}
\begin{remark}\label{rmk:HH'}
If $H'$ is another generating set of $\Cpx^m$ consisting of self-adjoint elements, then each element of $H'$ is a polynomial in elements of $H$.
Thus,
for every $N'\in\Nats$ and $\eps'>0$, there are $N\in\Nats$ and $\eps>0$ such that if
$\sigma$ approximates $\tau$ with tolerance $(N,\eps)$ for the generating set $H$,
then
$\sigma$ approximates $\tau$ with tolerance $(N',\eps')$ for the generating set $H'$.
\end{remark}
\begin{prop}\label{prop:tauh}
Suppose Connes' embedding conjecture is true.
Let $H$ be a finite generating set for $\Cpx^m$,
consisting of self-adjoint elements.
Let $\tau$ be a tracial state on $\CFnm$.
Let $N\in\Nats$ and $\eps>0$.
Then there exists $k\in\Nats$ and a unital $*$--homomorphism $\pi:\CFnm\to M_k(\Cpx)$
so that the trace $\tr_k\circ\pi$ approximates $\tau$ with tolerance $(N,\eps)$
for the generating set $H$.
\end{prop}
\begin{proof}
In light of Remark~\ref{rmk:HH'}, we may without loss of generality assume $H$ is a singleton set, $H=\{h\}$,
and we write $H_j=\{h_j\}$.
Take $0<\delta<\eps/(2N)$.
Let $N_j\in\Nats$ and $\eps_j>0$ be obtained from Lemma~\ref{lem:hdelta}, so that for every
$(N_j,\eps_j)$-microstate $a_j\in M_k(\Cpx)$ for $h_j$,
there is a unital $*$-homomorphism $\pi_j:\Cpx^m\to M_k(\Cpx)$
with
\begin{equation}\label{eq:piha}
\|\pi_j(h_j)-a_j\|_2<\delta.
\end{equation}
Let $N'=\max(N,N_1,\ldots,N_n)$ and $\eps'=\min(\eps/2,\eps_1,\ldots,\eps_n)$.
By the assumption that Connes' embedding conjecture is true, there exists
an $(N',\eps')$-microstate $(a_1,\ldots,a_n)$ for $(h_1,\ldots,h_n)$.
By the choice of $(N',\eps')$, there exist $*$-homomorphisms $\pi_j:\Cpx^m\to M_k(\Cpx)$ as above,
so that~\eqref{eq:piha} holds.
By the choice of $\delta$, it follows that $(\pi_1(h_1),\ldots,\pi_n(h_n))$ is an $(N,\eps)$-microstate for $(h_1,\ldots,h_n)$.
We have the universal free product $*$-homomorphism $\pi=*_1^n\pi_j:\CFnm\to M_k(\Cpx)$
and the previous statement implies that $\tr_k\circ\pi$ approximates $\tau$ with tolerance $(N,\eps)$
for the generating set $H$.
\end{proof}
\begin{thm}\label{connes=closure} Connes' embedding conjecture is true if and only if $C^s_{qc}(n,m)$ is the closure of $C^s_q(n,m)$ for all $n,m.$
\end{thm}
\begin{proof} Proposition~\ref{closureconnes} shows that equality of the closure implies that Connes' embedding conjecture is true. For the converse assume that Connes' embedding conjecture is true, and let $H= \{ e_1, \ldots, e_m \}$ be the coordinate projections for $\bb C^m$. Let $V$ be a set of cardinality $n,$ and for $v\in V$ let $H_v= \{ e_{v,1}, \ldots, e_{v,m} \}$ be a generating set for the $v$-th copy of $\bb C^m$ in $*_{v \in V} \bb C^m= \CFnm.$
We know that the closure of $C^s_q(n,m)$ is a subset of $C^s_{qc}(n,m),$ so it is enough to show the reverse inclusion.
Suppose that we are given $\big( p(i,j|v,w) \big) \in C^s_{qc}(n,m).$ By Theorem~\ref{synctr} there is a trace $\tau: \CFnm \to \bb C$ such that $p(i,j|v,w) = \tau(e_{v,i}e_{w,j}).$ Apply Proposition~\ref{prop:tauh} with $N=2$ to conclude that there is $k$ and a *-homomorphism $\pi: \CFnm \to M_k$ so that $\tr_k \circ \pi$ approximates $\tau$
with tolerance $(2, \epsilon)$ for $H.$
Hence, \[ |\tr_k \circ \pi(e_{v,i}e_{w,j}) - p(i,j|v,w)| < \epsilon.\]
Let $E_{v,i} = \pi(e_{v,i}) \in M_k,$ so that these are projections and if we set
$p_{\epsilon}(i,j|v,w) = \tr_k(E_{v,i}E_{w,j}),$ then $\big( p_{\epsilon}(i,j|v,w) \big) \in C^s_q(n,m)$ and
converges to $\big( p(i,j|v,w) \big)$ as $\epsilon \to 0.$ Hence, $C_{qc}^s(n,m)$ is contained in the closure of $C_q^s(n,m).$
\end{proof}
The above result characterizes the closure of $C^s_q(n,m)$, assuming that Connes' embedding conjecture is true. But can we say anything about the closure without assuming that the conjecture is true? In particular, we ask the following:
\begin{prob}[Synchronous Approximation Problem]\label{syncapprox} Is the closure of $C_q^s(n,c)$ equal to $C_{qa}^s(n,c)$ for all $n$ and $c$?
\end{prob}
If the answer to the above problem was affirmative, then it would give a new proof of Ozawa's result that Connes' embedding conjecture is true if and only if $C_{qc}(n,c)=C_{qa}(n,c)$ for all $n$ and $c.$ In fact, we would have that the following are equivalent:
\begin{enumerate}[(i)]
\item Connes' embedding conjecture is true,
\item $C^s_{qc}(n,c)=C^s_{qa}(n,c)$ for all $n,c,$
\item $C_{qc}(n,c)=C_{qa}(n,c)$ for all $n,c.$
\end{enumerate}
To see this, note that if the answer to Problem~\ref{syncapprox} is affirmative, then
the above result shows that (ii) implies (i).
The fact that (i) implies (iii) was proven in \cite{jnppsw, fritz2,
fkpt_dg}. We sketch
the proof. By Kirchberg's result, if Connes' is true, then $\CFnc
\otimes_{\min} \CFnc = \CFnc \otimes_{\max} \CFnc$ for every $n,c$. The
matrices in $C_{qc}(n,c)$ are all given by $p(i,j|v,w) \phi(e_{v,i}
\otimes e_{w,j})$ for some state on $\CFnc \otimes_{\max} \CFnc.$ But
$\phi$ is also a state on $\CFnc \otimes_{\min} \CFnc$ and all such
states can be shown to be limits of states given by finite
dimensional representations. In fact, this was first explicitly
shown in \cite{ws2008}.
It remains to show that (iii) implies (ii). But if the two sets are equal then their synchronous subsets are equal.
Now we consider a graph $G$ having vertex set $V$ consisting of $n$ vertices,
and without loops (so that every edge has two distinct vertices).
For $v,w\in V$, we will write $v\sim w$ when $v$ is connected to $w$ by an edge.
Let us fix $m\in\Nats$ and consider the C$^*$-algebra $\CFnm$ as in~\eqref{eq:A}, but written
\[
\CFnm=*_{v\in V}\Cpx^m
\]
Let $e_1,\ldots,e_m$ be the minimal projections
in $\Cpx^m$ and for $v\in\Gamma_0$, let $e_{v,1},\ldots,e_{v,m}$ be the copies of these in the corresponding
generating copy of $\Cpx^m$ in $\CFnc$.
We will say that a tracial state $\tau$ on $\CFnm$ satisfies
\begin{itemize}
\item[$\bullet$]
the {\em orthogonality condition} if $\tau(e_{v,i}e_{w,i})=0$
whenever $v,w\in V$, $v\sim w$ and $i\in\{1,\ldots,m\}$
\item[$\bullet$]
the {\em weak orthogonality condition} if $\tau(e_{v,1}e_{w,1})=0$ whenever
$v,w\in V$ and $v\sim w$.
\end{itemize}
Note that the weak orthogonality condition depends on our choice of ordering of the projections;
thus, we fix such an ordering.
In practice, we will only be concerned with the weak orthogonality condition when $m=2$.
Our next main goal is the following result.
\begin{prop}\label{prop:WOC}
Suppose Connes' embedding conjecture is true.
Let $m=2$, consider the generating set $H=\{e_1\}$ for $\Cpx^2,$
let $N\in\Nats$ and let $\eps>0$.
Suppose $\tau$ is a tracial state on $\CFnm$ that satisfies the weak orthogonality condition.
Then there exists $k\in\Nats$ and a unital $*$-homomorphism $\pi:\CFnm\to M_k(\Cpx)$ such that
the trace $\tr_k\circ\pi$ satisfies the weak orthogonality condition and
approximates $\tau$ with tolerance $(N,\eps)$ for the generating set $H$.
\end{prop}
Once we have proven the above result we see that:
\begin{cor} Suppose that Connes' embedding conjecture is true and let $G$ be a graph. Then $\xi_q(G) = \xi_{qc}(G),$ that is, the Mancinska-Roberson projective rank of $G$ is equal to the tracial rank of $G.$
\end{cor}
\begin{proof}
Applying Theorem~\ref{xichar}, we see that $\xi_{qc}(G)$ is the reciprocal of the largest $\lambda$ for which there exists a trace $\tau$ and projections $\{e_v \}_{v \in V}$ satisfying the weak orthogonality conditions, such that $\tau(e_v) \ge \lambda$ for all $v.$ But by the above result, whenever this happens, then for every $\epsilon >0$ there is a $k$ and projections $E_v \in M_k$ satisfying the weak orthogonality conditions with $tr_k(E_v) \ge \lambda - \epsilon.$
Thus, $\xi_q(G) \le \xi_{qc}(G).$ But since $C^s_q(n,2) \subseteq C^s_{qc}(n,2)$ the other inequality follows.
\end{proof}
For the next two lemmas we let $\Mcal$ be a finite von Neumann algebra equipped with a normal,
faithful tracial state $\tau$, and we let $\|x\|_2=\tau(x^*x)^{1/2}$ for $x\in\Mcal$ be the corresponding $2$-norm.
(Recall that $\Mcal$ is said to be a factor if its center is trivial; for example matrix algebras $M_k(\Cpx)$ are factors.)
In fact, we will apply the lemmas only in the case of $\Mcal$ being a matrix algebra,
but it seems just as easy and possibly useful to write the result in greater generality.
\begin{lemma}\label{lem:qqt}
Let $\Mcal$ be a von Neumann algebra with normal, faithful tracial state $\tau$ and with projections $p,q\in\Mcal$.
Let $\delta=\tau(pq)$.
Then there is a unitary $u\in\Mcal$
and there is a projection $q'\in\Mcal$ such that
\begin{enumerate}[(i)]
\item $q'\perp p$
\item $q'\le u^*qu$
\item $\tau(q)-\tau(q')\le\delta$
\item $\|u-1\|_2\le2\sqrt\delta$
\item $\|q-q'\|_2\le5\sqrt\delta$.
\end{enumerate}
Suppose, furthermore, that $\Mcal$ is either diffuse (i.e., has no minimal projections) or is a finite factor (i.e., a matrix algebra $M_k(\Cpx)$ for some $k$).
Then there is a projection $\qt\in\Mcal$ such that
\begin{enumerate}[(i)]
\setcounter{enumi}{5}
\item $q'\le\qt$
\item $\qt\perp p$
\item $\tau(\qt)=\min(\tau(q),1-\tau(p))$
\item $\|q-\qt\|_2\le6\sqrt\delta$.
\end{enumerate}
\end{lemma}
\begin{proof}
Note that we have $\delta\le1$.
To find $q'$ satisfying (i)-(v),
we may without loss of generality assume $\Mcal$ is generated by $\{1,p,q\}$.
As is well known, the universal, unital C$^*$-algebra $\Bfr$ generated by two projections $P$ and $Q$ is
the set of all continuous functions $f$ from $[0,1]$ into $M_2(\Cpx)$ whose values at the endpoints are diagonal,
where $P$
and $Q$ are represented by the functions
\[
P=\left(\begin{matrix} 1&0\\ 0&0\end{matrix}\right),\qquad
Q(t)=\left(\begin{matrix} t&\sqrt{t(1-t)}\\ \sqrt{t(1-t)}&1-t\end{matrix}\right).
\]
Furthermore, every tracial state $\sigma$ on $\Bfr$ is given by
\[
\sigma(f)=a_0f(0)_{11}+b_0f(0)_{22}+\int\tr_2(f(t))\,d\mu(t)+a_1f(1)_{11}+b_1f(1)_{22},
\]
for a Borel measure $\mu$ on the open interval $(0,1)$ and for nonnegative $a_0,b_0,a_1,b_1$, so that
$a_0+b_0+\mu((0,1))+a_1+b_1=1$.
Thus, the von Neumann algebra $\Mcal$ is the weak closure of the image of $\Bfr$ under the Gelfand--Naimark--Segal representation
of such a trace.
We get
\begin{equation}\label{eq:Mcal}
\Mcal=\smd{\Cpx}{a_0}\oplus\smd{\Cpx}{b_0}\oplus\big(L^\infty(\mu)\otimes M_2(\Cpx)\big)\oplus\smd{\Cpx}{a_1}\oplus\smd{\Cpx}{b_1},
\end{equation}
where $L^\infty(\mu)\otimes M_2(\Cpx)$ should be removed if $\mu$ is the zero measure, and is otherwise interpreted as being functions from $(0,1)$ into $M_2(\Cpx)$,
up to equivalence $\mu$-a.e.
The $a_i$ and $b_i$ are written in~\eqref{eq:Mcal} only to remind us about the trace.
To wit, we have
\[
\tau(r_0\oplus s_0\oplus f \oplus r_1\oplus s_1)=a_0r_0+b_0s_0+\int\tr_2(f(t))\,d\mu(t)+a_1r_1+b_1s_1.
\]
Of course, if any $a_i=0$ or $b_i=0$, then the corresponding summand in~\eqref{eq:Mcal} should be removed.
We also have
\[
\begin{matrix}
p&=&1\oplus0\,\oplus&\begin{pmatrix}1&0\\0&0\end{pmatrix}&\oplus\,1\oplus0 \\[4ex]
q&=&0\oplus1\,\oplus&\left(\begin{smallmatrix} t&\sqrt{t(1-t)}\\ \sqrt{t(1-t)}&1-t\end{smallmatrix}\right)&\oplus\,1\oplus0.
\end{matrix}
\]
We calculate
$\delta=\tau(pq)=\frac12\int t\,d\mu(t)+a_1$.
Letting
\[
q'=0\oplus1\oplus\begin{pmatrix}0&0\\0&1\end{pmatrix}\oplus0\oplus0,
\]
we have $q'\perp p$ and $\tau(q)-\tau(q')=a_1\le\delta$.
Moreover, we see that $q'$ is a subprojection of $u^*qu$ for the unitary
\[
u=1\oplus1\oplus\left(\begin{smallmatrix} \sqrt{1-t}&\sqrt{t}\\ -\sqrt{t}&\sqrt{1-t}\end{smallmatrix}\right)\oplus1\oplus1
\]
and we calculate
\[
\tau(|u-1|^2)=2\int\big(1-\sqrt{1-t}\big)\,d\mu(t)\le2\int t\,d\mu(t)\le4\delta,
\]
so~(iv) holds.
Now~(v) follows from (ii)--(iv).
We will now find $\qt$ satisfying (vi)--(viii).
If $1-\tau(p)\le\tau(q)$, then we simply let $\qt=1-p$.
If $\tau(q)<1-\tau(p)$, then
will let $\qt=q'+r$ for a projection $r\le(1-p)\wedge(1-q')$ such that $\tau(r)=\tau(q)-\tau(q')$.
Since $q'\le1-p$, $(1-p)\wedge(1-q')$ is a projection in $\Mcal$ of trace $1-\tau(p)-\tau(q')$;
moreover, the desired trace value, namely $\tau(q)-\tau(q')$, is less than $1-\tau(p)-\tau(q')$.
Now $\Mcal$ does contain a projection of trace $\tau(q)-\tau(q')$, namely, the projection $u^*qu-q'$.
Thus, assuming either $\Mcal$ is diffuse or a matrix algebra, we conclude that the desired projection $r$ exists.
Since $\tau(r)\le\delta$, we have $\|r\|_2\le\sqrt\delta$ and~(ix) follows from~(v).
\end{proof}
\begin{lemma}\label{lem:WOC}
Fix a graph $G$ as described above.
For every $\eps>0$ there is $\delta>0$ such that if
$(e_v)_{v\in V}$ are projections in $\Mcal$ satisfying
\begin{equation}\label{eq:tauee}
\tau(e_ve_w)<\delta,\quad(v,w\in V,\,v\sim w),
\end{equation}
then there exist projections
$(\et_v)_{v\in V}$ in $\Mcal$ satisfying
\begin{gather}
\et_v\perp\et_w,\quad(v,w\in V,\,v\sim w) \label{eq:eperpet} \\
\|e_v-\et_v\|_2<\eps\quad(v\in V). \label{eq:enearet}
\end{gather}
\end{lemma}
\begin{proof}
We proceed by induction on the number $n=|V|$ of vertices of the graph.
For $n=1$ there is nothing to prove, for we may take $\et_v=e_v$.
Suppose $n\ge2$ and the lemma has been proved for all smaller graphs.
Choose any vertex $v_0\in V$ and let $G'$ be the graph obtained from $G$
by removing the vertex $v_0$ (and all edges containing $v_0$).
We let $V'= V\backslash\{v_0\}$ denote the vertex set of $G'$.
Choose any $\eta$ satisfying
$0<\eta<\eps^2/(50(n-1))$.
By induction hypothesis, there is $\delta'>0$ such that whenever
$(e_v)_{v\in V'}$ are projections in $\Mcal$ satisfying
\[
\tau(e_ve_w)<\delta',\quad(v,w\in V',\,v\sim w),
\]
then there exist projections
$(\et_v)_{v\in V'}$ in $\Mcal$ satisfying
\begin{gather}
\et_v\perp\et_w,\quad(v,w\in V',\,v\sim w) \\
\|e_v-\et_v\|_2<\eta\quad(v\in V'). \label{eq:eet'}
\end{gather}
Let $\delta=\min(\delta',\eps^2/(50(n-1)))$ and
suppose $(e_v)_{v\in V}$ are projections in $\Mcal$ satisfying~\eqref{eq:tauee}.
Let $(\et_v)_{v\in V'}$ be projections obtained using the induction hypothesis as described above.
Then using also~\eqref{eq:eet'} we get
\[
\tau(e_{v_0}\et_w)<\delta+\eta,\quad(w\in V',\,v_0\sim w).
\]
Let
\[
f=\bigvee_{w\in V',\,v_0\sim w}\et_w.
\]
Then $f\le\sum_{v_0\sim w\in V'}\et_w$, so
$\tau(e_{v_0}f)=\tau(e_{v_0}fe_{v_0})<(n-1)(\delta+\eta)$.
By Lemma~\ref{lem:qqt}, there is a projection $\et_{v_0}\in\Mcal$ such that $\et_{v_0}\perp f$ and
\[
\|e_{v_0}-\et_{v_0}\|_2\le5\sqrt{(n-1)(\delta+\eta)}<\eps.
\]
This finishes the construction of the family $(\et_v)_{v\in V}$ of projections satisfying~\eqref{eq:eperpet} and~\eqref{eq:enearet}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:WOC}]
Let $N'=\max(N,2)$.
Let $\delta>0$ be as obtained from Lemma~\ref{lem:WOC}, but for $\eps/2$ instead of $\eps$.
Let $\eps'=\min(\eps/2,\delta)$.
By Proposition~\ref{prop:tauh}, there is $k$ and a $*$-homomorphism $\rho:\CFnc\to M_k(\Cpx)$ such that $\tr_k\circ\rho$ approximates
$\tau$ with tolerance $(N',\eps')$ for the generating set $H$.
Consider the projection $E_v=\rho(e_{v,1})\in M_k(\Cpx)$.
Since $\tau$ was assumed to satisfy the weak orthogonality condition, we have $\tr_k(E_vE_w)<\eps'$
whenever $v,w\in V$ and $v\sim w$.
Using $\|X\|_2=\tr_k(X^*X)^{1/2}$ for $X\in M_k(\Cpx)$,
by Lemma~\ref{lem:WOC}, there exist projections $(E'_v)_{v\in\Gamma_0}$, such that
\begin{gather*}
E'_v\perp E'_w,\quad(v,w\in V,\,v\sim w) \\
\|E'_v-E_v\|_2<\frac\eps2\quad(v\in V).
\end{gather*}
Now defining $\pi:\CFnc\to M_k(\Cpx)$ to be the unital $*$-homomorphism determined by $e_v\mapsto E_v'$,
we have that $\tr_k\circ\pi$ approximates $\tau$ with tolerance $(N,\eps)$ for the generating set $H$.
\end{proof}
\begin{lemma}\label{lem:OC<1}
Fix a graph $G$ as described above and let $m\in\Nats$.
For every $\eps>0$ there is $\delta>0$ such that if
$(e_{v,i})_{v\in V,\,1\le i\le m}$ are projections in $\Mcal$ satisfying
\begin{align}
\tau(e_{v,i}e_{w,i})&<\delta,\quad(v,w\in V,\,v\sim w,\,1\le i\le m), \label{eq:taueiei} \\
\sum_{i=1}^me_{v,i}&\le 1,\quad(v\in V),
\end{align}
then there exist projections
$(\et_v)_{v\in V}$ in $\Mcal$ satisfying
\begin{gather}
\et_{v,i}\perp\et_{w,i},\quad(v,w\in V,\,v\sim w,\,1\le i\le m) \label{eq:eiperpeit} \\
\sum_{i=1}^m\et_{v,i}\le 1,\quad(v\in V )\\[1ex]
\|e_{v,i}-\et_{v,i}\|_2<\eps\quad(v\in V,\,1\le i\le m). \label{eq:eineareit}
\end{gather}
\end{lemma}
\begin{proof}
This follows immediately from Lemma~\ref{lem:WOC} applied to the graph $G^{(m)}$ obtained from $G$
as follows.
The vertex set $V(G^{(m)})$ is $V\times\{1,\ldots,m\}$.
There is an edge between vertices $(v,i)$ and $(w,j)$ in $G^{(m)}$ if and only if either (a) $v=w$ and $i\ne j$
or (b) there is an edge between $v$ and $w$ in $G$ and $i=j$.
\end{proof}
For those familiar with products of graphs, if $K_m$ denotes the complete graph on $m$ vertices, then $G^{(m)} = G \Box K_m,$ which is often called the {\it Cartesian product} of the graphs.
\begin{remark} If we could prove that the projections $\{ \tilde{e}_{v,i} \}$ can also be chosen to satisfy $\sum_{i=1}^m \tilde{e}_{v,i} = 1,$ for all $v \in V,$ then the above results would imply that Connes' embedding conjecture true implies that $\chi_q(G) = \chi_{qc}(G).$
\end{remark}
\begin{ques}
Fix a graph $G$ and a rational number $\gamma$.
Is it true that
for every $\eps>0$ there is $\delta>0$ such that for all integers $k$ that are large enough and divisible by the denominator of $\gamma$,
if $(e_v)_{v\in V}$ are projections in $M_k(\Cpx)$ satisfying
\begin{gather*}
\tr_k(e_ve_w)<\delta,\quad(v,w\in V,\,v\sim w), \\
\tr_k(e_v)=\gamma,\qquad(v\in V),
\end{gather*}
then there exist projections
$(\et_v)_{v\in V}$ in $M_k(\Cpx)$ satisfying
\begin{gather*}
\et_v\perp\et_w,\quad(v,w\in V,\,v\sim w) \\
\|e_v-\et_v\|_2<\eps\quad(v\in V). \\
\tr_k(\et_v)=\gamma.
\end{gather*}
\end{ques}
\section{Values of Games}
Let $\cl G$ be a finite input-output game of the type described in the
introduction with inputs $X$, outputs $O,$ and with ``rules''
$\lambda: X \times X \times O \times O \to \{0,1\}.$ Suppose that in
addition we are given a probability distribution on the inputs. By
this we mean a set $\Gamma =( \gamma_{x,y}), $ such that $\gamma_{x,y}
\ge 0$ and $\sum_{x,y \in X} \gamma_{x,y} =1.$
Then for $t \in \{ loc, q, qa, qc \}$ we define the {\it value of the
game given the distribution} to be
\begin{multline*} \omega_t(\cl G, \Gamma) = \\ \sup \{\sum_{x,y \in X, i,j \in O}
\gamma_{x,y} \lambda(x,y,i,j) p(i,j|x,y) : \big( p(i,j|x,y) \big) \in C_t(n,c)\}.\end{multline*}
For $t \in \{ loc, qa, qc \}$ this supremum is actually attained, but,
since we do not know if $C_q(n,c)$ is closed the supremum is necessary
for this case. A major problem in the theory of non-local games,
related to the strong Tsirelson conjecture, is to
determine if $\omega_q(\cl G, \Gamma)$ is always attained. This is essentially the {\it bounded entanglement problem.}
Also, note that since $C_{qa}(n,c)$ is defined to be
the closure of $C_q(n,c)$ we have that $\omega_q(\cl G, \Gamma) =
\omega_{qa}(\cl G, \Gamma).$
We define the {\it synchronous value of the game given the
distribution} to be
\begin{multline*} \omega^s_t(\cl G, \Gamma) = \\
\sup \{ \sum_{x,y \in X, i,j \in O}
\gamma_{x,y} \lambda(x,y,i,j)\gamma_{x,y} p(i,j|x,y) : \big( p(i,j|x,y) \big) \in
C^s_t(n,c) \}.\end{multline*}
Note that if a game has a winning strategy then $\omega_t(\cl G,
\Gamma) =1,$ for every $\Gamma.$ Conversely, it is easy to see that,
if $\gamma_{x,y} \ne 0$ for all $x,y,$ then for $t \in \{ loc, qa,
qc \},$ $\omega_t(\cl G, \Gamma) =1$ implies that $\cl G$ has a
winning strategy.
We summarize a few consequence of our results in these terms.
\begin{prop} If Connes' embedding conjecture is true, then
$\omega_q(\cl G, \Gamma) = \omega_{qc}(\cl G, \Gamma)$ and
$\omega^s_q(\cl G, \Gamma) = \omega^s_{qa}(\cl G, \Gamma)= \omega^s_{qc}(\cl G, \Gamma)$ for every
$\cl G$ and every $\Gamma.$
\end{prop}
If the answer to our Synchronous Approximation Problem is affirmative,
then $\omega^s_q(\cl G, \Gamma) = \omega^s_{qa}(\cl G, \Gamma)$ for every $\cl G$ and every $\Gamma.$
\begin{prop} Given a game $\cl G$ with $n$ inputs $X$ and $m$ outputs $O$ and a distribution $\Gamma,$ set
\[B = \sum_{x,y \in X, i,j \in O} \gamma_{x,y} \lambda(x,y,i,j) e_{x,i}e_{y,j} \in \CFnm .\]
Then
\[ \omega^s_{qc}(\cl G, \Gamma) = \sup \{ \tau(B) \, | \, \tau: \CFnm \to \bb C \text{ is a tracial state } \},\] and this supremum is attained.
If we restrict the family of traces to those that have finite dimensional
GNS representations, then we obtain $\omega^s_q(\cl G, \Gamma).$ If we restrict the family of traces to those that have abelian GNS representations, then we obtain $\omega^s_{loc}(\cl G, \Gamma),$ and the supremum is attained.
\end{prop}
However, in the finite dimensional case, we can say even more.
\begin{prop}
$\omega^s_q(\cl G, \Gamma) = \sup \{ tr_k\big( \sum_{x,y,i,j} \gamma_{x,y}
\lambda(x,y,i,j) E_{v,i} E_{w,j}\big) \}$
where the supremum is taken over all $k \in \bb N$ and all sets of
projections in $M_k$ satisfying $\sum_{i} E_{v,i} = I.$
\end{prop}
\begin{proof}
Given a finite dimensional representation $\pi$ of $\CFnc$ write
$\pi(\CFnc) = M_{k_1} \oplus \cdots \oplus M_{k_r}$ and
$\pi(e_{v,i}) = E_{v,i,1} \oplus \cdots \oplus E_{v,i,r}$ so that
$\tau(e_{v,i}) = \alpha_1 tr_{k_1}(E_{v,i,1}) + \cdots + \alpha_r
tr_{k_r}(E_{v,i,r})$
for some weights $\alpha_l \ge 0$ with $\alpha_1 + \cdots + \alpha_r =1.$
Note that
\begin{multline*} \tau \big( \sum_{x,y,i,j} \gamma_{x,y}
\lambda(x,y,i,j) e_{v,i}e_{w,j} \big) = \\
\sum_{l=1}^r \alpha_l\,\, tr_{k_l} \big( \sum_{x,y,i,j} \gamma_{x,y}
\lambda(x,y,i,j) E_{v,i,l}E_{w,j,l} \big),
\end{multline*}
so this convex sum is dominated by
\[ \max \{ tr_{k_l} \big( \sum_{x,y,i,j} \gamma_{x,y} \lambda(x,y,i,j)
E_{v,i,l}E_{w,j,l} \big): 1 \le l \le r \},\]
from which the result follows.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,325
|
Andrzej Jerzy Zakrzewski (22 August 1941, in Warsaw – 10 February 2000, in Warsaw) was a Polish lawyer, historian, politician, journalist, and publicist. He was a member of the Conservative People's Party (Stronnictwo Konserwatywno-Ludowe), the Solidarity Electoral Action (Akcja Wyborcza Solidarność), and Minister of Culture and National Heritage of the Republic of Poland in cabinet of Jerzy Buzek (1999–2000).
References
Writers from Warsaw
Lawyers from Warsaw
Politicians from Warsaw
1941 births
2000 deaths
Government ministers of Poland
20th-century Polish historians
Polish male non-fiction writers
Polish publicists
Solidarity Electoral Action politicians
20th-century Polish lawyers
20th-century Polish journalists
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 513
|
Soothe your stresses away as you unwind with a relaxing aromatherapy massage and breathe in the aroma of your very own personalised essential oil blend.
Aromatherapy can be a fantastic treatment, particularly for anyone suffering from stress and tension. It's also great value as not only do you get to enjoy a wonderfully relaxing and therapeutic massage, you also get all the benefits of your own individual oil blend, which is tailored to your particular needs. During the massage, the oils will slowly begin to sink into the skin and absorb into the body, where they can get to work helping to bring the body back into balance again.
Aromatherapy is based on the use of essential oils. These are natural extracts derived from a wide variety of plants. Each oil each has its own properties and they can be blended together to help with both physical and emotional problems.
It can be effective for all sorts of conditions including high stress levels, back, neck and shoulder pain, muscle tension and many other health problems.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,202
|
import eachOfLimit from './internal/eachOfLimit';
import parallel from './internal/parallel';
/**
* The same as [`parallel`]{@link module:ControlFlow.parallel} but runs a maximum of `limit` async operations at a
* time.
*
* @name parallelLimit
* @static
* @memberOf module:ControlFlow
* @method
* @see [async.parallel]{@link module:ControlFlow.parallel}
* @category Control Flow
* @param {Array|Collection} tasks - A collection containing functions to run.
* Each function is passed a `callback(err, result)` which it must call on
* completion with an error `err` (which can be `null`) and an optional `result`
* value.
* @param {number} limit - The maximum number of async operations at a time.
* @param {Function} [callback] - An optional callback to run once all the
* functions have completed successfully. This function gets a results array
* (or object) containing all the result arguments passed to the task callbacks.
* Invoked with (err, results).
*/
export default function parallelLimit(tasks, limit, callback) {
parallel(eachOfLimit(limit), tasks, callback);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,253
|
\section{Introduction} \label{sec:intro}
Different corpus may present different language styles, featured by variations in attitude, tense, word choice, \textit{et cetera}. As human beings, we always have an intuitive perception of style differences in texts. In the literature of linguistics, there also developed a number of mature theories for characterizing style phenomena in our daily lives \citep{bell1984language, coupland2007style, ray2014style}. However, with decades of advancement of machine learning techniques in Natural Language Processing (NLP), an interesting and fundamental question still remains open: \textit{How is style information encoded by learning models?}
In this paper, we share our novel observations for this question in a specified version, that is, in what approach Sequence-to-Sequence (\textit{seq2seq}; \citealt{sutskever2014sequence}), a prestigious neural network architecture widely used in NLP and representation learning \citep{bahdanau2014neural, li2015hierarchical,kiros2015skip}, encodes language styles.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{arrow_plot.png}
\caption{Visualization of the first two eigenvectors of style matrices on four subcorpora with ratings ranging in $1$, $2$, $4$, $5$, collected from Yelp review dataset. The reviews with higher rating are more positive in attitude and vice versa.}
\label{fig:eig}
\end{figure}
In our preliminary studies, we applied a typical seq2seq model as an autoencoder to learn semantic vectors for sentences from Yelp review dataset \footnote{https://www.yelp.com/dataset/challenge} and we strikingly made the following observation: After calculating respectively the covariance matrices of the semantic vectors of reviews with different attitude polarity and intensity, we found their \textit{second eigenvectors}, i.e. eigenvectors with the second largest eigenvalues, were roughly grouped in two parts according to their polarity and meanwhile showed slight difference according to their intensity, as in the colored part of Fig. \ref{fig:eig}, while the \emph{first eigenvectors} illustrated in gray formed a single cluster, as if they captured certain common attributes of the Yelp corpus, e.g. casual word choices. This phenomenon suggests the covariance matrices have probably encoded the language style in an informative way. Based on this observation, we provide the notion of style matrix and investigate a number of far-reaching implications brought by style matrix in the remainder of this paper.
\begin{conjecture}[Style Matrix]\label{conj:style_mat}
The style of a corpus is encoded in the covariance matrix of its semantic vectors, which is called \emph{style matrix}.
\end{conjecture}
For the best of our knowledge, our research question and conjecture are quite novel and there barely exist any relevant works before. The most related works are probably the recent studies on text style transfer, a task which was first investigated by \citet{shen2017style} for converting a given corpus in one style to another. Although most of them have not discussed style at a fundamental level, we mainly identify three related perspectives in existing style transfer methods.
\begin{enumerate}[A.]
\item \emph{Discrete Label}: A major proportion of the state-of-the-art style transfer methods simply view the corpus style as discrete labels. For instance, the style of positive and negative reviews in Yelp dataset are respectively assigned with binary labels \citep{shen2017style,hu2017toward,chen2018adversarial}.
\item \emph{Style Embedding}: Instead of discrete labels, some methods propose to learn \emph{semantic-independent} embedding vectors as distributed representations of style \citep{fu2018style}.
\item \emph{Lexicon-Based}: Other works suggest use the most significant lexical units, i.e. those serving as major factors to style classification decision, as representatives of text style \citep{li2018delete,xu2018unpaired}.
\end{enumerate}
In principle, our main conjecture is better compatible with linguistic aspects of style phenomenon than the aforementioned perspectives, especially in the following aspects.
\begin{enumerate}
\item \textit{Style is a statistical phenomenon.} According to the variationist's view in sociolinguistics \cite{coupland2007style}, style emerges from variation of language usages and is always a global phenomenon rather than a property of single sentence. In our conjecture, style matrix by definition reflects the covariance of the corpus, while Perspective C could only characterize the sentence-level style.
\item \textit{Style is inherent in semantics.} As \citet{ray2014style} suggests, expression often helps to form meaning. In our conjecture, style matrix is an explicit function of semantic vectors, while Perspective B improperly assumed style embedding's independence on semantic.
\item \textit{Style is multi-modal.} Usually, we recognize style in texts from many different aspects \cite{bell1984language}. For instance, \textit{I am very unhappy with this place} is negative in attitude and meanwhile present in tense. Moreover, one can also recognize slight differences in style intensity for both negative sentences, i.e. with/without \textit{very} in the example above. With experiments, we show style matrix is able to distinguish various intensity level of style (\S \ref{sec:style_intensity}) and capture multiple styles in one corpus (\S \ref{sec:multi_style}), while Perspective A is impotent to characterize these delicate style differences by discrete labels.
\end{enumerate}
In practice, based on the notion of style matrix, we propose a novel algorithm called \emph{Neutralization-Stylization} (NS) for unpaired text style transfer. Given style matrices of source and target corpora obtained from a pre-trained seq2seq autoencoder, our algorithm works in a fully learning-free manner by first preparing a pair of matrix transform operators from the style matrices. After the preparation, it simply applies these operators to the style matrix of given corpus to accomplish text style transfer on the fly. By introducing additional style information as supervision on the learning process of the seq2seq autoencoder, we observe NS algorithm can achieve comparable performance with the state-of-the-art style transfer methods on each standard metric. Moreover, the flexibility of our method is further demonstrated by its ability to control the style of unlabeled sentences from other domains, i.e. \emph{out-of-domain text style transfer}, which we propose as a much challenging task to foster future researches.
In summary, our contributions are as follows:
\begin{itemize}[leftmargin=*]
\item We present the notion of style matrix as an informative delegate to language style and explains for the first time how seq2seq models encode language styles (\S \ref{sec:style_mat}).
\item With the aid of style matrix, we devise Neutralization-Stylization as a learning-free algorithm for text style transfer among binary, multiple and mixed styles (\S \ref{sec:transfer}), which achieves competitive performance compared with the state-of-the-art methods (\S \ref{sec:comparison}).
\item We introduce the challenging out-of-domain text style transfer task to further prove the flexibility of our proposed method (\S \ref{sec:ood}).
\end{itemize}
\if0
We talk about how we characterize language style in daily lives according to the theories in linguistic literature in section two. And We also enumerate the methods up to now for style transfer. In section three, the concept of the style matrix is given formally. In the next section, we propose the Neutralization-Stylization method for the task of unsupervised text style transfer. We verify the transfer method we proposed and our conjecture about the style matrix with experiments in a variety of scenarios in section five.
\fi
\if0
The remainder of this paper proceeds as follows: Section 2 reviews previous text style transfer methods and their implicit assumptions on style representations; Section 3 provides a formal introduction to style matrix and discusses its interesting implications on language style; In Section 4, we propose a learning-free style transfer algorithm, which is further augmented with prior knowledge of style as external supervision; Section 5 \& 6 are left for experiments and discussions.
To refresh the mind, seq2seq works by the following procedures: 1) taking in a sequence of tokens; 2) encoding the sequence into vector-valued embedding; 3) decoding out a sequence of tokens from the encoded vector.
\fi
\if0
(with an average eigenvalue $2.10$)
In our preliminary study, we applied a typical seq2seq model as an autoencoder to learn semantic vectors for sentences in Yelp dataset and we strikingly made the following observation: After calculating respectively the covariance matrices of the semantic vectors of reviews with different ratings, we found their first eigenvectors (with an average eigenvalue $6.31$) were grouped according to their indicated attitude polarity and meanwhile showed slight differences due to their intensity of polarity, as is shown in the colored part of Fig. \ref{fig:eig}. This phenomenon suggests the covariance matrices have probably captured the difference in attitude. Since \emph{attitude} is a typical style factor in Yelp dataset, we boldly generalize our aforementioned observation as the following conjecture.
\fi
\section{Style Matrix} \label{sec:style_mat}
\subsection{A General Framework for Style Matrix Extraction}
Given a corpus $X=\{x^{(1)},x^{(2)},\hdots,x^{(N)}\}$, where $x^{(i)}$ is a sequence of tokens, we wonder whether there exists an explicit way to extract the global style of $X$ with no other external knowledge. Inspired from the variationist's approach to language style in the context of sociolinguistic \citep{coupland2007style}, we suggest exploiting the second-order statistics of semantics, specifically the covariance matrix, as an informative representation of the corpus style (Conjecture \ref{conj:style_mat}). Somewhat coincidentally, a similar viewpoint on visual style has been investigated in the computer vision community recently \citep{gatys2015texture}.
Due to the discrete essence of language, to compute the covariance matrix is not directly applicable to raw representations such as those in one-hot scheme. In order to fulfill the statement in our main conjecture, we require the semantic vector to be both \emph{distributed} and nearly \emph{lossless}. The former property requires the semantic of the original sentence can be compressed into a latent vector, while the latter requires the original sentence can be near-optimally reconstructed from the semantic vector alone.
Formally, we first convert $X$ into distributed representations with a mapping $E$ (i.e. \emph{encoder}) from $X$ to $\mathbb{R}^{d}$, a $d$-dimensional real-valued vector space. In order to guarantee $E$ is lossless, we further require the existence of a reverse mapping $D$ (i.e. \emph{decoder}) from $\mathbb{R}^{d}\to{X}$ which satisfies $D\circ{E} = I_{X}$, the identity mapping on the corpus. Once these conditions satisfied, we call the distributed representations $Z$, which consists of $\{z^{(i)}\}_{i=1}^{N}$ s.t. $z^{(i)} = E(x^{(i)})$, the \emph{semantic vectors} of corpus $X$. As a slight abuse of notation, we also use $Z$ to represent the semantic vecotrs in matrix form, i.e. $Z \doteq [z^{(1)}, \hdots, z^{(N)}]\in\mathbb{R}^{d\times{N}}$. Based on these notations, we provide the formal counterpart to Conjecture \ref{conj:style_mat} as follows.
\begin{definition}[Style Matrix]\label{def:style_mat}
Given corpus $X$ with a semantic encoder $E$ satisfying the requirements above, we define the style matrix $S_{X}\in\mathbb{R}^{d\times{d}}$ as
\begin{equation}\label{eq:style_mat}
S_X = \frac{1}{N-1}\hat{Z}_X\hat{Z}_X^{T}
\end{equation}
where $\hat{Z}_X$ denotes $Z_X$ after being centered, i.e. $\hat{Z}_X \doteq Z_X - \bar{z}_X\mathbf{1}_{d}^{T}$ and $\bar{z}_X \doteq\frac{1}{N}\sum_{i=1}^{N}z_X^{(i)}$.
\end{definition}
In recent studies of sentence embedding (e.g. \citealt{le2014distributed, conneau2017supervised, pagliardini2017unsupervised}), there indeed exist various existing choices for implementing encoder $E$. However, as most of them do not have an explicit notion of the decoder, the extracted style matrix would therefore not be able to be further utilized in downstream style transfer tasks. Therefore, in the next section, we propose to leverage the power of seq2seq paradigm \citep{sutskever2014sequence} as a practical tool for extracting highly informative style matrix and meanwhile, facilitates style transfer tasks with the simultaneously trained decoder module. A detailed implementation is provided below.
\subsection{Case Study: Seq2seq for Style Matrix Extraction}\label{sec:style_mat_extraction}
As an overview, we implement the encoder $E$ in Definition \ref{def:style_mat} with the encoder module of a seq2seq model, while its decoder module $G$ learns alongside $E$ under the reconstruction loss to guarantee the original semantic is largely preserved in the obtained semantic vectors $E(X)$. Given a sentence $\mathbf{x} = \{w_1, \hdots, w_T\}$ with each token $w_i$ from a vocabulary $\mathcal{V}$, we propose the learning process for style matrix extraction below.
For the encoder module, we use a recurrent neural network with Gated Recurrent Unit (GRU; \citet{cho2014learning}), i.e. $\text{GRU}_{E}$, to encode x into hidden state vectors with
\begin{equation}
\mathbf{h}=\text{GRU}_{E}(\mathbf{x})
\end{equation}
where $\mathbf{h}=\{h_1,h_2,\ldots,h_T\}$ contains all the hidden states calculated by the GRU encoder.
Next, viewing the last hidden state $h_T$ of the encoder as the semantic vector $z$, we further require the decoder can reconstruct $x$ based on $z$ token by token with a GRU (denoted as $\text{GRU}_{D}$). Formally, at each step $t$, $\text{GRU}_{D}$ takes the generated token $y_{t}$ and the previous hidden state $s_{t-1}$ as input to calculate the current state by
\begin{equation}
s_t=\text{GRU}_{D}(y_t,s_{t-1})
\end{equation}
where the initial state is set as $z$.
Subsequently, with a linear projection layer followed by a softmax transformation, the distribution of the next token $y_{t+1}$ over the vocabulary is calculated as
\begin{equation}
P(y_{t}|\mathbf{x}, y_{<t}) = \text{softmax}(Ws_t)
\end{equation}
where $W$ is a learnable matrix in $\mathbb{R}^{|\mathcal{V}|\times{d}}$.
By convention of unsupervised learning \citep{lecun2015deep}, we set the reconstruction objective as the categorical cross entropy between the input sequence $\mathbf{x}$ and the distribution of the reconstructed sequence $\mathbf{y}$. In practice, we further apply the scheduled sampling technique to accelerate the aforementioned learning process \citep{bengio2015scheduled}.
It is worth to notice, in our implementation of seq2seq for autoencoding, we have intentionally avoided the usage of attention mechanism \citep{luong2015effective}. It is mainly because, with the attention mechanism, information flow from encoder to decoder is not limited to the semantic vector $z$. For example, the reconstruction process is otherwise also dependent on the context vector. Therefore, although attention mechanism can bring optimal reconstruction loss even with small hidden state size, it may cause potential semantic loss and therefore compromise the quality of the extracted style matrix.
As a final remark, we demonstrate our method above with GRU modules only for the sake of concreteness. Besides GRU, there are various available recurrent architectures for implementing $E, G$, such as vanilla recurrent unit \citep{rumelhart1985learning}, Long Short-Term Memory network (LSTM; \citet{hochreiter1997long}) and their bidirectional or stacked variants \citep{jurafsky2000speech}. In experiments, we also report results with several typical architectures as a comprehensive self-comparison.
\if0
By denoting the obtained
Formally, consider there exist a
Now that embedding a sentence to a semantic vector is common in NLP
tasks, is it possible that the high order statistics of these vectors can reflect the style of the corpus? It comes out that the covariance matrix of these vectors, which we called the style matrix, can reflect the variance of these semantic vectors along each dimension and the correlations between different dimensions on vector space, thus capture the style of corpus.
To define the style matrix formally, for corpus $X=\{x^{(1)},x^{(2)},\ldots,x^{(N)}\}$, we assume that there exist a mapping to K-dimensional space $E: X\rightarrow Y$ and a reverse mapping $G: Y\rightarrow X$, where N is the number of sentences and $Y=\{y^{(1)},y^{(2)},\ldots,y^{(N)}\}\in R^{K\times N}$ contains the semantic vectors. We firstly center Y by subtracting the mean vector $y_m$ and get $\hat Y$. $\hat Y\hat Y^{\mathrm{T}}$ is the style matrix of corpus $X$.
It is worth mentioning that the generated style matrix relies on both the corpus and the mapping functions. Bad mapping will reduce the style matrix's ability to capture the style information. In the following subsections, we will show how to transfer the styles using style matrices while providing a method to achieve ``good'' mapping.
We also tried other recurrent networks, e.g. Long Short-Term Memory network(LSTM), Bi-directional LSTM or GRU, and result in a similar performance. So we choose to use GRU in all our experiences because of its high calculate efficiency. We didn't use the attention based seq2seq model because the attention mechanism will cause the loss of semantic information from hidden vectors.
\begin{equation}
(S_X)_{ij} = \frac{1}{n-1}\sum_{k=1}^{n}(z^{(k)}_{i}- \bar{z}_{i})(z^{(k)}_{j}- \bar{z}_{j})
\end{equation}
In this section, we propose a general framework for style matrix extraction and then introduce style matrix extraction with seq2seq models in details as a case study.
\fi
\section{Style Transfer with Style Matrix} \label{sec:transfer}
In this section, we propose a novel algorithm called \emph{Neuralization-Stylization} (NS) for unpaired text style transfer by directly \emph{aligning} the style matrix of one corpus to the other with a pair of plug-and-play matrix operations. To achieve competitive performance as the state-of-the-art style transfer methods, we further augment the unsupervised style matrix extraction process in Section \ref{sec:style_mat_extraction} by introducing human-defined style information as external supervision.
\subsection{Neutralization-Stylization algorithm}
As a covariance matrix in essence, style matrix $S_X$ can be factorized into the following form due to its positive semi-definiteness
\begin{equation}\label{eq:eigen}
S_X=P_X \Lambda_X P_X^{T}
\end{equation}
where $\Lambda_X$ is a diagonal matrix consisting of its eigenvalues and $P_X$ is an orthogonal matrix formed by its eigenvectors \citep{meyer2000matrix} .
Given two corpora $X$ and $Y$ and a seq2seq autoencoder $(E, G)$ pretrained on $X \cup Y$ as a larger corpus, we calculate Eq. \ref{eq:style_mat} respectively on $X, Y$ to obtain the style matrices $S_X, S_Y$. Using eigenvalue decomposition in Eq. \ref{eq:eigen}, we next introduce a pair of Neutralization and Stylization operators, which can be easily used for on-the-fly text style transfer in a plug-and-play manner. Note both operators are defined on a set of semantic vectors rather than a single embedding, which highly corresponds to the statistical essence of language style \citep{coupland2007style}.
\subsubsection{Plug-and-Play Style Transfer Operators}
\textbf{Neutralization.} Neutralization operator is used to remove the style characteristic of corpus $X$ from a set of semantic vectors $Z$. Formally, in the spirit of Zero-phase Component Analysis (ZCA) \citep{bell1997edges}, neutralization operator is defined as
\begin{equation}
N_XZ =P_X\Lambda_X^{-\frac{1}{2}}P_X^{\mathrm{T}}(Z - \bar{z}_X\mathbf{1}_{d}^{T})
\end{equation}
An intuitive way to understand how it works is by replacing $Z$ directly with the semantic vectors of $X$. It is easy to check: $N_XZ_X$ has its style matrix as $\mathbf{I}_{d\times{d}}$, which means the dimensions of semantics become uncorrelated after neutralization.
\noindent\textbf{Stylization.} Stylization transformation is used to add the style characteristic of corpus $Y$ to
a set of semantic vectors $Z$ by reestabilishing the correlation among dimensions of semantics, which, with inspirations from \citet{hossain2016whitening}, is defined as
\begin{equation}
S_YZ=P_Y\Lambda_Y^{\frac{1}{2}}P_Y^{\mathrm{T}}Z + \bar{z}_Y\mathbf{1}_{d}^{T}
\end{equation}
Similarly, by stylizing a neutral set of semantic vectors $Z$ (i.e. $ZZ^{T} = \mathbf{I}_{d\times{d}}$), we can easily check $S_YZ$ has the same style matrix as that of corpus $Y$, which hence demonstrates the properness of $S_Y$.
\subsubsection{On-the-Fly Text Style Transfer}
With the well-defined neutralization and stylization operators, our proposed learning-free NS algorithm works straightforwardly by: (1) encoding $x$ with $E$; (2) applying prepared $(N, S)$ operators successively; (3) decoding the semantic vector with $D$. Formally, the target sentence $y$ is calculated as
\begin{equation}
y = D(S_YN_XE(x))
\end{equation}
Moreover, thanks to the flexibility of style matrix perspective and NS algorithm, we can even conduct \emph{out-of-domain style transfer}, where the input sentence $x$ not necessarily comes from corpus $X$ or has style labels. For details, we present an interesting case study on out-of-domain style transfer between Yelp and Amazon datasets in Section \ref{sec:ood}.
\subsection{Incorporate Human-Defined Style Label} \label{sec:style_label}
In practice, we notice the performance of NS algorithm with raw style matrix is not competitive with the state-of-the-art methods specified on this task. We speculate the main reason lies in: Style matrix is highly informative and probably incorporates even the most delicate aspect of style of the underlying corpus. Therefore, its unsatisfactory performance on style transfer task implies the corpus actually has other latent attributes of style besides the human-defined ones, as we have illustrated with the Yelp example in Section \ref{sec:intro} by its clustered first eigenvectors (gray arrows in Fig. \ref{fig:eig}).
To enhance the quality of style transfer, we suggest to augment the style matrix extraction process with human-defined attribute (e.g. attitude). Concretely, we propose to train the encoder $E$ of the seq2seq model in a semi-supervised way by adding a nonlinear binary classifier $C: \mathbb{R}^{d}\to\{0, 1\}$ on the semantic space, which provides supervision signal simultaneously with the original unsupervised reconstruction process. Formally, given semantic vector $z$, we define the classifier as
\begin{equation}
C(z)=\sigma(w^{T}\text{vec}(zz^T))
\end{equation}
where $w\in R^{d^2}$ is the trainable parameter and $\sigma$ is the sigmoid activation.
Noticeably, under both scenarios, our text style transfer algorithm is learning-free because: we only need to pretrain a seq2seq model, either in fully unsupervised or semi-supervised way, to obtain a pair of encoder and decoder and prepare the $(N, S)$ operators with several matrix operations. Without time-consuming adversarial training (e.g. \citet{shen2017style}), our augmented method achieved competitive transfer performance on each standard metric (\S \ref{sec:comparison}).
\if0
With the seq2seq model is pre-trained in a totally unsupervised way and has no idea about the styles each sentence belongs to, we justify our conjecture about style matrix.
These operations are similar to those proposed by \cite{hossain2016whitening} and have been verified valid at image style transfer\cite{li2017universal}.
To verify our assumption about the text style and the effect of the Neuralization-Stylization algorithm, we start with training E and G in an unsupervised way, which is equivalent to the normal seq2seq model. With the generated semantic vectors, we calculate the style matrices of the Yelp reviews with positive or negative attitude respectively and perform the pair of transformations on two sets of semantic vectors. Finally, with these vectors, we reconstruct the reviews with the decoder. We evaluate the transfer strength with a pre-trained fastText classifier as presented in the experimental part. Even though we train the seq2seq model in a totally unsupervised way, about a third of the transferred reviews can fool the classifier.
On the other hand, style information captured by the style matrix relies on all the sentences contained in the corpus. Yelp reviews with the rating 4 or 5 are both semantically positive but at different degrees. Recall we have trained a classifier C, the sub-corpora which considered to be positive or negative with high confidence are available. With their style matrices, we perform Neuralization-Stylization transformations on original corpora and improve the model ability to rewrite sentences about a specific attribute.
\fi
\section{Experiments\footnote{Code is provided at \url{https://bit.ly/2QgEUNE}}}
\subsection{Overall Settings}
\noindent\textbf{Datasets.} We used the following two standard benchmark datasets for empirical studies.
\textbf{Yelp:} The Yelp dataset collected the reviews to restaurants on Yelp. Each sentence is associated with an integer rating ranging from $1$ to $5$, where a higher score implies the more positive of the corresponding review's attitude and vice versa. We treated the attitude of reviews with ratings above $3$ as positive while those below $3$ as negative.
\textbf{Amazon:} The Amazon dataset contains the product reviews on Amazon. Each sentence is originally labeled with positive or negative attitudes \citep{he2016ups}.
With an automatic tense analysis tool \citep{ramm2017annotating}, we annotated the tense attribute for sentences in Yelp and Amazon as an additional style factor. We filtered out the sentences which were not in past and present tense and split each processed dataset into train, validation and test sets. For statistics, please refer to Appendix A.
\noindent\textbf{Evaluation Metric.}
We evaluated the performance of style transfer on the following two standard metrics.
\textbf{Accuracy (Acc.)}: In order to evaluate whether the transferred sentences have the desired style, we followed the evaluation method in \citet{shen2017style} by pretraining a style classifier on the training set and utilizing its classification accuracy on the transferred sentences as a metric. Specifically, we used the TextCNN model \citep{kim2014convolutional} as a style classifier.
\textbf{BLEU}: In order to evaluate the quality of content preservation, we used the BLEU score \cite{papineni2002bleu} between the generated and the source sentences as a measure. Intuitively, a higher BLEU score primarily indicates the model has a stronger ability to preserve content by copying style-neutral words from the source sentence.
To evaluate the overall performance of style transfer quality, we also calculated the geometric mean (i.e. \textit{G-Score}) and arithmetic mean (i.e. \textit{Mean}) of Acc. and BLEU metrics.
\begin{table}
\caption{Results of style transfer with NS algorithm on corpora pairs with different style contrast levels.}\label{tab:intensity_level}
\centering
\begin{adjustbox}{width=0.45\textwidth}
\begin{tabular}{c|c|c}
\hline
NS Operators From &Acc. (Baseline\footnote{The error rate of original sentences in validation set is provided as baseline, so as in Table \ref{tab:multiclass}})&BLEU\cr
\hline
(R1, R5)&39.48 (2.67)&36.97\cr
(R1 $\cup$ R2, R4 $\cup$ R5) &35.84 (2.67)&39.44\cr
(R2, R4)&32.82 (2.67)&41.08\cr
\hline
\end{tabular}
\end{adjustbox}
\label{tab_star}
\end{table}
\noindent\textbf{Implementation Details.} We embedded words into distributed representations (with dimension $d = 300$) using CBOW \cite{mikolov2013distributed} and froze the word embeddings during the training process. We implemented the seq2seq model with (1) GRU of $300$ hidden units, (2) LSTM of $150$ hidden units and (3) bi-directional GRU of both $150$ forward and backward hidden units. For (2), we concatenated the final hidden state and cell state to form the $300$-dimensional semantic vector, while for (3), we concatenated the forward and backward final hidden states. We trained each seq2seq model on the training set with Adam optimizer \cite{kingma2014adam} and performed style transfer on the validation set. We set the weight of reconstruction loss and classification loss as 10:1. As observed in Section \ref{sec:comparison}, the informativeness of style matrix was insensitive to different choices of recurrent architectures and hence we only report the results of GRU implementation in other parts.
\subsection{Explore the Styles of Yelp}\label{sec:4_3}
As is discussed in Section \ref{sec:intro}, the notion of style matrix conforms to the linguistic aspects that style is innate in semantics and is multi-modal. To demonstrate style matrix can indeed capture these delicate style phenomena, we first mixed up all the reviews on Yelp with different ratings and trained a seq2seq model with reconstruction loss only. We then divided the corpus into several sub-corpora with well-designed criteria. Finally, we performed text style transfer with $(N,S)$ operators prepared respectively with these pairs of sub-corpora. Detailed results and analyses are followed in each part.
\begin{table*}[ht]
\centering
\captionsetup{justification=centering}
\caption{Performance of different style transfer methods on standard benchmarks.}\label{tab:performance}
\begin{adjustbox}{width=0.8\textwidth}
\begin{tabular}{c|cccc|cccc}
\toprule
\multirow{2}{*}{Model}&
\multicolumn{4}{c}{ Yelp}&\multicolumn{4}{c}{ Amazon}\cr
\cmidrule(lr){2-5} \cmidrule(lr){6-9}
&Acc.&BLEU&G-Score&Mean&Acc.&BLEU&G-Score&Mean\cr
\midrule
Test Set&97.48&-&-&-&80.97&-&-&-\cr
Cross-Aligned&\textbf{83.78}&12.69&32.37&46.73&60.84&8.56&22.82&34.70\cr
Style-Embedding&6.34&\textbf{85.14}&23.23&45.74&29.21&\textbf{68.14}&\textbf{44.61}&48.68\cr
NS-GRU&80.33&13.43&\textbf{32.85}&\textbf{46.88}&\textbf{79.50}&12.97&32.11&46.23\cr
NS-LSTM&78.07&12.38&31.10&45.23&74.30&15.90&34.37&45.10\cr
NS-BiGRU&72.02&12.60&30.12&42.31&74.04&25.23&43.22&\textbf{49.64}\cr
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table*}
\subsubsection{Style Intensity} \label{sec:style_intensity}
We collected four corpora which contained sentences respectively with rating $1$, $2$, $4$ and $5$ (denoted as R1, R2, R4, R5) and discarded the neutral sentences with rating $3$. The former two sub-corpora have the same polarity of attitude (i.e. negative) but with different intensity and so as the latter two. For visualization, Fig. \ref{fig:heatmap} plots the first $50$ eigenvectors of each style matrix, which shows a recognizable color gradience from the most negative corpus (with rating $1$) to the most positive corpus (with rating $5$).
\begin{figure}
\centering{
\includegraphics[width=0.5\textwidth]{image.png}
}
\caption{Heatmap of the first $50$ eigenvectors of style matrices on Yelp subcorpora with different intensity levels of attitude, from R1 the most negative corpus to R5 the most positive one (better viewed in color).}
\label{fig:heatmap}
\end{figure}
Subsequently, we constructed three sets of $(N, S)$ operators respectively from stylistic pairs (R1, R5), (R1 $\cup$ R2, R4 $\cup$ R5) and (R2, R4), in the decreasing order of style contrast level. We performed style transfer on the same validation set with the three sets of prepared $(N, S)$ operators. The results are reported in Table \ref{tab:intensity_level}. As we can see, the style transfer quality of each set of $(N, S)$ operators were positively related to the degree of style contrast and we suggest this phenomenon as an implicit validation for the informativeness of style matrix on capturing slight difference in style intensity.
\subsubsection{Multiple Styles} \label{sec:multi_style}
\begin{table}[ht]
\caption{Results of style transfer with NS algorithm on corpora pairs with multiple styles.} \label{tab:multiclass}
\centering
\begin{adjustbox}{width=0.35\textwidth}
\begin{tabular}{c|c|c}
\hline
&Acc. (Baseline)&BLEU\cr
\hline
Attitude &29.56 (2.67)&62.04\cr
Tense &65.94 (2.73)&52.28\cr
\hline
\end{tabular}
\end{adjustbox}
\end{table}
Based on the attitude and tense annotations on Yelp, we partitioned the original corpus into two pairs of sub-corpora, namely the attitude pair \textit{(positive, negative)} and the tense pair \textit{(present, past)}. Correspondingly, we calculated attitude (tense) transform operators respectively on each pair and applied the prepared operators to transfer the target attribute with the other style attribute fixed. We report the transfer performance of NS algorithm in Table \ref{tab:multiclass}, which empirically proved style matrix can simultaneously capture multiple style attributes.
\subsection{Unpaired Text Style Transfer} \label{sec:comparison}
In this part, we compared the performance of NS algorithm with the state-of-the-art methods on Yelp and Amazon datasets. We trained a seq2seq model in the semi-supervised way as described in Section \ref{sec:style_label} and transferred attitude of sentences with NS algorithm. We chose the following representative state-of-the-art style transfer methods as baselines.
\textbf{Cross-Aligned:} This method assumes a shared latent content distribution across the corpora with different styles and leverages refined alignment of latent representations to perform style transfer \citep{shen2017style} .
\textbf{Style-Embedding:} This method learns separate content representations and style representations using adversarial networks. With the style information embed into distributed vector representations, one single decoder is trained for different corpora \citep{fu2018style}.
As observed in Sec. \ref{sec:style_intensity}, the transform operators have a stronger transfer capability when generated from a pair of corpora with higher style contrast, which inspires us to further enhance the performance of NS algorithm by removing sentences with low confidence judged by the simultaneously trained style classifier. Fig. \ref{fig:line} plots the model performance on different metrics over drop rates ranging from $0$ to $0.9$ with a fixed stride $0.15$.
As we can see, the increase in drop rate caused an increase of Acc. and decrease of BLEU score. We speculate it is inevitable due to the tight interdependence between style and semantics. The result at $90\%$ drop rate provides further evidence on this phenomenon, that is, to change the style of the validation set to a corpus with extreme style feature would largely change their semantics.
\begin{figure}[ht]
\centering{
\includegraphics[width=0.4\textwidth]{line_plot.png}
}
\caption{Performance of NS algorithm with different sentences' drop rates for preparing transfer operators.}
\label{fig:line}
\end{figure}
\begin{table*}[ht]
\centering
\captionsetup{justification=centering}
\caption{Sampled out-of-domain sentences transferred by NS algorithm.}
\label{tab:ood_visualization}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{ll|ll}
\hline
\multicolumn{2}{c}{Yelp (Tense)} & \multicolumn{2}{c}{Amazon (Attitude)}\cr
\hline
\hline
Source&she did not finish the liver . & Source&the edger function did not work well for me .
\cr
Past&she did not finish the liver . & Neg&the edger function did not work well for me .
\cr
Pres&she does not finish the liver . & Pos&the edger function work well for me .
\cr
\hline
Source&however , i think i would try somewhere else to dine . & Source&my daughter was just frustrated with this toy .\cr
Past&however , i thought i would try somewhere to dine . & Neg&my daughter was just upset with this toy .\cr
Pres&however , i think i would try somewhere else to dine . & Pos&my daughter was just happier with this toy .\cr
\hline
\end{tabular}
\end{adjustbox}
\end{table*}
Since the trade-off between the transfer ability and content preservation can be controlled, it is hard to select one balanced point to fully characterize the performance of our method. As a complement, we suggest to use Mean as an overall performance measure, which is more stable than G-Score as observed in Fig. \ref{fig:line}. Table \ref{tab:performance} shows the performance of our methods with different recurrent architectures and baselines. As we can see, our method achieved comparable performance with two baselines while averagely outperformed them on Amazon, the benchmark with a larger vocabulary size. For an illustrative comparison, we further provide some generated samples from each method in Appendix A.
\subsection{Out-of-Domain Style Transfer} \label{sec:ood}
\begin{table}[ht]
\centering
\caption{Performance of NS algorithm on out-of-domain style transfer tasks.}\label{tab:ood}
\begin{adjustbox}{width=0.35\textwidth}
\begin{tabular}{cccc}
\toprule
\multicolumn{4}{c}{Yelp (Tense)}\cr
\cmidrule(lr){1-4}
Acc. (Test)&BLEU&G-Score&Mean\cr
86.35(94.80)&32.64&53.09&59.49\cr
\midrule
\multicolumn{4}{c}{Amazon (Attitude)}\cr
\cmidrule(lr){1-4}
Acc. (Test)&BLEU&G-Score&Mean\cr
87.94(97.48)&22.05&44.03&55.00\cr
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
In the final part, we propose \textit{out-of-domain style transfer} as a much challenging task for text style transfer, where, given a corpus with style labels, the style transfer models are required to control the style of unlabeled sentences coming from out-of-domain corpora. For validation of NS algorithm's performance on this task, we use the Yelp with attitude labels only and Amazon with tense labels only to control their style on the other pair of attributes which is not observed by them. In other words, we would transfer tenses of sentences in Yelp with the $(N, S)$ operators prepared from Amazon and vice versa.
In this scenario, we only need a slight modification on our proposed method in Sec. \ref{sec:transfer}, that is, to train the seq2seq model on Yelp $\cup$ Amazon with two style classifiers, namely attitude classifier on Yelp and tense classifier on Amazon. After preparing the style transform operators on the other domain, it is straightforward to out-of-domain style transfer. The results are reported in Table \ref{tab:ood}. It is worth to notice, even though the validation set is unlabeled in the style attribute we want to transfer, our NS algorithm can still achieve superior performance in both cases, which further validated the flexibility of style matrix perspective and the effectiveness of NS algorithm. We also provide some illustrative results in Table \ref{tab:ood_visualization}. Noticeably, the capability of out-of-domain style transfer allows us to leverage several corpus annotated with single style attributes for controlling multiple styles on each corpus.
\if0
We trained the seq2seq model on the training set and performed style transfer on the test set.
In several well-designed cases, we trained the seq2seq model with reconstruction loss only and combined the pretrained model with NS algoritm for style transfer back and forth between different style domains.
Respectively on Yelp and Amazon datasets, We trained the seq2seq model with style labels from datasets (\S \ref{sec:style_label}) and combined the pretrained model with NS algorithm for style transfer. By comparing the performance of NS algorithm on each standard metric with typical baseline methods.
our Neutralization-Stylization algorithm can achieve style transfer on multiple styles while the style labels are kept the secret to seq2seq model.
In this section, we evaluate the informativeness of style matrix and the performance of our proposed NS algorithm on unpaired text style transfer task. Before diving into the details for each part of experiments, we describe the general setups.
\subsection{Summary of Results}
After appling our proposed style matrix extraction and NS algorithm under three representative scenarios, we highlight some experimental findings as follows.
\begin{itemize}[leftmargin=*]
\item \textit{Informativeness -} We empirically validated the style matrix could indeed capture the multi-modality of style phenomenon, especially the difference in intensity level of style features and the co-occurrence of style attributes (\S \ref{sec:4_3}).
\item \textit{Effectiveness -} We empirically proved our proposed NS algorithm can achieve comparable performance with the state-of-the-art methods (\S \ref{sec:comparison}).
\item \textit{Flexibility -} We successfully applied NS algorithm to out-of-domain style transfer in a straightforward approach. We observed the transferred out-of-domain sentences by NS algorithm showed easily recognizable style feature ($\text{Acc.}>80\%$), preserved content well ($\text{BLEU}>22\%$) and meanwhile were highly legible (\S \ref{sec:ood}).
\end{itemize}
\fi
\section{Related work}
\noindent\textbf{Unparalled Text Style Transfer.} A major proportion of works proposed to learn the style-independent semantic representations of sentences for downstream transfer tasks \citep{fu2018style,shen2017style,hu2017toward,chen2018adversarial}. These works minimized reconstruction loss of a variational autoencoder \citep{kingma2013auto} to compress the sentences and align the distributions of these vectors by adversarial training. Some other works utilized heuristic transformation to accomplish style transfer by explicitly dividing the sentence into semantic words and style words \citep{li2018delete,xu2018unpaired}. Essentially different from these previous works, our work focuses on studying how seq2seq models perceive language styles and the competitive performance of our proposed style transfer algorithm is therefore better to be considered as an implicit justification to our style matrix view on language style.
\noindent\textbf{Style in Other Domains.} Style phenomenon is also studied in other domains, especially in computer vision. The groundbreaking works by \citet{gatys2015texture,gatys2016image} showed the Gram matrices of the feature maps extracted by a pre-trained convolution neural network are able to capture the visual style of an image, which was immediately followed by numerous works have been developed to transfer the style by matching the generated Gram matrices (e.g. \citealt{ulyanov2016texture,ulyanov2017improved, johnson2016perceptual,chen2017stylebank}) and \citet{li2017demystifying} theoretically proves that it's equivalent to minimize the maximum mean discrepancy of two distributions.
\if 0
Almost all the current studies consider the style of the corpus as a fixed label assigned manually other than the representations abstracted from the corpus automatically. For example, in the commonly used Yelp reviews dataset, all the reviews with a rating above three are considered semantically positive. From this point of view, the majority of these studies decompose a sentence to independent style and semantic information. One solution is to learn the style-independent content representation of sentences and then reconstruct the sentences with the given style labels.\cite{fu2018style,shen2017style,hu2017toward,chen2018adversarial} Generally, these works optimize reconstruction loss which is classical in auto-encoder to compress the sentences to a shared vector space and align the distribution of these vectors from different corpora with an adversarially trained classifier. However, it's hard to successfully transfer the style and preserve the original content simultaneously with these adversarial methods according to\cite{li2018delete,xu2018unpaired,lample2018multiple}. These observations indicate the difficulty of separating the style information from the semantic content as they are complicatedly mixed together. Another solution proposed by \cite{li2018delete,xu2018unpaired} implements heuristic transformation, concretely, they explicitly classify the words in a sentence into content words and sentiment words(when taking the sentiment as style). Then with an auto-encoder fed with the former and specific styles, they can synthesize new sentences. As mentioned by \cite{li2018delete}, the deleted sentiment words also contain non-sentiment information, e.g. we can't use sentiment word ``delicious'' to complete ``The customer service is'' because their semantics are not related. Thus deleting the sentiment words will inevitably cause the loss of semantic information.
In the field of text style transfer...
Similarly, image style transfer aims to reconstruct an image with some characters of the style image while preserving its content. The groundbreaking works proposed by Gatys et al. \cite{gatys2015texture,gatys2016image} show that the Gram matrices (or covariance matrices) of the feature maps, which are extracted by a frozen convolution neural network trained on classification task, are able to capture the visual style of an image. After that numerous works have been developed by matching the Gram matrices and \cite{li2017demystifying} theoretically proves that it's equivalent to minimize the Maximum Mean Discrepancy with the second order polynomial kernel. Now that \cite{gatys2016image} synthesize the target image through iterative optimization which is inefficient and time-consuming, \cite{ulyanov2016texture,johnson2016perceptual} propose to train a feed-forward network per style and significantly speed up the image reconstruction. And \cite{chen2017stylebank,dumoulin2017learned} further improve the flexibility and efficiency by incorporating multiple styles with only one network. In order to reconstruct images to arbitrary styles(unseen at the training stage) with a single network forward pass, \cite{} proposes whitening and coloring transformations to directly match the feature covariance to the style image at intermediate layers of a pre-trained auto-encoder network.
\fi
\section{Conclusion}
In this paper, we have investigated the style matrix encoded by seq2seq models as an informative delegate to language style. The notion of style matrix conforms well to human experiences and existing linguistic theories on language style. In practice, we have also proposed NS algorithm as a plug-and-play solution to unpaired text style transfer which achieved competitive transfer quality with the state-of-the-art methods and meanwhile showed superior flexibility in various use cases. In the future, we plan to discuss how the quality of semantic vectors impacts the informativeness of style matrix and study what is encoded in higher-order statistics of semantic vectors.
\clearpage
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| null | null |
\section*{Notation}
The notation used in this paper is included below.
\subsection{Indices/Set}
\begin{supertabular}{p{0.1\textwidth}p{0.78\textwidth}}
$d/D$ & Index/set of consumers \\
$D_{n}$ & Set of consumers located in bus $n$ \\
$g/G$ & Index/set of generating units \\
$G^{C/R}$ & Set of conventional/renewable generating \newline units \\
$G_{n}$ & Set of generating units located in bus $n$ \\
$k/K$ & Index/set of contingencies \\
$l/L$ & Index/set of transmission line \\
$L_{n}^{O/F}$ & Set of lines whose origin/destination bus is \newline $n$ \\
$n/N$ & Index/set of buses \\
$N^{O/F}_{l}$ & Set of origin/destination bus of line $l$ \\
$t/T$ & Index/set of time periods \\
$T_{v}$ & Set of periods in which PEVs in group $v$ \newline can be charged or discharged to the grid \\
$v/V$ & Index/set of PEV groups \\
\end{supertabular}
\subsection{Variables}
\begin{supertabular}{p{0.1\textwidth}p{0.78\textwidth}}
$c_{gt}^{SU/SD}$ & Startup/shutdown cost of generator $g$ in \newline period $t$ \\
$c_{t}^{P}$ & Operation cost in period $t$ \\
$c_{t}^{SP}$ & Spillage cost of renewable units in period $t$ \\
$c_{t}^{SU/SD}$ & Startup/shutdown cost in period $t$ \\
$c_{t}^{UD}$ & Unserved demand cost in in period $t$ \\
$c_{t}^{UD,PR}$ & Unserved demand cost in period $t$ after \newline contingency $k$ \\
$cc_{t}^{V}$ & Frequency reserve schedule by PEVs group \newline cost in period $t$ \\
$cp_{t}^{V}$ & Deployed energy cost in period $t$ \\
$c_{t}^{\Delta f}$ & Frequency deviation cost in period $t$ \\
$c_{vnt}^{V,PR}$ & Scheduled capacity that can be used for PFR \newline by PEVs group $v$ in bus $n$ and period $t$ \\
$e_{vnt}^{C/D}$ & Energy charged/discharged by PEVs group $v$ \newline in bus $n$ and period $t$ \\
$e_{vntk}^{C/D,PR}$ & Energy charged/discharged by PEVs group $v$ \newline in PFR in bus $n$, period $t$ and after a \newline contingency $k$ \\
$e_{vntk}^{V}$ & Energy stored by PEVs group $v$ in bus $n$, \newline period $t$ and after a contingency $k$ \\
$p_{dt}^{UD}$ & Unserved demand of consumer $d$ in period $t$ \\
$p_{dtk}^{UD,PR}$ & Unserved demand of consumer $d$ in period $t$ \newline after contingency $k$ \\
$p_{gt}$ & Power produced by unit $g$ in period $t$ \\
$p_{lt}^{L}$ & Power flow in line $l$ in period $t$ \\
$p_{vntk}^{V,PR}$ & Primary frequency response provided by PEVs \newline in group $v$ in bus $n$, period $t$ and after a \newline contingency $k$ \\
$p_{vntk}^{V,PRC}$ & Primary frequency response provided by PEVs \newline in group $v$ in charging mode in bus $n$, period \newline $t$ and after a contingency $k$ \\
$p_{vntk}^{V,PRD}$ & Primary frequency response provided by PEVs \newline in group $v$ in discharging mode in bus $n$, \newline period $t$ and after a contingency $k$ \\
$p_{gtk}^{PR}$ & Primary reserve output of generator unit $g$ in \newline period t after contingency $k$ \\
$s_{gt}$ & Spillage of renewable unit $g$ in period $t$ \\
$u_{gt}$ & Binary variable that is equal to 1 if the unit \newline $g$ is online, and 0 otherwise \\
$\Delta f_{tk}$ & Frequency deviation in period $t$ after \newline contingency $k$ \\
\end{supertabular}
\subsection{Parameters}
\begin{supertabular}{p{0.1\textwidth}p{0.78\textwidth}}
$A_{gt}$ & Availability factor of renewable unit $g$ in \newline period $t$, varying between 0 and 1 \\
$C_{g}$ & Production cost of generator $g$ \\
$C_{g}^{SU/SD,F}$ & Startup/shutdown cost factor of generator $g$ \\
$Cc_{vnt}^{V,PR}$ & PFR capacity cost offer of PEV group $v$ in \newline bus $n$ and period $t$ \\
$Cp_{vnt}^{V,PR}$ & Deployed reserve cost offer of PEV group $v$ \newline in bus $n$, period $t$ and after contingency $k$\\
$C^{P}$ & Penalization cost of forced intermittent \newline power spillage \\
$C^{UD}$ & Penalization cost of unserved demand \\
$C^{\Delta F}$ & Penalization cost of frequency deviation \\
$D^{PR}$ & Maximum duration of primary frequency \newline response in hours \\
$DR_{g}$ & Frequency droop of generator unit $g$ \\
$DR_{v}$ & Frequency droop of PEV group $v$ \\
$DT_{g}$ & Minimum down time of unit $g$ \\
$E_{max,v}^{V}$ & Capacity of batteries of PEVs in group $v$ \\
$E_{min,v}^{V}$ & Minimum value of energy that must \newline remain in the batteries of PEVs in group $v$ \newline in each charging/discharging period \\
$E_{vn}^{VF}$ & Minimum status of PEVs batteries in \newline group $v$ and period $t$ at the end of charging \newline period \\
$E_{vn}^{V0}$ & Initial status of the batteries of PEVs in \newline group $v$ and period $t$ at the beginning of the \newline charging period \\
$N_{vn}^{V}$ & Number of PEVs in group $v$ \\
$P_{dt}^{D}$ & Power demand of supplied to consumer $d$ \newline in period $t$ \\
$P_{max,g}$ & Maximum power output of unit $g$ \\
$P_{max,lt}^{L}$ & Capacity of line $l$ in period $t$ \\
$P_{max}^{V}$ & Maximum power charging/discharging rate \newline of PEVs \\
$P_{min,g}$ & Minimum power output of unit $g$ \\
$PR_{g}^{U/D}$ & Ramp-up/down limit of unit $g$ \\
$S_{gk}$ & Availability parameter, equal to 1 if the unit \newline $g$ is out after contingency $k$ and 0 otherwise \\
$TC_{g}$ & Nº of periods that unit $g$ must be initially \newline offline \\
$TG_{g}$ & Nº of periods that unit $g$ must be initially \newline online \\
$T_{v}^{0/F}$ & Initial/final period which PEVs group $v$ can \newline be charged or discharged \\
$UT_{g}$ & Minimum up time of unit $g$ \\
$X_{l}$ & Reactance of line $l$ \\
$\eta_{v}$ & Efficiency of PEVs in group $v$\\
\end{supertabular}
\section{Introduction}
The presence of renewable energy sources (RES) in electrical power systems has increased due to environmental concerns. The emission of Greenhouse Gases is one of the biggest concerns of our society due to global warming \cite{Paris, ODS}. The electricity production sector is one of the main emitters of carbon dioxide ($CO_2$). For this reason, there is a great need for the increase of carbon-free sources in the electrical system to reduce the use of conventional fossil energy \cite{Irena}.
In this context, the Federal University of Amapá (UNIFAP) started in August 2020 the installation of a photovoltaic solar plant, as part of the project "UNIFAP SOLAR: Implantação de Geração Distribuída Fotovoltaica no Campus Marco Zero do Equador". This project aims to implement environmental sustainability programs at the university, as well as promote actions for the institutional community through the reduction and reuse of resources and energy \cite{unifap-solar}. The installation that has started foresees the installation of about 1.3 MWp in solar photovoltaic power plants at UNIFAP.
Despite the efforts to increase generation through renewable sources, it can be seen that the presence of renewable sources in isolated systems has occurred on a smaller scale than in interconnected systems \cite{Carrion-2019}. One of the main reasons is that the intermittency of energy resources such as solar and wind generates energy quality problems. The insertion of renewable sources in the system causes a decrease in the system's equivalent inertia, which may result in threads of instability.
In contrast, the use of energy storage systems has proven to be an effective tool to increase the flexibility of the energy system operation \cite{Dunn-2011}, facilitating the installation of renewable sources in isolated power systems. The Vehicle-to-Grid ($V2G$) capability of Plug-in Electric Vehicles (PEVs), allows vehicles to inject the energy stored in the batteries into the grid. As a consequence of this, PEVs can provide ancillary services to the electric power system when connected to the grid. To balance power consumption and production, PEVs can be used as both a load and a generating sources to maintain the system frequency at acceptable values by charging their batteries when there is too much generation in the grid and acting as generators by discharging the batteries when there is too much load in the system \cite{Kempton-2008}.
Despite the numerous advantages of renewable technologies, the production coming from these sources can be considered as non-dispatchable, variable, and uncertain. Traditional synchronous generators can provide inertia and primary frequency response (PFR) to power systems. However, renewable energy is connected to the grid by the power converters, which are unable to provide inertia to the system in a similar manner than synchronous generators \cite{Fanglei2020}. In a context in which renewable units are supplying a great part of the demand, the determination of the day-ahead scheduling is more important and complex than in thermal-dominated power systems. The day-ahead schedule is a large-size mixed-integer linear programming (MILP), the determination of the day-ahead scheduling of a power system is a complex mathematical problem that is based on the formulation of the economic dispatch or unit commitment problems.
The insertion of renewable energy sources and electric vehicles in power systems has been extensively studied in recent years. In \cite{Restrepo-2005} the UC that simultaneously accounts for both primary and tertiary reserve constraints is formulated. In Reference \cite{Carrion-2006} a mixed-integer linear formulation for the thermal UC problem is presented in order to reduce the computational burden of existing MILP approaches. A two-stage stochastic UC model with high renewable penetration is present in \cite{asensio2015stochastic}, the model proposed is applied in the power system of the Canary Islands and Crete.
Reference \cite{mercier} studies the support of the battery energy storage system in the dynamic stability of an isolated power system with low grid inertia. The participation of electric vehicles providing frequency control is examined in \cite{almeida2011electric}. In \cite{Yoo2019} the frequency-support parameters of energy storage systems are calculated for achieving stable frequency response from a power system with high penetration of renewable generators. An economic feasibility study of V2G frequency regulation is performed in \cite{han2013economic}, in consideration of battery wear. Reference \cite{zhang2016day} propose an algorithm that optimizes the charging/discharging of energy storage devices in order to minimize the total system day-ahead operating cost.
Reference \cite{aziz2018electric} presents a study of electric vehicle utilization to support a small-scale energy management system, showing that the utilization is feasible and deployable. In Reference \cite{bellekom2012electric}, the Dutch power system operation is analyzed when the penetration of EV and RES is expected to increase significantly due to the goals to reduce CO2 emissions. In \cite{carrion2015operation} the participation of PEVs in a renewable-dominated power system based on the isolated power system of Lanzarote-Fuerteventura, Spain, is analyzed.
In this context, considering the need of introducing carbon-free technologies in the energy system, this work will assess the participation of PEVs in the day-ahead energy generation market and the provision of ancillary services in a predominantly renewable isolated system. In this work, the mathematical modeling is based on \cite{Carrion-2019}, with some simplifications for a less complex system. The microgrid to be used in this work corresponds to the electrical system of the Marco Zero Campus of the Federal University of Amapá. For the development of the analysis, it will consider that it is a system connected to the grid with isolated operation capacity, with about $40\%$ of the energy mix being renewable sources.
\section{Model}
A linear programming formulation is used in this work, where the objective function to be minimized is the operating cost of an electrical power system. Among the restrictions to which the objective function is subject, stand out those constraints related to PEVs, renewable energy sources, load flow, frequency deviation, frequency regulation, and operation of the generating units.
\subsection{Problem Formulation}
\label{section-equations}
The formulation described in this work was based in \cite{Carrion-2019}. The mathematical formulation of the proposed unit commitment problem is the following:
{
\fontsize{7pt}{\baselineskip}\selectfont
\begin{equation}
\begin{split}
\MoveEqLeft
\emph{Min} \hskip 0.2cm \sum_{t \in T} \sum_{g \in G} (C_{g}\cdot p_{gt} + c_{gt}^{SU} + c_{gt}^{SD}) + \sum_{t \in T} \sum_{d \in D} C^{UD} \cdot p_{dt}^{UD} \\
&+ \sum_{t \in T} \sum_{g \in G^R} C^{P} \cdot s_{gt}
+ \sum_{d \in D} \sum_{t \in T} \sum_{k \in K} (C^{UD} \cdot p_{dtk}^{UD,PR} - C^{\Delta F} \cdot \Delta f_{tk}) \\
&+ \sum_{t \in T} \sum_{n \in N} \sum_{v \in V} (Cc_{vnt}^{V,PR} \cdot c_{vnt}^{V,PR}
+ \sum_{k \in K} Cp_{vnt}^{V,PR} \cdot p_{vntk}^{V,PR})
\end{split}
\label{objfunction}
\end{equation}
}
Subject to:
{
\fontsize{7pt}{\baselineskip}\selectfont
\begin{equation}
\begin{split}
\MoveEqLeft
\sum_{g \in G_{n}} p_{gt} + \sum_{l \in L_{n}^{F}} p_{lt}^{L} - \sum_{l \in L_{n}^{O}} p_{lt}^{L} + \sum_{v \in V} (e_{vnt}^{D}-e_{vnt}^{C}) \\
&= \sum_{d \in D_{n}} (p_{dt}^{D} - p_{dt}^{UD}),
\hskip 0.3cm \forall t \in T, \forall n \in N
\end{split}
\label{energybalance}
\end{equation}
\begin{equation}
- P_{max,lt}^{L} \leq p_{lt}^{L} \leq P_{max,lt}^{L},
\hskip 0.3cm \forall l \in L, \forall t \in T
\label{flowlimits}
\end{equation}
\begin{equation}
p_{lt}^{L} = \frac{1}{X_{l}} \cdot (\theta_{N^{O}_{lt}}- \theta_{N^{F}_{lt}}),
\hskip 0.3cm \forall l \in L, \forall t \in T
\label{flowline}
\end{equation}
\begin{equation}
P_{min,g} \cdot u_{gt} \leq p_{gt} \leq P_{max,g} \cdot u_{gt},
\hskip 0.3cm \forall g \in G^C, \forall t \in T
\label{limitsC}
\end{equation}
\begin{equation}
p_{gt} + s_{gt} = P_{max,g} \cdot A_{gt},
\hskip 0.3cm \forall g \in G^R, \forall t \in T
\label{limitsR}
\end{equation}
\begin{equation}
p_{gt} - p_{g t-1} \leq PR_{g}^{U},
\hskip 0.3cm \forall g \in G^C, \forall t \in T
\label{prampup}
\end{equation}
\begin{equation}
p_{gt-1} - p_{gt} \leq PR_{g}^{D}
\hskip 0.3cm \forall g \in G^C, \forall t \in T
\label{prampdown}
\end{equation}
\begin{equation}
c_{gt}^{SU} \geq C_{g}^{SU,F} (u_{gt}-u_{gt-1}),
\hskip 0.3cm \forall g \in G^C, \forall t \in T
\label{startup}
\end{equation}
\begin{equation}
c_{gt}^{SD} \geq C_{g}^{SD,F} (u_{gt-1}-u_{gt}),
\hskip 0.3cm \forall g \in G^C, \forall t \in T
\label{shutdown}
\end{equation}
\begin{equation}
c_{gt}^{SU} \geq 0, c_{gt}^{SD} \geq 0 \hskip 0.3cm \forall g \in G^{C}, \forall t \in T
\label{startup-shutdown}
\end{equation}
\begin{equation}
\sum_{t=1}^{TG_{g}} (1-u_{gt}) = 0,
\hskip 0.3cm \forall g \in G^C
\label{uptime01}
\end{equation}
\begin{equation}
\begin{split}
&\sum_{\tau =t}^{t-UT_{g}-1} u_{g\tau} \geq UT_{g}(u_{gt} - u_{gt-1}),\\
\forall g& \in G^C, \forall t = TG_g + 1 \cdots T-UT_g+1
\end{split}
\label{uptime02}
\end{equation}
\begin{equation}
\begin{split}
&\sum_{\tau=t}^{T} [u_{g\tau} - (u_{gt}-u_{gt-1})] \geq 0, \\
\forall &g \in G^C, \forall t = T - UT_g +2 \cdots T
\end{split}
\label{uptime03}
\end{equation}
\begin{equation}
\sum_{t=1}^{TL_{g}} u_{gt} = 0,
\hskip 0.3cm \forall g \in G^C
\label{downtime01}
\end{equation}
\begin{equation}
\begin{split}
&\sum_{\tau=t}^{t+DT_g-1} (1-u_{g\tau}) \geq DT_g(u_{gt-1} - u_{gt}), \\
& \forall g \in G^C, \forall t = TL_g+1 \cdots T-DT_g+1
\end{split}
\label{downtime02}
\end{equation}
\begin{equation}
\sum_{\tau=t}^{T} [1-u_{g\tau}-(u_{gt-1}-u_{gt})] \geq 0,
\hskip 0.3cm \forall g \in G^C, \forall t = T-DT_g+2 \cdots T
\label{downtime03}
\end{equation}
\begin{equation}
0 \leq p_{gtk}^{PR} \leq - \frac{1}{DR_{g}} \cdot \Delta f_{tk},
\hskip 0.3cm \forall g \in G^C \hskip 0.2cm | \hskip 0.2cm g \notin S_{gk}, \forall t \in T, \forall k \in K
\label{pr01}
\end{equation}
\begin{equation}
p_{gtk}^{PR} + p_{gt} \leq P_{max,g} \cdot u_{gt},
\hskip 0.3cm \forall g \in G^C \hskip 0.2cm | \hskip 0.2cm g \notin S_{gk}, \forall t \in T, \forall k \in K
\label{pr02}
\end{equation}
\begin{equation}
p_{gtk}^{PR} = 0,
\hskip 0.3cm \forall g \in G^{R} \hskip 0.2cm | \hskip 0.2cm g \notin S_{gk}, \forall t \in T, \forall k \in K
\label{pr03}
\end{equation}
\begin{equation}
p_{gtk}^{PR} = - p_{gt},
\hskip 0.3cm \forall g \in G \hskip 0.2cm | \hskip 0.2cm g \in S_{gk}, \forall t \in T, \forall k \in K
\label{pr04}
\end{equation}
\begin{equation}
\sum_{d \in D} p_{dtk}^{UD,PR} + \sum_{g \in G} p_{gtk}^{PR} + \sum_{v \in V} \sum_{n \in N} p_{vntk}^{V,PR} = 0,
\hskip 0.3cm \forall t \in T, \forall k \in K
\label{pr05}
\end{equation}
\begin{equation}
e_{vntk}^{V} = N_{vn}^{V} \cdot E_{vn}^{V0},
\hskip 0.3cm \forall v \in V, \forall n \in N , t=T_{v}^{0}-1, \forall k \in K
\label{statusbat01}
\end{equation}
\begin{equation}
e_{vntk}^{V} \geq N_{vn}^{V} \cdot E_{vn}^{VF},
\hskip 0.3cm \forall v \in V, \forall n \in N , t=T_{v}^{F}, \forall k \in K
\label{statusbat02}
\end{equation}
\begin{equation}
\begin{split}
e_{vntk}^{V} = e_{vnkt-1}^{V} &+ \eta_{v} (e_{vnt}^{C} - e_{vntk}^{C,PR}) - \frac{1}{\eta_{v}} (e_{vnt}^{D} + e_{vntk}^{D,PR}),\\
\forall v &\in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\end{split}
\label{energystoredpevs}
\end{equation}
\begin{equation}
\begin{split}
N_{vn}^{V} &\cdot E_{min,v}^{V} \leq e_{vntk}^{V} \leq N_{vn}^{V} \cdot E_{max,v}^{V}, \\
& \forall v \in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\end{split}
\label{limitsenergybat}
\end{equation}
\begin{equation}
e_{vnt}^{C}, e_{vnt}^{D} \leq N_{vn}^{V} \cdot P_{max,v}^{V},
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T, \forall k \in K
\label{limitsenergybat2}
\end{equation}
\begin{equation}
e_{vntk}^{C,PR} = D^{PR} \cdot p_{vntk}^{V,PRC},
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\label{pfrpev01}
\end{equation}
\begin{equation}
D^{PR} \cdot p_{vntk}^{V,PRC} \leq e_{vnt}^{C} ,
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\label{pfrpev02}
\end{equation}
\begin{equation}
e_{vntk}^{D,PR} = D^{PR} \cdot p_{vntk}^{V,PRD},
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\label{pfrpev03}
\end{equation}
\begin{equation}
\begin{split}
e_{vnt}^{D} &\leq N_{vn}^{V} \cdot P_{max}^{V} - D^{PR} \cdot p_{vntk}^{V,PRD},\\
\forall v &\in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\end{split}
\label{pfrpev04}
\end{equation}
\begin{equation}
0 \leq p_{vntk}^{V,PRC} \leq N_{vn}^{V} \cdot P_{max}^{V},
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\label{pfrpev05}
\end{equation}
\begin{equation}
0 \leq p_{vntk}^{V,PRD} \leq N_{vn}^{V} \cdot P_{max}^{V},
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\label{pfrpev06}
\end{equation}
\begin{equation}
p_{vntk}^{V,PR} = p_{vntk}^{V,PRC} + p_{vntk}^{V,PRD},
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\label{pfrpev07}
\end{equation}
\begin{equation}
0 \leq p_{vntk}^{V,PR}\leq - \frac{1}{DR_{v}} \cdot \Delta f_{tk},
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T_{v}, \forall k \in K
\label{pfrpev08}
\end{equation}
\begin{equation}
\begin{split}
p_{vntk}^{V,PR}&, e_{vnt}^{C}, e_{vnt}^{D}, e_{vntk}^{C,PR}, e_{vnt}^{D,PR} = 0, \\
\forall v &\in V, \forall n \in N , \forall t \notin T_{v}, \forall k \in K
\end{split}
\label{pfrpev09}
\end{equation}
\begin{equation}
0 \leq p_{vntk}^{V,PR}\leq c_{vnt}^{V,PR},
\hskip 0.3cm \forall v \in V, \forall n \in N , \forall t \in T, \forall k \in K
\label{pfrpev10}
\end{equation}
}
\vskip-0.7cm
The objective function (\ref{objfunction}) represents the expected costs considering production, startup and shutdown costs ($C_{g}\cdot p_{gt}$, $c_{gt}^{SU}$ and $c_{gt}^{SD}$, respectively) and penalization for unserved energy ($C^{UD}\cdot p_{dt}^{UD}$), unserved primary reserve ($C^{UD} \cdot p_{dtk}^{UD,PR}$), frequency deviation ($C^{\Delta F} \cdot \Delta f_{tk}$), forced intermittent units spillage ($C^{P} \cdot s_{gt}$) and costs related to PEVs. Observe that the costs associated with frequency deviations and forced spillage are fictitious penalization costs intended to avoid frequency deviations and forced intermittent units spillage if possible.
Constraint (\ref{energybalance}) presents the energy balance in the pre-contingency state. Constraints (\ref{flowlimits}) and (\ref{flowline}) represent the power flow in line $l$. The power limits of the generating units are presented by constraints (\ref{limitsC}) and (\ref{limitsR}). Constraints (\ref{prampup}) and (\ref{prampdown}) formulate the power ramps of generating units. Constraints (\ref{startup})-(\ref{startup-shutdown}) formulate startup and shutdown costs of generating units. Constraints (\ref{uptime01})-(\ref{uptime03}) represent the minimum up time of unit $g$ and the minimum down time is formulated by constraints (\ref{downtime01})-(\ref{downtime03}). The PFR is expressed by constraints (\ref{pr01})-(\ref{pr05}). Constraints (\ref{statusbat01}) and (\ref{statusbat02}) represent the status of batteries of PEVs. The energy stored by PEVs in each period $t$ is formulated by constraints (\ref{energystoredpevs}), (\ref{limitsenergybat}) and (\ref{limitsenergybat2}). Finally, the participation of PEVs in PFR is expressed by constraints (\ref{pfrpev01})-(\ref{pfrpev10}). Problem (\ref{objfunction})-(\ref{pfrpev10}) is a MILP problem that can be solved by commercial solvers.
\section{Case Study}
\subsection{System Description}
\label{section-input}
The microgrid used in this work corresponds to the Marco Zero Campus electrical system of the Federal University of Amapá. UNIFAP can be modeled as a consumer unit served at a voltage of 13.8 kV, with a contracted demand of 1000 kW off-peak and 1400 kW at peak hours, and an average monthly consumption of 341.780 kWh in 2019 \cite{diagnostico-pdp}.
The microgrid comprises 64 buses, 63 lines, 32 consumer units, and 5 generators, where $G1$, $G2$ and $G3$ are conventional generators, and $G4$ and $G5$ are renewable energy sources. The technical characteristics of thermal and renewable generators are listed in Tables \ref{input-nonren-tab} and \ref{input-ren-tab}, respectively. Six PEV charging points were considered, which are located at strategic points of the university campus. In this work, three models of electric vehicles were used. Each model characterizes a group of vehicles. Groups 1 and 2 represent the PEVs consisting on the academic community's own transport, whereas Group 3 corresponds to the bus belonging to the university's vehicle fleet. Table \ref{input-pevs-tab} provides the technical characteristics of PEVs.
\begin{table*}[!h]
\centering
\caption{Technical characteristics of dispatchable units}
\hfill\par
\begin{tabular}{ccccccccccc}
\hline & $C_{g}$ & $P_{max,g}$ & $P_{min,g}$ & $P0_{g}$ & $C_{gt}^{SU,F}$ & $C_{gt}^{SD,F}$ & $PR_{g}^{U}$ & $PR_{g}^{D}$ \\
& (R\$/$MWh$) & ($MW$) & ($MW$) & ($MW$) & (R\$) & (R\$) & ($MW$) & ($MW$) \\ \hline
Unit 1 & 505.0 & 0.60 & 0.12 & 0.30 & 909.00 & 9.09 & 0.15 & 0.15\\
Unit 2 & 505.0 & 0.60 & 0.12 & 0.30 & 909.00 & 9.09 & 0.15 & 0.15\\
Unit 3 & 505.0 & 0.60 & 0.12 & 0.30 & 909.00 & 9.09 & 0.15 & 0.15\\ \hline
\end{tabular}
\label{input-nonren-tab}
\end{table*}
\begin{table}[!h]
\centering
\caption{Technical characteristics of intermittent units}
\hfill\par
\begin{tabular}{ccc}
\hline
& $C_{g}$ & $P_{max,g}$ \\
& (R\$/$MWh$) & ($MW$) \\ \hline
Unit 4 & 0.000 & 0.554\\
Unit 5 & 0.000 & 0.720\\ \hline
\end{tabular}
\label{input-ren-tab}
\end{table}
\begin{table}[!h]
\centering
\caption{Technical characteristics of PEVs}
\hfill\par
\begin{tabular}{ccccc}
\hline & $E_{max,v}^{V}$ & $E_{min,v}^{V}$ & $Cc_{v}^{V,PR}$ & $Cp_{v}^{V,PR}$ \\
& ($MWh$) & ($MWh$) & (R\$/$MW$) & (R\$/$MWh$) \\ \hline
Group 1 & 0.052& 0.0052& 50& 300\\
Group 2 & 0.066& 0.0066& 50& 300\\
Group 3 & 0.324& 0.0324& 50& 300\\ \hline
\end{tabular}
\label{input-pevs-tab}
\end{table}
The production costs of conventional generators included in Table \ref{input-nonren-tab}, $C_{g}$, were defined according to the ANEEL Tariff Ranking \cite{ranking} and it was considered that there is no production cost for intermittent sources. The cost of unserved demand is set at R\$10,000/$MWh$, aiming at a high penalty. The maximum allowed frequency deviation is considered 1 Hz.
The system demand in each period is shown in Figure \ref{demand}. It is important to highlight that the time period $t=1$ corresponds to the time 12 am, $t=2$ at 1 am, $t=3$ at 2:00, and so on until $t=24$ corresponding to 11 pm.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.4]{figures/demand.png}
\caption{Energy demand in each period}
\label{demand}
\end{figure}
The availability factor of renewable units is presented in Figure \ref{avail-factor}, this value will limit the generation of these energy sources, that is, only in period 14 (when the availability factor is equal to 1) will the sources be able to generate the equivalent of their generation capacity.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.4]{figures/aval.png}
\caption{Availability factor of intermittent unit}
\label{avail-factor}
\end{figure}
\subsection{Results}
The model proposed in Section \ref{section-equations} was tested on the system described in Section \ref{section-input}. All simulations are performed with GAMS using a laptop with $2.4$GHz processors and $4$GB of RAM.
Three cases will be analyzed, these are:
\begin{itemize}
\item Case 1: Day-ahead scheduling without considering frequency reserve constraints;
\item Case 2: Day-ahead scheduling with frequency reserve constraints. Only generation units participate in this service;
\item Case 3: Day-ahead scheduling with frequency reserve constraints. Generation units and PEVs participate in this service;
\end{itemize}
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.45]{figures/power-prod.png}
\caption{Power produced by units in each time period}
\label{generation}
\end{figure*}
Figure \ref{generation} shows the power produced by the generating units ($p_{gt}$) in the three considered cases. Please, observe that the first three units (Units $1$, $2$ and $3$) are conventional generating units, while the others (Units $4$ and $5$) are intermittent units. Note that when frequency regulation is not considered (Case 1), most of the energy produced in hours 1-7 is provided by Unit 1. Then, a failure of this unit may put at risk the operation of the system. However, if frequency regulation is considered, (Cases 2 and 3), the energy schedule considers the possibility of failures in the units and the demand in each hour is provided by several units. Observe that when units are operating at a low capacity level, and another unit fails, the low level unit cannot instantaneously increase to a high value of output power, due to ramp-up and ramp-down limits.
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.45]{figures/energycharg.png}
\caption{Energy charged by electric vehicles for each case}
\label{energycharg-fig}
\end{figure*}
Figure \ref{energycharg-fig} represents the energy charged ($e_{vnt}^{C}$) by electric vehicles before contingencies. It is noteworthy that when contingencies are not considered, the period in which electric vehicles are most charged is between $t=9$ and $t=16$, as it is a period with high availability of generation through renewable sources. When frequency reserve is considered, the system becomes more flexible, allowing electric vehicles to charge in other periods beyond the period in which renewable sources are available. Thus, these vehicles will provide frequency support in eventual failures of generating units.
Related to the discharged energy by PEVs, only around the instant $t=22$ do the vehicle's discharge in all cases. Since it is a period during the night when the intermittent sources (Units $4$ and $5$) are no longer available for generation and still have relatively high demand, then PEVs can use stored energy to help meet demand.
Figure \ref{pfrbypevsc3-fig} shows the PFR provided by electric vehicles ($p_{vntk}^{V,PR}$). The contingency $k=1$ represents the failure of the Unit $1$, the contingency $k=2$ the Unit $2$ failure and so on.
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.35]{figures/prfbypevs-case3.png}
\caption{PFR provided by electric vehicles in case 3}
\label{pfrbypevsc3-fig}
\end{figure*}
It is important to note that for the first three contingencies, the vehicles provide support in frequency in the periods referring to the night period when there is no availability of intermittent sources. At $k=4$, the PEVs did not provide PFR as the generating units could provide what was needed. In the $k=5$ contingency, the vehicles provided PFR during the daytime period, as this is the contingency that represents the unit's failure with the highest generation capacity (Unit 5). In addition, it is an intermittent unit that operates during the day.
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.35]{figures/prfbygenunits-case1.png}
\caption{PFR provided by generating units in case 1}
\label{prfbygenunitsc1-fig}
\end{figure*}
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.35]{figures/prfbygenunits-case2.png}
\caption{PFR provided by generating units in case 2}
\label{prfbygenunitsc2-fig}
\end{figure*}
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.35]{figures/prfbygenunits-case3.png}
\caption{PFR provided by generating units in case 3}
\label{prfbygenunitsc3-fig}
\end{figure*}
The Figures \ref{prfbygenunitsc1-fig}, \ref{prfbygenunitsc2-fig} and \ref{prfbygenunitsc3-fig}, show the PFR provided by each generating unit ($p_{gtk}^{PR}$) in each of the 3 cases studied and after each of the contingencies. The negative values in Figures \ref{prfbygenunitsc1-fig}, \ref{prfbygenunitsc2-fig} and \ref{prfbygenunitsc3-fig} represent the power that the missing generator was providing before the contingency, and the positive values represent the PFR provided by the other generating units. Thus, it is ideal that there is symmetry concerning the positive and negative values, which would mean that the system can supply the power of the missing generator. Thus, note that the case with the best performance was case 3, where frequency regulation is considered, and electric vehicles participate in this service.
The frequency deviation ($\Delta f_{tk}$) is shown in Figure \ref{energystinterm-fig}. The post-contingency states are shown on the horizontal axis, referring to each period for each contingency considered. This axis was reordered so that the frequency deviations are in ascending order, in order to obtain a better view of each case. Note that a greater frequency deviation was allowed to ensure that the unserved demand is as close to zero as possible. It is known that high-frequency variations can cause power quality problems, but as previously mentioned, a maximum frequency deviation of 1 Hz was considered. Note that Case 1 has a large number of post-contingency states with negligible frequency deviations. These post-contingency states correspond to contingencies of units with very small production.
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.45]{figures/freq-deviation.png}
\caption{Frequency deviation in each period for each contingency considered for each analyzed case}
\label{energystinterm-fig}
\end{figure*}
Finally, post-contingency unserved demand ($p_{dtk}^{UD,PR}$) is shown in Figure \ref{unserveddem-fig}. Note there is not unserved demand in Case 3. In Case 2, there is unserved demand only in period $t=20$ after a contingency of one of the conventional generating units ($k=1$, $k=2$ and $k=3$). Note that, in this period, the intermittent units are no longer available for generation and there is a need to charge Group 2 of electric vehicles. Finally, observe that Case 1 has a significantly higher unserved demand than the rest of cases.
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.55]{figures/unservdem-area.png}
\caption{Unserved demand post-contingency}
\label{unserveddem-fig}
\end{figure*}
A presentation of the day-ahead schedule costs for each case studied is shown in Tabele \ref{tab-costs}, it is possible to observe that the largest portion of the costs in case 1 is due to unserved demand after the contingencies, and in the other cases, this value decreases considerably.
\begin{table*}[!h]
\centering
\caption{Costs for each case studied (R\$)}
\hfill\par
\begin{tabular}{ccccccc}
\hline & $c_{t}^{P}$ & $c_{t}^{UD,PR}$ & $c_{t}^{\Delta f}$ & $cc_{t}^{V}$ & $cp_{t}^{V}$ & Total \\ \hline
Case 1 & 14223.26& 165684.05& 5.41& 0.00& 0.00& 179.91 M\\
Case 2 & 14591.18& 8669.48& 8.25& 0.00& 0.00& 23.27 M\\
Case 3 & 14526.60& 0.00& 8.28& 33.55& 432.46& 15.00 M \\ \hline
\end{tabular}
\label{tab-costs}
\end{table*}
\section{Conclusion}
This work focused on analyzing the participation of PEVs in the day-ahead energy generating and reserve capacity scheduling, specifically, the participation of PEVs in the PFR of systems with high penetration of renewable energy sources. The proposed approach consists in a mathematical model that represents the day-ahead scheduling of a power system, which is formulated as a unit commitment problem.
The model was applied to a case study based on electrical system of the Marco Zero Campus of the Federal University of Amapá. The results obtained make it possible to verify the importance of planning the system's operation considering the possibility of generating units failure concerning operating costs and system stability, thus making electrical power systems more flexible for the insertion of renewable sources. In addition, it was possible to estimate quantitatively the impact of the participation of PEVs in reducing the commitment of generating units that operate with a low capacity factor in the event of an unexpected generating units failure. It was possible to verify that there was a great improvement in the system operation when electric vehicles participated in different services in the system. This work also shows that in the scenario where there is a penalty for unserved demand at the Federal University of Amapá and where electric vehicles are available to provide ancillary services, the system's operating cost reduces considerably when these PEVs participate in frequency support.
It is important to note that this work is considered a microgrid. In a large system, the contribution of electric vehicles in the system is expected to be more relevant.
\bibliographystyle{IEEEtran}
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It's Not About the Dinosaurs. It's About Stockwell Day's Insanity
During the 2000 Canadian Election campaign, when Stockwell Day, as leader of the newly formed Alliance Party, was running as Prime Minister; the media and political opponents had a field day.
Here was a man right out of the dark ages, with views somewhere between those of Ross Perrot and Attilla the Hun, but without the brains or brass of either.
Lawyer Virginia May probably described him best. "He is a mirage: the closer you get to him, the more you realize there is nothing there."
In the review for the book, REQUIEM FOR A LIGHTWEIGHT Stockwell Day and Image Politics, by Trevor Harrison: "For all of the 'hype' about Stockwell Day's accomplishments by the various media, in the spring of 2000 he was an untested politician of modest accomplishments, a meager national profile, and enough controversial baggage to fill a Ryder truck. Yet, a few months later, he was leader of Canada's newest political party, a rising star in the new right firmament...."
So were the assaults of Mr. Day justifiable? For the most part yes, and in fact we are seeing the evidence of what narrow minded opinions can do. From the attacks on the scientific community from this so-called Conservative Party, to their international embarrassments and Stalin like handling of Immigration, they have completely transformed this country from a source of pride to something barely recognizable.
Canadians should have paid attention.
Stephen Harper may have replaced Stockwell Day as the leader of the Alliance Party, but it's the same old ideology, shared by a minority of Canadians, but being worn by us all. A person's religious beliefs should be private, but when they want the top job in the land, we have to be sure that those beliefs don't cross over into the governing of a population with mixed faiths.
University professor Jeffrey O. Shallit, discussed the matter back in 2000, and gave his views.
"Day's religious beliefs were fair game"
by Jeffrey Shallit
The recent flap over Stockwell Day's fundamentalist beliefs raises a serious question: To what extent should a political candidate's religious beliefs be subject to public scrutiny?
It's not a question with an easy answer. As the current religious conflicts in Northern Ireland and the former Yugoslavia remind us, nothing could be more damaging to democracy than the factionalism induced by religion. Would we want to live in a Canada where Christians vote only for the Christian party, Jews only for the Jewish party, and Muslims only for the Islamic Party?
Obviously not. But then again, the way a candidate approaches religion tells us a lot about their personality. For example, American TV evangelist and political candidate Pat Robertson has stated that only Christians and Jews should be allowed to hold public office. Although Robertson claims Biblical support for his view, it is clearly incompatible with the guarantees in the American Bill of Rights. On the other hand, former US President Jimmy Carter lives his beliefs by working actively for charitable organizations such as Habitat for Humanity, building houses for poor people. Both men claim to be Christians, but evidently their opinions differ on what that means.
Given a choice between Robertson and Carter, should voters simply ignore what they know about their religious beliefs?
I think the answer is that a candidate's religious beliefs should be off limits to the extent that they do not seriously impact public policy. Believing that God specially created human beings or a refusal to work on Sundays should not be cause for an inquisition by the press. But when religious beliefs cause a candidate to call for changes in the abortion law, for example, the public is entitled to know the basis for those beliefs.
Stockwell Day apparently adheres to a fundamentalist interpretation of the Bible, where the earth is only 6,000 years old and human beings co-existed with dinosaurs. These beliefs have been roundly criticized in the Canadian media.
Some fundamentalist Christians have been quick to defend Day, labelling any inquiry into a candidate's religion as bigotry, and claiming Day is being singled out because he is Christian. But is this really true?
The scientific evidence for a 4.6-billion-year old earth, the extinction of dinosaurs about 65 million years ago, and the evolution of modern humans about 2 million years ago couldn't be more firm. With virtually unanimous agreement, scientists point to multiple independent lines of evidence that support these conclusions.
Day certainly has the right to believe the earth is young. This alone should not disqualify him for the position of Prime Minister. After all, he might hold such a discredited position because he hasn't studied the evidence. Although I personally might wonder at such a lack of intellectual curiosity, most people don't consider geology and paleontology as litmus tests for public office.
But when Day advocates the teaching of the pseudoscience known as creationism in public schools, the situation changes. Now it is no longer just a case of a privately-held religious belief, but rather forcing children to be indoctrinated in a narrow sectarian interpretation of one particular religious text.
The Charter of Rights and Freedoms guarantees every Canadian the right to freedom of religion. But this important guarantee doesn't mean that a candidate's positions are immune from criticism merely because they are religiously based.
The defence of Stockwell Day by fundamentalists Christians would be far more credible if accompanied by an expressed willingness to support candidates who are atheists, Muslims, and Wiccans. Somehow I don't think this is likely to happen.
Posted by Emily Dee at 12:25 PM
Labels: Alliance Party, Conservatives, Dinosaurs, Extremism, Stockwell Day
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\section{Introduction} \label{sec:introduction}
Two-dimensional (2D) crystals are currently of great interest for both applied and fundamental research. The most prominent example of this kind materials is graphene. However, the class of 2D materials is very large and is continuously growing. It contains single-layered materials such as those belonging to the group IV of the periodic table (silicene, germanene and stanene) and to the group V (arsenene, antimonene, bismuthene), as well as layered 2D materials like Transition Metal Dichalcogenides and MXenes.
Among them silicene seems to be a promising material due to its compatibility with existing silicon-based electronic devices. Because of sp3 hybridization, silicene is the 2D buckled hexagonal lattice of silicon atoms, and is considered as a one of alternative materials to graphene \cite{2,3,4,5,6}. Research activities on silicene significantly increased after its successful synthesis under UHV conditions on several substrates, like for instance on Ag(111), ZrB2(0001), MoS2(0001), and Ir(111) \cite{7,8,9,10,11}.
Silicene displays several interesting characteristics, which have been revealed by recent experimental and theoretical investigations. These include for instance: (i) a remarkable spin–orbit coupling parameter, that leads to the energy gap of 1.5 meV \cite{16,17,18} at the Dirac point, which is much larger than that in graphene (24 $\mu$eV) \cite{3,19,21}; (ii) electrically tunable bandgap; (iii) the phase transition from a spin Hall topological insulator to a band insulator~\cite{31}; (iv) the strain-induced tunable bandgap \cite{22,23,24}, and (v) promising electric and thermoelectric characteristics ~\cite{33,34,36,85,86,87,88}. The energy gap in silicene makes it promising for applications, however the realization of stabile monolayer of silicene is still problematic \cite{1}. For example, to achieve high on-to-off current ratios and a perfect switching capability, the material exploited for Field-Effect-Transistors (FETs) is usually required to have a fairly large bandgap \cite{11}, significantly larger than that mentioned above for silicene.
Electronic and magnetic properties of silicene can be tuned by impurities, i.e., magnetic atoms built into the monolayer structure.
Importantly, recent achievements of nanotechnology enable precise arrangements of impurities (including regular lattices), so by doping one can modify electronic properties in a controllable way.
The single-side adsorption of alkali metal atoms on silicene has been reported to give rise to a bandgap of approximately 0.5 eV \cite{25,26}, that is much larger than that due to intrinsic spin-orbit interaction. If both sides of the single-layer silicene have been saturated with hydrogen atoms, the formed composite system has been shown to be a kind of a nonmagnetic semiconductor \cite{27}. Based on the previous studies, the ferromagnetic characteristics of silicene can be attributed to the single-side hydrogenation \cite{28}.
\begin{figure}[t]
\centerline{\includegraphics[width=0.8\columnwidth]{Fig1.pdf}}
\caption{(a) Hexagonal crystal structure of silicene, with $d$ being the atom spacing and $\Delta$ denoting the buckling parameter. The unit cell is indicated by he red dashed lines, and the basis consists of two Si atoms, labelled with A and B. (b-d) Types of impurity substitutions: monomer (b), horizontal dimer (c), and vertical dimmer (d). }
\label{fig:fig1}
\end{figure}
Strain engineering enables external control of electronic characteristics of semiconductor heterostructures and nanomaterials. This technique is widely applied to nano-electro-mechanical and nano-opto-mechanical systems, as well as to MOSFETs \cite{29,30}.
In turn, the in-plane strain in silicene leads to a modification of the electronic structure and its transport characteristics \cite{66,32}.
An important question is whether a significant gap in the spectrum of silicene can be induced by the strain, especially at low values of strain. If this is the case, tunable silicene devices would be of great practical importance. According to earlier density functional theory (DFT) calculations, a gap in the spectrum of silicene can be open under arbitrary uniaxial strain. Its magnitude varies non-monotonically with the strain. These findings were supported by other ab-initio (without empirical parameters and from first principles) calculation \cite{68}. For similar strain magnitudes, however, the two gap calculations were not in agreement. Recently, Pereira et al. \cite{39} have questioned the accuracy of those conclusions by applying the Tight Binding (TB) approach. According to their findings, a spectral gap can only be achieved for +20\% uniaxial deformations. Furthermore, this effect highly depends on the deformation route regarding the rudimentary lattice. The aforementioned findings are in accordance with the studies of Hasegawa et al. \cite{40}, indicating that there is a robust gapless Dirac spectrum with regard to arbitrary and not extremely large changes in the nearest neighbour hopping parameters. Moreover, employing the TB model, Wunsch et al. \cite{41} found that the semi-metallic phase appears for hopping parameter expansion.
The results of ab-initio calculations \cite{42} are in agreement with the gapless situation presented in \cite{43}.
The inconsistency between various ab-initio calculations ware partially related to the fact that due to strain the Dirac points shift from the high symmetry points of the Brillouin zone. This resulted in arriving at the wrong conclusion that a bandgap is achievable for any strain.
In addition, Faccio et al. also performed DFT research and calculated the impact of $ \approx 2\% $ strain in silicene nanoribbon [43].
Keeping in mind that mechanical strain can substantially change the physical properties of silicene, we have performed detailed ab-initio calculations of the magnetic and electronic characteristics of silicene with inserted dimers and monomers of Cr atoms.
Then, we have analysed the influence of a biaxial strain for up to $ \approx \pm 8\% $ deformations.
We have shown that magnetic and electronic characteristics of silicene with substitutional impurity atoms can be easily controlled by various kinds of strains (i.e., substrate-induced strains or external mechanical forces).
It is worth to note, that the gap due to strain and doping is significantly larger than the gap induced by the intrinsic spin-orbit interaction in undoped and unstrained silicene.
\section{Methodology and strain-dependent structural properties} \label{sec:methodology}
To study electronic and magnetic properties of silicene under mechanical strain we have used the DFT method \cite{44} within the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) \cite{45} form of the exchange-correlation functional, as implemented in the
QuantumATK code package ver.~S-2021.06~\cite{45-1,45-2}. The PseudoDojo collection of optimized norm-conserving Vanderbilt (ONCV) pseudopotentials and ultra-basis set have been used for the optimization of structures and for further calculations \cite{46}. For the Brillouin zone integration we have taken 8$\times$8$\times$1 Monkhorst-Pack k-points in self-consisting calculation (SCF), and the mesh cut-off of energy has been set to 450 Ry. Structures ware relaxed until the forces on each atom were less than 0.05 eV/$\AA$ and relative convergence for the Self-Consistent Field (SCF) energy is reached until $10^{-5}$ eV/$\AA$. We have used 15 $\AA$ vacuum region to prevent interaction of two neighboring layers along the $c$-axis \cite{47,48,49}.
The atomic structure of silicene should be characterized before modelling the electronic structure \cite{50}. The first structure optimization process for a single layer of Si was reported by Takeda and Shiraishi \cite{51}. By analogy to graphite, they defined a hexagonal lattice for Si atoms (with a periodicity perpendicular to the plane with a large vacuum layer of minimally 10 $\AA$), and then varied the in-plane lattice constant and positions of the basis atoms (Figure 1(a)) inside the unit cell, while keeping constant the imposed D3d symmetry \cite{52}. According to their results, the buckled structure has a lower total energy, with a local minimum for $a = 3.855$ $\AA$ and a deformation angle of 9.9$^{\circ}$, when compared to the energy of a flat structure \cite{50}.
In the present study, the 4$\times$4 monolayer of silicene was characterized systematically (thus the unit cells were repeated up to four times in the $x$ and $y$ directions) using DFT \cite{53}.
The Si atom bounded to three nearest neighbour surrounding Si atoms with the Si-Si bond length of $2.28\,\AA$ and lattice parameter $a$ of $3.84\,\AA$ was assumed prior to structural relaxation.
The optimized lattice parameter, $a = 3.86\,\AA$, correlates quite well with other data, even though the standard GGA functional method overestimates it. The buckling parameter $\Delta$ is $0.46\,\AA$, which is also consistent with other studies \cite{55,59,62,63,64}.
In this paper we analyse three different types of substitution in silicene monolayer by Chromium adatoms as presented in Figure~\ref{fig:fig1}. These are: (i) monomer substitution, where one Si atom in the supercell is substituted by Cr atom ($3.2\% $ substitution), as presented in Figure~\ref{fig:fig1} (b); (ii) horizontal dimer (HDimer) substitution with two neighbouring Si atoms substituted by Cr atoms ($6.25\%$ substitution), see Figure~\ref{fig:fig1} (c); (iii) vertical dimer (VDimer) substitution, where one Si atom is replaced by two Cr atoms ($3.2\%$ substitution), as shown in Figure~\ref{fig:fig1} (d). These three structures were modeled within the optimized 4$\times$4 supercell geometries, and we analysed behaviour of the electronic and magnetic properties with the strain.
\begin{figure}[t]
\centerline{\includegraphics[width=0.65\columnwidth]{Fig2.pdf}}
\caption{Variation of the Cr-Si and Cr-Cr bond lengths, $d_{X-X}$ (Fig.2.(a)), as well as of the buckling parameter (b) under the biaxial tensile and compressive strain for the monomer, horizontal dimer, and vertical dimer substitutions.}
\label{fig:fig2}
\end{figure}
The strain is defined as a deformation resulting from external loads or forces, that may be calculated by the following equation:
${\epsilon={\Delta a} /a_0}$ with ${\Delta a = a - a_{0}}$, where $a$ is the lattice parameter of the strained silicene and ${a_0=3.86\,\AA}$ is the lattice constant of the unstrained silicene \cite{66,68,67}. The model of strained unit cell for the tensile and compression strains is achieved by varying the lattice constant along the lattice vectors by the following substitution $a \rightarrow \epsilon a$.
Accordingly, the uniaxial tensile strain or compression ($\Delta a \gtrless 0$ respectively) is realized by modification of the lattice constant in the
$x$ direction, that is by modification of the lattice vector $\mathbf{a}_{1}$ (mechanical force applied along the lattice vector $\mathbf{a}_{1}$), whereas the biaxial tensile or compressive strain is described by modification of both $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ lattice vectors (forces are oriented along both lattice vectors).
The presence of biaxial strain affects the buckling parameter, $\Delta$, as well as the Cr-Si and Cr-Cr bonds in the silicene monolayer with monomer and dimer substitutions, respectively.
Figure \ref{fig:fig2} presents the basic parameters of the relaxed structures as a function of strain and also for all the substitutions under consideration.
For the monomer substitution (see Figures \ref{fig:fig2}(a)), the bond length between Cr and Si atoms, $d_{\mathrm{Cr-Si}}$, slightly increases with the strain. However, there is no clear universal behavior of the buckling parameter with strain, though one can see that this parameter reaches a maximum for a specific strain equal $2\%$, and then decreases with increasing magnitude of either tensile or compressive strain.
For the silicene with HDimer substitution, the tensile strain reduces monotonically the bond length $d_{\mathrm{Cr-Cr}}$ and the buckling parameter. However, for the compressive strain one observes increase of the bond length for Cr-Cr dimers. In turn, the buckling parameter for Cr-Cr HDimers varies nonmonotonously with the magnitude of compressive strain, i.e., it reaches a minimum at some magnitude of the strain, see Figure \ref{fig:fig2}(b).
\begin{figure*}[t]
\centerline{\includegraphics[width=1.0\columnwidth]{Fig3.pdf}}
\caption{The electronic energy spectrum of silicene under selected values of biaxial strain (from $-4\%$ to $+4\%$) in the presence of spin-orbit interaction, and for Cr doping with the three types of substitution: (a-e) monomer, (f-j) HDimer, and (k-o) VDimer. }
\label{fig:fig3}
\end{figure*}
In turn, for the VDimer substitution, the strain only very slightly affects the bound lengths reducing it when the strain takes absolute values larger than 2$\%$.
The corresponding buckling parameter decreases with increasing tensile strain and increases with increasing magnitude of the compressive strain up to $4\%$, where it reaches a maximal value, and then decreases with a further increase of the magnitude of compressive strain.
At this point, one should note that high strain values are experimentally difficult to obtain. \cite{68,69,70,71,72,73}.
The recent reports about graphene-like monoatomic crystals indicate that the strain around $4\%$ can be relatively easily obtained.
Accordingly, in this paper we have restricted our considerations to the strain ranging from $-6\%$ to $8\%$.
\section{Strain-dependent electronic and magnetic properties} \label{sec:transport}
It is well known that strain has a significant impact on electronic and magnetic properties of 2D crystals \cite{74,76,77,89,90}.
One of the consequences of the strain in 2D crystal is the electronic bandgap engineering, i.e., strain-induced bandgap opening \cite{33} or a direct–indirect–direct bandgap transition in green phosphorene \cite{79,80}.
Figure \ref{fig:fig3} presents the band structures of strained and unstrained silicene doped with Cr for the three different types of substitution (Monomer, HDimer and VDimer), as discussed in Section 2. The band structure has been calculated along the high-symmetry points of the Brillouin zone, i.e., along the $\Gamma$-M–K–$\Gamma$ path.
In turn, Table 1 collects information about the band gap in silicene under strain and with different types of Cr substitution. The electronic band structure of undoped silicene monolayer under strain is presented in \ref{AA}. Here one needs to remind, that strain in undoped silicene does not open a gap. A small gap appears only due to spin-orbit interaction. When neglecting this interaction, the gap in undoped silicene remains equal to zero (at the Dirac points), see also Figure \ref{fig:fig5} in Appendix A.
Apart from this, due to hybridization of the 3d-impurity states and those of pure silicene, the band structure becomes remarkably modified by doping. Each state of the doped system includes in general contributions due to 3d-transition metals as well as due to silicon atoms. To show this explicitly, we have presented the so-called {\it fat-bands} structure, where the two contributions are indicated explicitly with different colors. In \ref{AB} we present {\it fat-bands} calculated for silicene monolayer doped with Cr-atoms (i.e., the band structure projected over orbitals of Silicone and Chromium atoms). The corresponding results are shown in Figure \ref{fig:fig6}, and from this figure one can estimate whether a particular band contributes to conductivity or not. If it is due to impurities only and is dispersionless around the Fermi level, it does not contribute to conductivity. If however it is dispersive around the Fermi level, then even though the silicon contribution is small, it contributes to conductivity. These features have been taken into account when determining the band gaps. In \ref{AB} we show the {\it fat-bands} for silicene with Cr-Monomers, Cr-HDimers and Cr-VDimers.
The silicene with Cr-monomers is either metallic or semimetallic, with the Fermi energy crossing the valence bands. In the presence of compressive strain the energy gap becomes opened in the spectrum above the Fermi energy, while the Fermi level is still inside the valence bands. In the presence of tensile strain the system moves from semimetallic to metallic one.
In turn, for silicene with Cr-HDimer substitution, the electronic structure is only slightly affected by the strain.
The most promising effect of strain on the electronic structure can be observed for silicene monolayer with Cr-VDimer substitution. Without strain the system is a semiconductor with the energy gap equal 0.13 eV. Applying compressive strain one can close the energy gap and move the Fermi level to the valence bands. The tensile strain, in turn, leads to reduction of the energy gap for the 4$\%$ strain, and leads to its complete closure for strain $\ge 6$ $\%$.
\begin{figure*}[t]
\centerline{\includegraphics[width=1\columnwidth]{Fig4.pdf}}
\caption{Magnetic moment behaviour as a function of the applied biaxial strain for the Cr substitution obtained based on calculations with and without spin-orbit coupling (SOC). The green curves (no SOC) correspond to the collinear calculations discussed in \ref{AC}.}
\label{fig:fig4}
\end{figure*}
\begin{table}[]
\caption{Band Gap (~eV) changing for Cr substitution during the biaxial strain and applied spin-orbit interaction}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Strain (\%) & -6\% & -4\% & -2\% & 0\% & 2\% & 4\% & 6\% & 8\% \\ \hline
Monomer-Cr & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
HDimer-Cr & 0.1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
VDimer-Cr & 0 & 0 & 0.12 & 0.13 & 0 & 0.05 & 0 & 0 \\ \hline
\end{tabular}
\end{table}
Our calculations also reveal the impact of biaxial strain on the magnetic characteristics of the silicene monolayer doped with the specified above Monomers, HDimers and VDimers of Cr atoms.
The corresponding results are presented in
Figure \ref{fig:fig4}, where
spin polarization of silicene monolayer doped with Cr atoms is shown as a function of strain (the three types of substitution are presented).
Situations in the presence of spin-orbit interaction and that in the absence of spin-orbit coupling are shown there.
In the latter case, the calculation procedure is described in \ref{AC}.
For silicene with the Cr-monomers, the magnetic moment varies monotonously with the strain when the spin-orbit coupling is included, while in its absence, the magnetic moment increases with the magnitude of strain, both tensile and compressive. Moreover, the difference between these two situations is relatively large.
In turn, for the case of Cr-HDimer, the magnetic moment decreases with the compressive and tensile strain, and the difference between the cases with and without spin-orbit interaction is small. For the silicene monolayer doped with Cr-VDimers, the magnetic moment only weakly depends on the strain for strain larger than $-4\%$, and the difference between the case with and without spin-orbit interaction is small except the strain below $-4\%$, where this difference is relatively large.
It is worth noting that the largest magnetic moment for the unstrained system is for silicene monolayer with Cr-HDimers, where the changes in the magnetic moment due to strain are also most pronounced.
\section{Conclusions} \label{sec:summary}
In this work we presented detailed study of electronic and magnetic properties of silicene doped with Cr atoms in one of the three doping schemes, i.e., monomer, HDimer, and VDimer substitutions. Numerical results based on the DFT calculations clearly show that the way of substitution may substantially change the structural, electronic and magnetic behaviour of the silicene under strain. The interplay of doping and strain may be used to engineer band gap, and thus also character of transport properties from metallic to half-metallic or semiconducting.
Such a strain-induced engineering of transport properties is important from the practical point of view as it may be used in various spintronics and/or logic devices. It is expected, that controlling current and magnetic state with a strain opens new route for nanoelectronics of future generation.
\\
\section{Acknowledgments}
We thank Prof. J. Barnaś for valuable discussions and reading this manuscript.
This work has been supported by the Norwegian Financial Mechanism 2014-
2021 under the Polish-Norwegian Research Project NCN GRIEG "2Dtronics"
no. 2019/34/H/ST3/00515.
\begin{figure*}[h]
\centerline{\includegraphics[width=0.7\columnwidth]{Fig5.pdf}}
\caption{The electronic band structure of undoped silicene monolayer under biaxial strain from $-8\%$ to $+8\%$ and in the absence of spin-orbit interaction.}
\label{fig:fig5}
\end{figure*}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,384
|
{"url":"http:\/\/amrita.olabs.edu.in\/?sub=1&brch=6&sim=150&cnt=1","text":"Metre bridge- Law of combination of resistors\n\n# Our Objective:\n\nTo verify the laws of resistances in series and parallel.\n\n# The Theory:\n\n## Metre Bridge\n\nThe metre bridge, consists of a one metre long wire of uniform cross sectional area, fixed on a wooden block. A scale is attached to the block. Two gaps are formed on it by using thick metal strips in order to make the Wheat stone\u2019s bridge. The terminal B between the gaps is used to connect galvanometer and jockey.\n\nThe metre bridge is operates under Wheatstone\u2019s principle. Here, four resistors P, Q, R, and S are connected to form the network ABCD.\n\nIn the balancing condition, there is no deflection on the galvanometer. Then,\n\n$\\frac{P}{Q} =\\frac{R}{S}$\nA resistance wire is introduced in gap G1 and the resistance box is in gap G2. One end of the galvanometer is connected to terminal D and its other end is connected to a jockey. As the jockey slides over the wire AC, it shows zero deflection at the balancing point (null point).\n\nIf the length AB is l, then the length BC is ( 100-l ).\n\nThen, according to Wheatstone\u2019s principle;\n\n$\\frac{X}{R} =\\frac{l}{(100-l)}$\nNow, the unknown resistance can be calculated as,\n\n## $X =\\frac{Rl}{(100-l)}$ Resistors in Series\n\nWhen two or more resistors are connected such a way that one end of one resistor is connected to the starting end of the other, then the circuit is called a Series Circuit.\n\nWhen the two resistors X1 and X2 are connected in series in a circuit, the current I passing through each resistor is same.\n\nUsing Ohm\u2019s Law, the potential difference V1 across X1 is:\n\n$V_{1}=IX_{1}$\n\n$V_{2}=IX_{2}$\nLet Xs be the effective resistance of the two resistors in series, and V be the potential difference across the ends.\n\n$V=V_{1}+V_{2}$\n\n$V=IX_{s}$\n\n$IX_{s}=IX_{1}+IX_{2}$\n\n$IX_{s}=I(X_{1}+X_{2})$\n\n$Hence, \\;X_{s}=X_{1}+X_{2}$\n\nThus, when a number of resistors are connected in series, the effective resistance is equal to the sum of the individual resistances. This is called the law of combination of resistances in series.\n\n$i.e, \\;X_{s}=X_{1}+X_{2}+X_{3}+X_{4}............+X_{n}$\nAdding resistors in series always increases the effective resistance.\n\n## Resistors in Parallel\n\nIf the starting ends of two resistors are joined to a point and the terminal ends of the two are combined and given connection to a source of electricity,those circuits are called Parallel Circuit.\n\nWhen the two resistors X1 and X2 are connected in parallel in a circuit, the potential difference across X1 and X2 are the same.\n\nThen the current passing through the circuit is,\n\n$I_{p}=I_{1}+I_{2}$\n\n$i.e,\\;\\frac{V}{X_{p}} = \\frac{V}X_{1}+\\frac{V}{X_{2}}$\n\n$i.e,\\;\\frac{1}{X_{p}} = \\frac{1}X_{1}+\\frac{1}{X_{2}}$\nIf there are \u2018n\u2019 number of resistors with different resistances connected in parallel, then we have,\n\n$i.e,\\;\\frac{1}{X_{p}} = \\frac{1}X_{1}+\\frac{1}{X_{2}}+\\frac{1}{X_{3}}+\\frac{1}{X_{4}}.................+\\frac{1}{X_{n}}$\nThat is, for a set of parallel resistors, the reciprocal of their equivalent resistance equals the sum of the reciprocals of their individual resistances. Thus, resistance decreases in parallel combination.\n\n# Learning Outcomes:\n\n\u2022 The student learns the following concepts:\n\u2022 Resistance in a circuit.\n\u2022 When two resistors are connected in series, its equivalent resistance increases.\n\u2022 Law of combination of resistors connected in series.\n\u2022 When two resistors are connected in parallel, its equivalent resistance decreases.\n\u2022 Law of combination of resistors connected in parallel.\n\nCite this Simulator:","date":"2022-01-21 14:28:22","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\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 15, \"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\": 0.7316356301307678, \"perplexity\": 760.717737834509}, \"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-2022-05\/segments\/1642320303385.49\/warc\/CC-MAIN-20220121131830-20220121161830-00298.warc.gz\"}"}
| null | null |
Sassacus (* um 1560; † Juni 1637) war ein Sachem der Pequot, einem Indianerstamm im heutigen US-Bundesstaat Connecticut. Er war der Sohn und Nachfolger von Sachem Wopigwooit, der als erster Häuptling des Stammes mit den weißen Kolonisten Kontakt aufnahm und um 1632 von den Holländern bei Hartford getötet wurde.
Nach dem Tod von Sachem Wopigwooit wollten sowohl Sassacus als auch sein Rivale Uncas dessen Nachfolge antreten. Das Herrschaftsgebiet der Pequot erstreckte sich zu dieser Zeit von der Narragansett-Bay im Osten bis zum Hudson River im Westen und umfasste im Süden den größten Teil von Long Island. Die Entscheidung des Stammesrats fiel auf Sassacus, und obwohl Uncas mit Sassacus' Tochter verheiratet war, akzeptierte er diese Entscheidung nicht. Die Pequot spalteten sich daraufhin in zwei Parteien, von denen die eine die holländischen und die andere die englischen Kolonisten favorisierte. In der Folge kam es zu Angriffen auf niederländische oder englische Pelzhändler, wenn sie der falschen Pequotgruppe begegneten. Uncas verweigerte Sassacus die Gefolgschaft und verließ schließlich mit 50 Kriegern und ihren Familien die Pequot-Dörfer. Sie siedelten in einem neuen Dorf am Connecticut River nördlich des heutigen Lyme und nannten sich jetzt Mohegan. Uncas gelang es, seine Gruppe mit der Aufnahme benachbarter kleiner Stämme derart zu vergrößern, dass sie von Sassacus nicht mehr zur Rückkehr gezwungen werden konnte.
1634 kam es zu einem Zwischenfall, als der als Sklavenjäger bekannte Bostoner Kapitän und Händler John Stone von Westlichen Niantic beim Versuch getötet wurde, indianische Frauen und Kinder als Sklaven zu fangen. Sein Tod sorgte für Empörung unter den Kolonisten. Da die Niantic Verbündete der Pequot waren, machte sich Sassacus zu Versöhnungsgesprächen auf den Weg. Die Puritaner ließen sich jedoch nicht durch Pelze und Wampum besänftigen, sondern verlangten die Auslieferung der Schuldigen. Es kam zu keiner Einigung, Sassacus und die Puritaner gingen im Zorn auseinander. Im Sommer 1635 entstand an der Mündung des Connecticut Rivers das neue englische Fort Saybrook und die Niederländer mussten ihren Handelsposten bei Hartford schließen, weil sie keinen Zugang mehr zum Long-Island-Sund hatten. Damit verloren die Pequot ihren holländischen Handelspartner.
Im folgenden Winter ersuchten die Pequot die Mohegan und Narraganset um Beistand für einen bevorstehenden Krieg gegen die Briten; beide Stämme jedoch lehnten nicht nur ab, sondern stellten sich auf die Seite der britischen Kolonisten. Sassacus entschied sich daraufhin, den Krieg allein zu führen. Im April 1637 unternahm er einen Vergeltungsangriff auf Wethersfield und Hartford und tötete 30 Kolonisten. Am 1. Mai 1637 erklärte die Führung der Kolonie von Connecticut den Offensiv-Krieg gegen die Pequot. Trotz der bestehenden Differenzen waren viele Mohegan nicht bereit, gegen ihre Pequot-Verwandten zu kämpfen. Uncas ließ die meisten seiner Anhänger zum Schutz der Dörfer zurück und zog mit nur 70 seiner treuesten Krieger nach Hartford, um die nur 90 Mann umfassende Kolonialtruppe unter John Mason zu verstärken. Man plante, das stark befestigte Pequot-Fort am Mystic River zu zerstören. Die kleine Armee wurde in Boote verladen und fuhr den Connecticut River hinab bis nach Fort Saybrook, wo weitere Truppen aufgenommen wurden. Anschließend führte der Weg die Küste entlang bis in die Nähe von Mystic, einem großen, befestigten Dorf der Pequot am gleichnamigen Fluss.
John Mason ließ am 25. Mai 1637 das Fort der Pequot umstellen und in Brand schießen. Danach drangen seine Truppen in das brennende Dorf ein und richteten ein Blutbad an, das später als Mystic-Massaker bekannt wurde. Die meisten der rund 700 Einwohner versuchten zu fliehen, wurden in die Flammen zurückgetrieben und über 500 von ihnen starben einen qualvollen Tod. Captain Mason, der die Pequot befehlsgemäß ausrotten sollte, verfolgte mit seinem Kommando die geflohenen Indianer, tötete sie oder legte sie in Ketten. Die Nachricht von diesem Massaker erreichte in Windeseile die übrigen Pequot-Dörfer, die daraufhin in kleinen Gruppen nach Westen flohen. Sassacus flüchtete mit einer Gruppe an der Connecticut-Küste entlang nach Westen und versuchte, die Mohawk-Dörfer zu erreichen. Die Briten nahmen mit Hilfe von Uncas und seinen Mohegan-Scouts die Verfolgung auf und entdeckten die gesuchten Pequot in der Nähe des heutigen Fairfield. In einem Sumpf wurden die Pequot umzingelt, lehnten jedoch eine kampflose Aufgabe ab. Daraufhin wurde den Frauen und Kindern erlaubt, den Sumpf zu verlassen. Im anschließenden Gefecht fanden 180 Pequot den Tod oder wurden gefangen. Sassacus floh mit den Überlebenden zu den Mohawk im heutigen Bundesstaat New York. Diese jedoch fürchteten die Vergeltung der Briten, töteten Sassacus im Juni 1637 und schickten seinen Kopf als Loyalitätsbeweis nach Hartford, der Hauptstadt der jungen Kolonie Connecticut. Der Ort in der Nähe von Guilford, in der das geschah, heißt noch heute "Sachem's Head" (dt. Kopf des Sachem).
Der Pequot-Krieg endete mit einer Serie von kleineren Gefechten, in denen die Engländer, Mohegan und Narraganset die meisten Pequot zur Strecke brachten. Schließlich kapitulierten die verbliebenen Pequot und baten um Frieden für ihr geschlagenes Volk.
Einzelnachweise
Indianischer Häuptling
Geboren im 16. Jahrhundert
Gestorben 1637
Mann
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,649
|
Q: How to bypass the Firebase Database refresh in Android While I am currently in other activity if my database changes my app get me into the list view page which is as shown bellow:
I wanna know how to I stop refreshing the data while I am not currently looking at that activity
Operating Code
DatabaseReference myRef = FirebaseDatabase.getInstance().getReference().child("new");
myRef.addValueEventListener(new ValueEventListener() {
@Override
public void onDataChange(DataSnapshot dataSnapshot) {
// This method is called once with the initial value and again
// whenever data at this location is updated.
// Object key = dataSnapshot.getValue();
if (dataSnapshot.hasChild(ch) ) {
Map<String ,Object> key = (Map<String, Object>) dataSnapshot.child(ch).getValue();
sname = (String) key.get("sname");
// Log.v("ABCD ", "Value is: " + key.get("sname"));
if(dataSnapshot.getValue() != null)
{
Intent intent = new Intent("com.example.sunny.new.Selectclass");
startActivity(intent);
finish();
}
} else{
AlertDialog.Builder builder1 = new AlertDialog.Builder(MainActivity.this);
builder1.setMessage("Check your Code");
builder1.setCancelable(true);
builder1.setPositiveButton(
"Ok",
new DialogInterface.OnClickListener() {
public void onClick(DialogInterface dialog, int id) {
dialog.cancel();
}
});
AlertDialog alert11 = builder1.create();
alert11.show();
}
}
@Override
public void onCancelled(DatabaseError error) {
// Failed to read value
Log.v("XYZ", "Failed to read value.", error.toException());
}
});
A: At onPause method of your activity, remove your event listener. And add it at onResume method.
Detail:
Define your DatabaseReference and ValueEventListener as reachable variables because you are going to reference them at onPause and onResume methods:
ValueEventListener myValueEventListner;
DatabaseReference myRef;
// Below may be inside onCreate method:
myRef = FirebaseDatabase.getInstance().getReference().child("new");
myValueEventListner= new ValueEventListener() {
// your event listener logic here
};
Add & remove at onPause and onResume methods:
@Override
public void onResume() {
super.onResume();
myRef.addValueEventListener(myValueEventListner);
}
@Override
public void onPause() {
super.onPause();
myRef.removeEventListener(myValueEventListner);
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 958
|
{"url":"https:\/\/standards.globalspec.com\/std\/1523666\/ds-en-iso-10961","text":"# DS\/EN ISO 10961\n\n## Gas cylinders - Cylinder bundles - Design, manufacture, testing and inspection\n\nactive, Most Current\n Organization: DS Publication Date: 25 April 2012 Status: active Page Count: 34 ICS Code (Pressure vessels): 23.020.30\n##### scope:\n\nThis International standard specifies the requirements for the design, construction, testing and initial inspection of a transportable cylinder bundle. It is applicable to cylinder bundles containing compressed gas, liquefied gas and mixtures thereof. It is also applicable to cylinder bundles for acetylene. It does not apply to packages in which cylinders are manifolded together in a support frame which is designed to be fixed permanently to a road vehicle, to a railway wagon or to the ground as a customer storage vessel. It does not apply to cylinder bundles which are designed for use in extreme environmental or operational conditions when additional and extraordinary requirements are imposed to maintain safety standards, reliability and performance, e.g. offshore cylinder bundles. Specific requirements for acetylene cylinder bundles containing acetylene in a solvent are included in an annex. ISO 10961 does not, however, cover acetylene cylinder bundles with solvent-free acetylene cylinders. ISO 10961 is intended primarily for industrial gases other than liquefied petroleum gases (LPGs), but it may also be used for LPGs.\n\n### Document History\n\nDS\/EN ISO 10961\nApril 25, 2012\nGas cylinders - Cylinder bundles - Design, manufacture, testing and inspection\nThis International standard specifies the requirements for the design, construction, testing and initial inspection of a transportable cylinder bundle. It is applicable to cylinder bundles containing...\nGas cylinders \u2013 Cylinder bundles \u2013 Design, manufacture, testing and inspection (ISO\/DIS 10961:2018)\nThis document specifies the requirements for the design, construction, testing and initial inspection of a transportable cylinder bundle. It is applicable to cylinder bundles containing cylinders or...\nGas cylinders - Cylinder bundles - Design, manufacture, testing and inspection (ISO 10961:2010)\nThis International Standard specifies the requirements for the design, construction, testing and initial inspection of a transportable cylinder bundle. It is applicable to cylinder bundles containing...","date":"2019-02-22 02:01:25","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\": 0.808325469493866, \"perplexity\": 8168.715377524084}, \"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-2019-09\/segments\/1550247512461.73\/warc\/CC-MAIN-20190222013546-20190222035546-00135.warc.gz\"}"}
| null | null |
<?php
namespace eig\APIAuth\Contracts;
/**
* Interface JWTPersistenceInterface
* @package eig\APIAuth\Contracts
*/
interface JWTPersistenceInterface
{
/**
* create
*
* @param array|null $params
*
* @return object
*/
public function create(array $params = null);
/**
* save
*
* @param array $params
*/
public function save(array $params);
/**
* get
*
* @param array $params
*
* @return mixed
*/
public function get(array $params);
/**
* all
* @return mixed
*/
public function all();
/**
* exists
*
* @param array $params
*
* @return boolean
*/
public function exists(array $params);
/**
* id
* @return string
*/
public function id();
/**
* issued
*
* @param null $issued
*
* @return mixed
*/
public function issued($issued = null);
/**
* notBefore
*
* @param null $notBefore
*
* @return mixed
*/
public function notBefore($notBefore = null);
/**
* expiration
*
* @param null $expiration
*
* @return mixed
*/
public function expiration($expiration = null);
/**
* token
*
* @param null $token
*
* @return mixed
*/
public function token ($token = null);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,246
|
Monsieur N. is a 2003 British-French film directed by Antoine de Caunes. It tells the story of the last years of the life of the Emperor Napoléon (played by Philippe Torreton), who was imprisoned by the British on St Helena. Napoléon retained a loyal entourage of officers who helped him plot his escape, and evaded the attentions of Major-General Sir Hudson Lowe (Richard E. Grant), the island's overzealous Governor.
The film suggests that Napoléon could have escaped to Louisiana, where he died, and that the body exhumed and now at Les Invalides is that of Napoléon's officer Cipriani. The film also suggests that Napoléon and his young new English wife, Betsy Balcombe, could have attended the ceremony of "Napoléon's" burial in the Invalides.
Plot
Napoleon is imprisoned on the island of Saint Helena in the South Atlantic Ocean. Here he dreams of how to escape from his captivity in his last "battle".
Reception
The film was well-received. , 71% of the 21 reviews compiled by Rotten Tomatoes are positive, with an average rating of 6.27/10. The website's critics' consensus reads: "Fueled by performances as polished as its visuals, Monsieur N. is a flawed yet largely absorbing look at an imagined chapter of Napoleon's exile."
The film received a positive but guarded review in The New York Times, which praised Philippe Torreton's performance but thought the narrative too complex for an audience not initiated in Napoléon's history.
Cast
References
External links
Official website
2003 films
2003 drama films
2003 multilingual films
2000s British films
2000s English-language films
2000s French films
2000s French-language films
French drama films
Films about Napoleon
Corsican-language films
Films directed by Antoine de Caunes
British drama films
Films with screenplays by René Manzor
Films set on Saint Helena
French multilingual films
British multilingual films
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,481
|
Il singolare del torneo di tennis RBC Bank Women's Challenger 2013, facente parte della categoria ITF Women's Circuit, ha avuto come vincitrice Asia Muhammad che ha battuto in finale Chalena Scholl 6-2, 6-2.
Teste di serie
Samantha Crawford (semifinali)
Adriana Pérez (semifinali)
Petra Rampre (primo turno)
Maria-Fernanda Alvarez-Teran (primo turno)
Allie Will (secondo turno)
Al'ona Sotnikova (quarti di finale)
Mayo Hibi (quarti di finale)
Ashley Weinhold (secondo turno)
Tabellone
Finale
Parte alta
Parte bassa
Collegamenti esterni
RBC Bank Women's Challenger 2013
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,832
|
Shep Landy was awarded a Neighborhood Homes Project-based Voucher contract by the Dallas Housing Authortity, which will allow 4 adults with IDD diagnoses and receiving SSI to live in this home. The residents will pay no more that 30% of their income on rent. Shep will receive a subsidy for the difference between the rent that is paid by the tenants and the fair market rent for the home. Tenants will be selected in the near future. The Neighborhood Homes Project is a collaboration between families, DHA and CPSH to show that neighborhood housing for adults living with IDD is desired in North Texas.
Individuals Living with Disabilities: Could you see yourself living in a house like this? Which bedroom do you like? Which friends would you like to live with?
Parents: Take a look at this home for your family member. Would you consider purchasing a home for adults to rent?
Private and Medicaid Service Providers: Would you consider providing services in this home or one like it?
Come tour this house! Bring a future independent resident. People with IDD are good neighbors and need housing that meets their unique situations, just like everyone else. Shep and CPSH will be available to answer your questions.
Post this on your refridgerator so you don't forget!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,105
|
goog.provide('shaka.test.Dash');
goog.require('goog.asserts');
goog.require('shaka.dash.DashParser');
goog.require('shaka.test.FakeNetworkingEngine');
goog.require('shaka.test.ManifestParser');
goog.require('shaka.test.Util');
goog.require('shaka.util.PlayerConfiguration');
goog.require('shaka.util.StringUtils');
goog.requireType('shaka.media.SegmentReference');
goog.requireType('shaka.util.Error');
/** @summary Utilities for working with the DASH parser. */
shaka.test.Dash = class {
/**
* Constructs and configures a very simple DASH parser.
* @return {!shaka.dash.DashParser}
*/
static makeDashParser() {
const parser = new shaka.dash.DashParser();
const config = shaka.util.PlayerConfiguration.createDefault().manifest;
parser.configure(config);
return parser;
}
/**
* Tests the segment index produced by the DASH manifest parser.
*
* @param {string} manifestText
* @param {!Array.<shaka.media.SegmentReference>} references
* @return {!Promise}
*/
static async testSegmentIndex(manifestText, references) {
const buffer = shaka.util.StringUtils.toUTF8(manifestText);
const dashParser = shaka.test.Dash.makeDashParser();
const networkingEngine = new shaka.test.FakeNetworkingEngine()
.setResponseValue('dummy://foo', buffer);
const playerInterface = {
networkingEngine: networkingEngine,
modifyManifestRequest: (request, manifestInfo) => {},
modifySegmentRequest: (request, segmentInfo) => {},
filter: () => {},
makeTextStreamsForClosedCaptions: (manifest) => {},
onTimelineRegionAdded: fail, // Should not have any EventStream elements.
onEvent: fail,
onError: fail,
isLowLatencyMode: () => false,
isAutoLowLatencyMode: () => false,
enableLowLatencyMode: () => {},
};
const manifest = await dashParser.start('dummy://foo', playerInterface);
const stream = manifest.variants[0].video;
await stream.createSegmentIndex();
shaka.test.ManifestParser.verifySegmentIndex(stream, references);
}
/**
* Tests that the DASH manifest parser fails to parse the given manifest.
*
* @param {string} manifestText
* @param {!shaka.util.Error} expectedError
* @return {!Promise}
*/
static async testFails(manifestText, expectedError) {
const manifestData = shaka.util.StringUtils.toUTF8(manifestText);
const dashParser = shaka.test.Dash.makeDashParser();
const networkingEngine = new shaka.test.FakeNetworkingEngine()
.setResponseValue('dummy://foo', manifestData);
const playerInterface = {
networkingEngine: networkingEngine,
modifyManifestRequest: (request, manifestInfo) => {},
modifySegmentRequest: (request, segmentInfo) => {},
filter: () => {},
makeTextStreamsForClosedCaptions: (manifest) => {},
onTimelineRegionAdded: fail, // Should not have any EventStream elements.
onEvent: fail,
onError: fail,
isLowLatencyMode: () => false,
isAutoLowLatencyMode: () => false,
enableLowLatencyMode: () => {},
};
const p = dashParser.start('dummy://foo', playerInterface);
await expectAsync(p).toBeRejectedWith(
shaka.test.Util.jasmineError(expectedError));
}
/**
* Makes a simple manifest with the given representation contents.
*
* @param {!Array.<string>} lines
* @param {number=} duration
* @param {number=} startTime
* @return {string}
*/
static makeSimpleManifestText(lines, duration, startTime) {
let periodAttr = '';
let mpdAttr = 'type="dynamic" availabilityStartTime="1970-01-01T00:00:00Z"';
if (duration) {
periodAttr = 'duration="PT' + duration + 'S"';
mpdAttr = 'type="static"';
}
if (startTime) {
periodAttr += ' start="PT' + startTime + 'S"';
}
const start = [
'<MPD ' + mpdAttr + '>',
' <Period ' + periodAttr + '>',
' <AdaptationSet mimeType="video/mp4">',
' <Representation bandwidth="500">',
' <BaseURL>http://example.com</BaseURL>',
];
const end = [
' </Representation>',
' </AdaptationSet>',
' </Period>',
'</MPD>',
];
return start.concat(lines, end).join('\n');
}
/**
* @param {shaka.extern.Manifest} manifest
* @return {!Promise.<shaka.media.SegmentReference>}
*/
static async getFirstVideoSegmentReference(manifest) {
const variant = manifest.variants[0];
expect(variant).not.toBe(null);
if (!variant) {
return null;
}
const video = variant.video;
expect(video).not.toBe(null);
if (!video) {
return null;
}
await video.createSegmentIndex();
const position = video.segmentIndex.find(0);
goog.asserts.assert(position != null, 'Position should not be null!');
const reference = video.segmentIndex.get(position);
goog.asserts.assert(reference != null, 'Reference should not be null!');
return reference;
}
/**
* Calls the createSegmentIndex function of the manifest. Because we are
* returning fake data, the parser will fail to parse the segment index; we
* swallow the error and return a promise that will resolve.
*
* @param {shaka.extern.Manifest} manifest
* @return {!Promise}
*/
static async callCreateSegmentIndex(manifest) {
const stream = manifest.variants[0].video;
await expectAsync(stream.createSegmentIndex()).toBeRejected();
}
/**
* Makes a set of tests for SegmentTimeline. This is used to test
* SegmentTimeline within both SegmentList and SegmentTemplate.
*
* @param {string} type The type of manifest being tested; either
* 'SegmentTemplate' or 'SegmentList'.
* @param {string} extraAttrs
* @param {!Array.<string>} extraChildren
*/
static makeTimelineTests(type, extraAttrs, extraChildren) {
describe('SegmentTimeline', () => {
const Dash = shaka.test.Dash;
const ManifestParser = shaka.test.ManifestParser;
const baseUri = 'http://example.com/';
/**
* @param {!Array.<string>} timeline
* @param {string} testAttrs
* @param {number=} dur
* @param {number=} startTime
* @return {string}
*/
function makeManifestText(timeline, testAttrs, dur, startTime) {
const start = '<' + type + ' ' + extraAttrs + ' ' + testAttrs + '>';
const end = '</' + type + '>';
const lines = [].concat(start, extraChildren, timeline, end);
return Dash.makeSimpleManifestText(lines, dur, startTime);
}
// All tests should have 5 segments and have the relative URIs:
// s1.mp4 s2.mp4 s3.mp4 s4.mp4 s5.mp4
it('basic support', async () => {
const timeline = [
'<SegmentTimeline>',
' <S d="12" t="34" />',
' <S d="21" />',
' <S d="44" />',
' <S d="10" />',
' <S d="10" />',
'</SegmentTimeline>',
];
const source = makeManifestText(timeline, '');
const references = [
ManifestParser.makeReference('s1.mp4', 34, 46, baseUri),
ManifestParser.makeReference('s2.mp4', 46, 67, baseUri),
ManifestParser.makeReference('s3.mp4', 67, 111, baseUri),
ManifestParser.makeReference('s4.mp4', 111, 121, baseUri),
ManifestParser.makeReference('s5.mp4', 121, 131, baseUri),
];
await Dash.testSegmentIndex(source, references);
});
it('supports repetitions', async () => {
const timeline = [
'<SegmentTimeline>',
' <S d="12" t="34" />',
' <S d="10" r="2" />',
' <S d="44" />',
'</SegmentTimeline>',
];
const source = makeManifestText(timeline, '');
const references = [
ManifestParser.makeReference('s1.mp4', 34, 46, baseUri),
ManifestParser.makeReference('s2.mp4', 46, 56, baseUri),
ManifestParser.makeReference('s3.mp4', 56, 66, baseUri),
ManifestParser.makeReference('s4.mp4', 66, 76, baseUri),
ManifestParser.makeReference('s5.mp4', 76, 120, baseUri),
];
await Dash.testSegmentIndex(source, references);
});
it('supports negative repetitions', async () => {
const timeline = [
'<SegmentTimeline>',
' <S d="8" t="22" />',
' <S d="10" r="-1" />',
' <S d="12" t="50" />',
' <S d="10" />',
'</SegmentTimeline>',
];
const source = makeManifestText(timeline, '');
const references = [
ManifestParser.makeReference('s1.mp4', 22, 30, baseUri),
ManifestParser.makeReference('s2.mp4', 30, 40, baseUri),
ManifestParser.makeReference('s3.mp4', 40, 50, baseUri),
ManifestParser.makeReference('s4.mp4', 50, 62, baseUri),
ManifestParser.makeReference('s5.mp4', 62, 72, baseUri),
];
await Dash.testSegmentIndex(source, references);
});
it('supports negative repetitions at end', async () => {
const timeline = [
'<SegmentTimeline>',
' <S d="5" t="5" />',
' <S d="10" r="-1" />',
'</SegmentTimeline>',
];
const source = makeManifestText(timeline, '', /* duration= */ 50);
const references = [
ManifestParser.makeReference('s1.mp4', 5, 10, baseUri),
ManifestParser.makeReference('s2.mp4', 10, 20, baseUri),
ManifestParser.makeReference('s3.mp4', 20, 30, baseUri),
ManifestParser.makeReference('s4.mp4', 30, 40, baseUri),
ManifestParser.makeReference('s5.mp4', 40, 50, baseUri),
];
await Dash.testSegmentIndex(source, references);
});
it('gives segment times relative to the presentation', async () => {
const timeline = [
'<SegmentTimeline>',
' <S t="0" d="10" r="-1" />',
'</SegmentTimeline>',
];
const source =
makeManifestText(timeline, '', /* duration= */ 50, /* start= */ 30);
const references = [
ManifestParser.makeReference('s1.mp4', 30, 40, baseUri),
ManifestParser.makeReference('s2.mp4', 40, 50, baseUri),
ManifestParser.makeReference('s3.mp4', 50, 60, baseUri),
ManifestParser.makeReference('s4.mp4', 60, 70, baseUri),
ManifestParser.makeReference('s5.mp4', 70, 80, baseUri),
];
for (const ref of references) {
ref.timestampOffset = 30;
}
await Dash.testSegmentIndex(source, references);
});
it('supports @timescale', async () => {
const timeline = [
'<SegmentTimeline>',
' <S d="4500" t="18000" />',
' <S d="9000" />',
' <S d="31500" />',
' <S d="9000" />',
' <S d="9000" />',
'</SegmentTimeline>',
];
const source = makeManifestText(timeline, 'timescale="9000"');
const references = [
ManifestParser.makeReference('s1.mp4', 2, 2.5, baseUri),
ManifestParser.makeReference('s2.mp4', 2.5, 3.5, baseUri),
ManifestParser.makeReference('s3.mp4', 3.5, 7, baseUri),
ManifestParser.makeReference('s4.mp4', 7, 8, baseUri),
ManifestParser.makeReference('s5.mp4', 8, 9, baseUri),
];
await Dash.testSegmentIndex(source, references);
});
});
}
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,374
|
{"url":"http:\/\/tex.stackexchange.com\/questions\/20654\/length-between-nested-lists-in-beamer","text":"# Length between nested lists in beamer\n\nHow do I adjust the length between nested lists? For example, in the following, I would like to adjust the distance where it states \"distance here\"\n\n\\begin{itemize}\n\\item one\n%% distance here\n\\begin{itemize}\n\\item\n\\end{itemize}\n%% distance here\n\\item two\n\\end{itemize}\n\n\nUPDATE: I forgot to mention I am using beamer, and it appears traditional methods can go wrong in beamer, as frame defines its own definition of itemize.\n\n-\nYou can use \\begin{itemize}\\addtolength{\\itemsep}{-0.5\\baselineskip}. See this link about tweaklist for more options. \u2013\u00a0Peter Grill Jun 14 '11 at 0:05\n@Peter Grill: but changing \\itemsep will affect the vertical spacing between items, and Vinh Nguyen (if I understand the question correctly) needs to increase the vertical spacing before the first and after last items of the inner itemize. \u2013\u00a0Gonzalo Medina Jun 14 '11 at 1:32\nThis would help: Is there a picture showing all the lengths used in lists? \u2013\u00a0Leo Liu Jun 14 '11 at 4:36\n\nYou shouldn't use enumitem with beamer. This will probably break all the beamer settings for lists (e.g. overlays, alerts etc). Use either simply vspace (for a local solution) or the responsible beamer template. And next time show a complete example instead of a code snippet. The class is a crucial information.\n\n\\documentclass{beamer}\n\\begin{document}\n\\begin{frame}\n\n\\begin{itemize}\n\\item one\n%% distance here\n\\begin{itemize}\n\\item\n\\end{itemize}\n%% distance here\n\\item two\n\\end{itemize}\n\n\\end{frame}\n\n\\begin{frame}\n\n\\begin{itemize}\n\\item one\n\\vspace{1cm}\n\\begin{itemize}\n\\item\n\\end{itemize}\n\\vspace{1cm}\n\\item two\n\\end{itemize}\n\n\\end{frame}\n\n\\begin{frame}\n\\setbeamertemplate{itemize\/enumerate subbody begin}{\\vspace{1cm}}\n\\setbeamertemplate{itemize\/enumerate subbody end}{\\vspace{1cm}}\n\\begin{itemize}\n\\item one\n%% distance here\n\\begin{itemize}\n\\item\n\\end{itemize}\n%% distance here\n\\item two\n\\end{itemize}\n\n\\end{frame}\n\n\\end{document}\n\n-\nThanks! I used placed the setbeamertemplate commands in the preamble so that its effects are global in the document. \u2013\u00a0Vinh Nguyen Jun 14 '11 at 17:38\n\nThe vertical space that you want to modify is given by the sum of the lengths \\topsep, and \\parskip; the enumitem package offers you a simple mechanism to control these (and some other attributes) of the list-like environments:\n\n\\documentclass{article}\n\\usepackage{enumitem}\n\n\\begin{document}\n\n\\begin{itemize}\n\\item one\n\\begin{itemize}[topsep=20pt]\n\\item\n\\end{itemize}\n\\item two\n\\end{itemize}\n\n\\end{document}\n\n\nIf you want to suppress all the vertical spacing in a list you can say something like\n\n\\usepackage{enumitem}\n...\n\\begin{itemize}[nolistsep]\n\\item\n\\end{itemize}\n\n\nEDIT: an example with beamer:\n\n\\documentclass{beamer}\n\\usepackage{enumitem}\n\n\\begin{document}\n\n\\begin{frame}\nNormal spacing\n\\begin{itemize}\n\\item one\n\\begin{itemize}\n\\item First subitem\n\\item Second subitem\n\\end{itemize}\n\\item two\n\\end{itemize}\nIncreased spacing\n\\begin{itemize}\n\\item one\n\\begin{itemize}[topsep=20pt]\n\\item First subitem\n\\item Second subitem\n\\end{itemize}\n\\item two\n\\end{itemize}\n\\end{frame}\n\n\\end{document}\n\n\nThe result:\n\nHowever, using enumitem with beamer is not the best choice. See Ulrike's answer for a proper solution with beamer.\n\n-\nI forgot to mention that I'm doing this in Beamer, and enumitem does not work there (I don't see bullets). Also, as seen here, the frame environment has its own definition of itemize. What to do... \u2013\u00a0Vinh Nguyen Jun 14 '11 at 1:30\n@Vinh Nguyen: yes, you forgot crucial information. Can you please add to your question that you are using beamer? \u2013\u00a0Gonzalo Medina Jun 14 '11 at 1:34\n@Vinh Nguyen: in fact, I just tested my example code using beamer instead of article and my code works (see my updated answer in a few minutes). If it is not working for you, please add a minimal working example illustrating your problem. \u2013\u00a0Gonzalo Medina Jun 14 '11 at 1:36\n@Gonzalo Medina Thank you for your help. Your working example is fine. Although the spacing does seem to work, as mentioned in my comments, the bullets disappear (try without enumitem). I would like to preserve the bullets in the lists. Thank you so much. \u2013\u00a0Vinh Nguyen Jun 14 '11 at 4:42\n@Vihn Nguyen: see Ulrike's answer for a proper solution with beamer. \u2013\u00a0Gonzalo Medina Jun 14 '11 at 12:05\n\nUlrike's solution for a global change isn't working for me. Putting these lines\n\n\\setbeamertemplate{itemize\/enumerate subbody begin}{\\vspace{1cm}}\n\\setbeamertemplate{itemize\/enumerate subbody end}{\\vspace{1cm}}\n\n\neither within a frame or in the preamble is doing nothing. (I can provide details about my configuration, file, etc., but not sure what's relevant.) In any event, I have found other (almost complete) solutions, although not as ... beamerly.\n\nSolution 1: Put this in the preamble:\n\n% space between items:\n\\newlength{\\wideitemsep}\n% reconfigure itemize lists:\n\\let\\olditem\\item\n\\renewcommand{\\item}{%\n\\setlength{\\itemsep}{\\wideitemsep}%\n\\olditem}\n\n\nDrawback (or advantage): Although the items are farther apart, lines of text before an after the list are still very close. (I found this idea at http:\/\/blog.nguyenvq.com\/2011\/05\/01\/spacing-between-items-in-itemize-or-enumerate-environments-lists\/.)\n\nSolution 2: Almost the same thing, but this also adds space before the list:\n\n\\let\\olditem\\item\n\\renewcommand{\\item}{%\n\\olditem\\vspace{4pt}}\n\n\nHowever, it does not add space after the list. I'd like to find a global way to do that. I've experimented with some of the other list parameters without success using this sort of method, which makes sense: The only effect you can have by redefining \\item will be local to a particular item, in general.\n\n-\nIf you indent your code with four spaces, it'll be put into a code block. There is also a button that looks like {} in the editor which will do this to selected text when you click on it. \u2013\u00a0qubyte Feb 13 '12 at 5:25\nThanks Mark. I am indeed new here. I actually tried both! It didn't seem to work either way. Don't know why. (Maybe because I used both four spaces and {}?) I'm glad Werner edited it. \u2013\u00a0Mars Feb 14 '12 at 19:18","date":"2015-11-28 22:18:41","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\": 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\": 0.8833356499671936, \"perplexity\": 2054.4112559375612}, \"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-2015-48\/segments\/1448398454160.51\/warc\/CC-MAIN-20151124205414-00139-ip-10-71-132-137.ec2.internal.warc.gz\"}"}
| null | null |
Review and Giveaway: More Than a Touch
More Than a Touch (Snowberry Creek - Book 2) by Alexis Morgan
Publisher: Penguin Group
Imprint: Signet Eclipse
(Received for an honest review from Signet Eclipse)
Alexis Morgan on the WEB: website, twitter, facebook, goodreads
Books in the series:
A Time for Home (2013), More Than a Touch (2014)
Coverart: Click the Image for a larger, clearer view of the covers in this series.
Excerpt from, More Than a Touch, courtesy of the author's website (half way down the page).
Welcome to the small rustic town of Snowberry Creek....
As Nick Jenkins travels from the war zones of Afghanistan to his comrade's hometown, the kinds of wounds he carries with him have little to do with the shrapnel damage to his arm. Burdened with the guilt of failing to save his friend Spence, Nick is nonetheless determined to find a home for the dog that had been Spence's constant companion.
Callie Redding, Spence's childhood best friend, was shocked to learn he left her his old Victorian home. She's even more surprised when one of his war buddies shows up with a dog at his side—and a heavy weight on his shoulders. As a tribute to their friend's life, Nick agrees to help Callie turn her inheritance into a welcoming bed-and-breakfast for the town of Snowberry Creek.
But as they work through their grief together, they also share something far more precious—the belief that love is worth fighting for....
Alexis Morgan's second book in the Snowberry Creek series, More than a Touch, is sweet contemporary romance about a soldier dealing with PTSD and survivor's guilt. Morgan does a fantastic job of showing the difficulties of a member of the military re-entering normal life and the adjustments they have to make. Leif and Zoe make a great couple who have some obstacles standing in their. Both are wounded in their own way and must help each other to heal. Another great book in this quaint series with a great hometown feel.
What I liked:
This was my first experience with this series and Snowberry Creek was a new town for me. I liked how similar this town was to my own small town, where many people know each other and their lives were intertwined. After loosing a hometown hero, Spence, the town is still recovering when this book starts. I liked the fact that Zoe was already a part of the town and how Leif very quickly became at home there. Mitch the hometown football star, and other characters really bring the town to life. A great new place for readers to visit.
Leif was such a great hero. He was military threw and threw. Everything they did had a commanding air and he just has that presence of a someone who has sworn to protect and serve. The fact that he had survivor's guilt even goes to show how much he cared about his fellow soldiers. I thought his struggles with PTSD seemed very realistic and heart felt. He also had a severe injury to his leg and that's how he meets Zoe.
Zoe is assigned to help Leif with his physical therapy. Even though there is a spark between them from the first time they meet Zoe refuses to let it go beyond the professional because she is determined it is not good business to date patients. But as their friendship grows it's apparent that these two were made for each other. I wasn't quite sure they were going to actually make it, until close to the end. I wished Zoe had figured things out a little sooner, but it was sweet and romantic when she did.
What I didn't like:
This one had a slow pace. It took awhile for the couple to get together and the reader doesn't get much time with them as a couple. That was my only qualm. It made the pacing seem a little draggy at times but it was still a very good love story.
This was just a very sweet romantic novel that Morgan shares with readers. It wasn't one of those books that knocks your socks off but it was definitely a feel good novel that many readers will really enjoy. I look forward to reading more about this little small town and the people who live there.
More Than a Touch is available NOW from your favorite bookseller.
Alexis Morgan has a BA in English from the University of Missouri–St. Louis. She and her husband make their home in the beautiful Pacific Northwest. Alexis shares her office with two parakeets, who rock out to her favorite music and keep her company while she's writing.
A lifelong avid reader, Alexis loves spending her days creating worlds filled with sexy warriors and the strong women who love them. She has been nominated for numerous industry awards, including the RITA from the Romance Writers of America, the top award in the romance genre.
The publisher is sponsoring a giveaway for one copy of More Than a Touch by Alexis Morgan.
~ You must be an email subscriber to participate.
~ US addresses only.
~ The deadline to enter this giveaway is Midnight EST, February 14th.
1. Please leave a comment about what you think of military or former military heroes, as heroes in a novel. Good or bad? And why.
Review: Dark Wolf
Dark Wolf (Dark - Book 25) by Christine Feehan
Publication Date: 01/07.2014
Imprint: Berkley Books
(Received for an honest review from Berkley Books)
Christine Feehan on the WEB: website, twitter, facebook, goodreads
Dark Prince (1999), Dark Desire (1999) Dark Gold (2000), Dark Magic (2000), Dark Challenge (2000), Dark Fire (2001), Dark Dream {Short Story} (2001), Dark Dream(2010), Dark Legend (2002), Dark Guardian(2002), Dark Symphony (2003), Dark Descent {Short Story} (2003), Dark Descent(2010),
Dark Melody (2003), Dark Destiny (2004),Dark Hunger {Short Story} (2004), Dark Secret (2004), Dark Demon (2005), Dark Celebration: A Carpathian Reunion (2006)Dark Possession (2007), Dark Curse (2008),Dark Slayer (2009), Dark Peril (2010), Dark Predator (2011), Dark Storm (2012), Dark Lycan (2013), Dark Wolf (2014)
Excerpt from, Dark Wolf, courtesy of the author's website.
WHERE DARK LYCAN ENDED, DARK WOLF BEGINS...
BOOK NUMBER 25 IN CHRISTINE FEEHAN'S ACCLAIMED CARPATHIAN SERIES
#1 New York Times bestselling author Christine Feehan now delivers her most eagerly awaited novel of all—ten years in the making—in the "the erotic, gripping series that has defined an entire genre" (J. R. Ward).
In Dark Lycan, Christine Feehan journeyed into the heart of the Carpathians, and into the souls of two lifemates stirred by the flush of passion and the threat of annihilation. In Dark Wolf, the breathtaking story continues as the bonds of family are imperiled, and the fate of two lovers lies hidden in the seductive shadows between life and death.
Skyler Daratrazanoff always recognized the miracle that was Dimitri Tirunul, a man beyond any dream that had ever engaged her nights. But she was human. Vulnerable. He was Carpathian. Nearly immortal. She was nineteen. He was an ancient. Yet she held half his soul, the light to his darkness. Without her, he would not survive. Caught between the two warring species, Dimitri has spent centuries hunting the undead to keep his people free, and humans safe. He had survived honorably when others had chosen to give up their souls. But now, marked for extermination by the Lycans, Dimitri found himself alone, and fearing for his life. But salvation was coming…
No Lycan would ever suspect someone like Skyler to dare mount a secret rescue operation. A teenage girl. A human of untested abilities. But she had something no one else had. She was predestined for Dimitri—as he was for her. And there was nothing stronger for Skyler than her desire to see her life-dream come true. Whatever the risk.
Christine Feehan brings readers the long awaited story of Dimitri and Skylar in her latest Dark novel, Dark Wolf. This romance has been a while in the making and readers will be excited to see how their happy ever after comes about. Feehan does and excellent job of bringing a woman with a tragic past full circle and developing her character to accept love and affection. Readers who are looking for an adventure packed paranormal romance will love this latest addition to a long standing series.
Finally, long time readers of Christine Feehan's Dark series get the romance of one of their most beloved couples, Skylar and Dimitri, but it doesn't come the way most readers expected. I was really happy that this book was to focus on this couple, one of my favorites who haven't quite made it together yet. But it didn't happen at all like I had envisioned it.
When Dark Wolf begins Dimitri is being held captive by the lycans and tortured because of his mixed heritage. Skylar is trying desperately to find and rescue him. But Dark Wolf is not a book about how Skylar and Dimitri fall in love or anything like that. Instead the couple is already together and all of the build up and the fall has already taken place before the book even begins. Certainly not what I expected. Feehan goes a totally different direction by bringing readers in late to party so to speak. And I'm sure that most long time readers of the series will be impressed.
Skylar is of course a great character no matter what circumstances we find her in. She is courageous and spunky. She's been through so many horrible experiences in her past and now she has finally found love with Dimitri. I liked that part very much. She deserved happiness, but now Dimitri is about to be killed and she has to protect him. The role reversal was quite interesting and I think Feehan did a great job with that aspect of the book. Dimitri is not your typical alpha male Carpathian. He understands Skylar need to rescue him and appreciates what she has to offer in order to help protect him this time.
I loved Dimitri's attitude toward Skylar. He was determined not to die though his torture was nearly unbearable, because he knew she needed him and this time he needed her. He is always respectful of Skylar and what shes been through in everything he does including the way he treats her physically and romantically. He is a hero that readers can get behind and appreciate.
I was a little taken aback by how Skylar responded to Dimitri emotionally and sexually to be honest. Her past is tragic and she has had some pretty horrific sexual experiences due to it, and though Dimitri is patient he still wants to be with Skylar. Feehan has made a great deal about how scarred and traumatized Skylar is, so I was really surprised at how quickly she threw caution to the wind and just went for it. I thought she might have a really hard time with things, but not so. Not sure I quite liked it, but I understood that eventually a person has to try again and in a way that was a good message.
I felt a little short changed by not being able to read how Dimitri and Skylar fell in love, but I think it was a plot to shake up the readers of the series. Just not sure it was a good shake. I thought Sklyar was a little too easily turned from a tragic victim to a woman in love. It didn't exactly ring true to me.
Overall I liked this book, but it was not the best or the worst of this series. Sometimes when a series continues for a long, long time it starts to loose it's luster and readers aren't treated to something new, but just a rehashing of old stories. I think this was Feehan's attempt to thwart that, but it didn't quite work. It might be time to think about wrapping up this long standing series and trying something altogether new. I love reading Christine Feehan's book, but this one didn't quite hold up to her normal standards. Not a bad book, but not a great book either.
Dark Wolf is available NOW from your favorite bookseller.
I'm giving this one 4 out of 5 apples from my book bag.
I live on the beautiful Northern California coast. I have always loved hiking, whale watching, and being outdoors. My camping days are over but I might consider glamping. LOL! I am surrounded by my family, my beloved grandchildren and my pack of dogs.
In addition to her Dark Carpathian novels, #1 New York Times bestselling author Christine Feehan is the author of the Ghost Walkers series, the Leopard series, and the Sea Haven series which include the novels of the Drake Sisters and the Sisters of the Heart.
Review: Hope Flames
Hope Flames (Hope - Book 1) by Jaci Burton
Imprint: Berkley Sensation
(Received for an honest review from Berkley Sensation)
Jaci Burton on the WEB: website, blog, twitter, facebook, goodreads
Excerpt from, Hope Flames, courtesy of the author's website.
When it comes to love, they already know the rules…by heart.
Thirty-two and finally setting up her veterinary practice in the town she once called home, Emma Burnett is on her own and loving it. Independent and driven, she's not letting any man get in the way of her dreams. Not again.
That's fine with Luke McCormack. Divorced and hardly lacking in female company when he needs it, he's devoted to the only faithful companion in his life—his police dog. Still, there's something about Emma he can't shake.
When a series of local break-ins leaves Emma vulnerable, she seeks help from the first man to spark her desire in years. And now they're giving each other something they thought they'd lost forever…hope.
Hope Flames is the first book in Jaci Burton's new contemporary romance series. Most readers are familiar with Burton's highly acclaimed Play-by-Play erotic romance series, but this is a change of direction and will appeal to much broader audience. Burton fans may miss some of the over-the-top sexual exploits they are used to with her writing, but she proves she is just as at home in the contemporary genre with this novel. Featuring a home town cop and a lady vet, Hope Flames is sure to be a new kind of hit for Jaci Burton.
If readers are looking for a typical Jaci Burton novel, this one is not it... Written for a wider range of readers Hope Flames has a more realistic and simple feel to it. I was really impressed with how Burton was able to switch gears and still write a romance that well worth my time to read. She still writes her trademark steamy scenes but will a little less graphic language and the book certainly doesn't suffer from it. Burton proves that she can easily write just about anything she wants and it will be very good.
I liked the fact that Burton makes both her lead characters people that readers can look up to. The hero, Luke is a cop with a canine sidekick and the heroine, Emma is a veterinarian who also works with animal rescue. Both are from a small town and have that small town appeal going for them. Working with the community whether it be with the humans or the animals was realistic and engaging.
Luke was super sexy and I liked the fact that he got a bit frustrated with his job from time to time, who doesn't, right? He had a protective nature and it was easy to see that it carried over from his career choice to how he dealt with everyone including Emma. I loved the way he cared for his dog and how he seemed in tune with what he needed and felt. A man who loves animals is always a plus in my book. Readers will be drawn to Luke instantly, he has that good guy feel to him.
What can I say about Emma. She thought she had it all figured out. She is just getting started with her veterinary practice and she doesn't have time for romance, but when she and Luke get thrown together with some local burglaries going on, it's easy to see that she might have forgotten about something... love. Burton gives Emma such a giving spirit. She offers everything she has to animal community, but still has some left over. I thought she was commendable character who deserved her happy ever after.
There wasn't too much I didn't like about this one, it did have some pacing issues and stalled out a couple of times, but Burton picks the action back up quickly and gets the reader back on track. I think Burton will probably get better and better with contemporary romance as she continues writing this series and her pacing will be more on point.
Burton crafts a great romance with a touch of mystery that will have readers flipping those pages until long into the night. It may not be what readers have come to expect from Burton, but perhaps it's time to expect something new.
Hope Flames is available NOW from your favorite bookseller.
Jaci Burton is a USA Today and New York Times bestselling author who lives in Oklahoma with her husband and dogs. She has three grown children who are all scattered around the country having lives of their own. A lover of sports, Jaci can often tell what season it is by what sport is being played. She watches entirely too much television, including an unhealthy amount of reality TV. When she isn't on deadline, which is often, Jaci can be found at her local casino, trying to become a millionaire (so far, no luck). She's a total romantic and loves a story with a happily ever after, which you'll find in all her books.
Review and Giveaway: The Smuggler Wore Silk
The Smuggler Wore Silk (Spy in the Ton - Book 1) by Alyssa Alexander
Alyssa Alexander on the WEB: website, blog, twitter, facebook, goodreads
Excerpt from, The Smuggler Wore Silk, courtesy of Amazon's Look Inside feature.
A thrilling, seductive adventure from award-winning author Alyssa Alexander…
After he is betrayed by one of his own, British spy Julian Travers, Earl of Langford, refuses to retire without a fight, vowing to find the traitor. But when the trail leads to his childhood home, Julian is forced to return to a place he swore he'd never see again, and meet a woman who may be his quarry—in more ways than one.
Though she may appear a poor young woman dependant on charity, Grace Hannah's private life is far more interesting. By night, she finds friendship and freedom as a member of a smuggling ring. But when the handsome Julian arrives, she finds her façade slipping, and she is soon compromised, as well as intrigued.
As she and Julian continue the hunt, Grace finds herself falling in love with the man behind the spy. Yet Julian's past holds a dark secret. And when he must make a choice between love and espionage, that secret may tear them apart.
Alyssa Alexander's debut novel, The Smuggler Wore Silk is the first book in the Spy in the Ton series. Readers are given a tale of action, adventure, mystery and intrigue, with a dose of romance to cap it off. Alexander has a way with dialogue that is exceptional. The witty banter she gives her hero was both entertaining and sexy. With a resourceful heroine, a few spies, a traitor and a smuggling ring led by a woman, what more could readers ask for in a historical romance?
First of all, one of the intriguing things about The Smuggler Wore Silk was the fact that it was a debut novel. It certainly didn't read like one. Often times the debut novel of most authors is not always their best work. If that's the case with this book, readers are in for one very good series. This book was well written, had engaging characters with colorful back stories and had enough mystery to keep readers on the edge of their seats. Alexander surprised me with her writing. She writes like a seasoned pro.
Julian, the hero of the story is a wayward earl who doesn't take the responsibilities of his title very seriously. He has spent an inordinate amount of time working as a spy for the British court. Known as Shadow, Julian's work has successful kept him from home for a very long time, keeping his secrets and pain at bay. I thought Julian was an excellent brooding hero. He has issues and flaws, but that was part of what made him so interesting. I loved the way Alexander wrote the way he talked and interacted with Grace, the heroine.
For me one of the best parts of this novel was the banter between Grace and Julian. He has a way with words that is very enjoyable. Not only does Alexander allow him to use a lot of humor, but she always gives Julian a sexiness that is unmistakable. It was a joy to read. Grace also has quite a bit of dialogue that is fun and keeps the reader guessing as to what she's really up to. Alexander's dialogue skills were excellent.
I loved Grace, she was so fun to read about. She is basically a servant by day, smuggler by night and now she's being accused of being a traitor. Julian believes she is the person who has jeopardized his work as a spy and who is trading in secrets. I loved the fact that Alexander shows Grace as a very good hearted person who would help anyone she could. The fact that she resorts to smuggling for that reason did not seem like a stretch at all. She was resourceful character and her relationship with Julian was very complicated and very hot.
The chemistry between these two main characters was simply sizzling. It is obvious though they are leery of each other, something else is at work, their own longings for someone to care about them. I loved how they worked together to draw out the real traitor and how Alexander kept readers guessing with some well placed clues.
There were plenty of people who could have done it and when the reader finds out the real culprit, they will probably kick themselves for not figuring it out sooner. The mystery aspect of the book was very well done.
The only thing I could mention here is that Grace allowed her Uncle to treat her in such a terrible manner. She is a strong woman, but even she takes a beating from the other women in her circle. I just thought she should have stood up for herself a little more. I mean she's a smuggler by night, why take the crap from the ton by daylight? Not a deal breaker, just a pet peeve.
This was an excellent debut novel. I look forward to seeing where Alexander will take this series next. She introduced several characters including, Angel, another spy who will probably be featured in the next book in the series. An a very choice for readers who like some extra pizazz to their historical romances.
The Smuggler Wore Silk is available NOW from your favorite bookseller.
I'm giving this one 5 out 5 apples from my book bag!
Despite being a native Michigander, Alyssa Alexander is pretty certain she belongs somewhere sunny. And tropical. Where drinks are served with little paper umbrellas. But until she moves to those white, sandy beaches, she survives the cold Michigan winters by penning romance novels that always include a bit of adventure. She lives with her own set of heroes, aka an ever-patient husband who doesn't mind using a laundry basket for a closet and a small boy who wears a knight-in-shining-armor costume for such tasks as scrubbing potatoes.
The publisher is sponsoring a giveaway for one copy of The Smuggler Wore Silk by Alyssa Alexander.
1. Please leave a comment about what you think of female spy's past, present, and future.
Review and Giveaway: Bitter Spirits
Bitter Spirits (Roaring Twenties - Book 1) by Jenn Bennett
Imprint: Berkley Sensations
Jenn Bennett on the WEB: website, blog, twitter, facebook, goodreads
Excerpt from, Bitter Spirits, courtesy of the author's website (bottom of the page).
It's the roaring twenties, and San Francisco is a hotbed of illegal boozing, raw lust, and black magic. The fog-covered Bay Area can be an intoxicating scene, particularly when you specialize in spirits…
Aida Palmer performs a spirit medium show onstage at Chinatown's illustrious Gris-Gris speakeasy. However, her ability to summon (and expel) the dead is more than just an act.
Winter Magnusson is a notorious bootlegger who's more comfortable with guns than ghosts—unfortunately for him, he's the recent target of a malevolent hex that renders him a magnet for hauntings. After Aida's supernatural assistance is enlisted to banish the ghosts, her spirit-chilled aura heats up as the charming bootlegger casts a different sort of spell on her...
On the hunt for the curseworker responsible for the hex, Aida and Winter become drunk on passion. And the closer they become, the more they realize they have ghosts of their own to exorcise…
Jenn Bennett's first foray into Paranormal Romance is sure to be an instant hit! Bitter Spirits is the first book in the Roaring Twentie's series featuring a lush historical landscape and multi-faceted characters. Urban Fantasy fans have long loved Bennett's Arcadia Bell series and will definitely be impressed with Bitter Spirits. Bennett's voice is strong and full of mystique and magic in this novel that will allow readers to journey back to 1927, San Franciso and the Prohibition Era. The paranormal aspects of the book only heighten the allure. A fantastic new series!
In short, I liked every single word of this book. Jenn Bennett writes a brilliant and intelligent novel with an incredible setting and characters who live up to the readers expectations. I have not read the author's Arcadia Bell series, but after reading Bitter Spirits it is a given that I will read her other books sooner rather than later. Bennett writes with passion for her subject, her setting and her characters and it can be felt by reader on every page. This is more than a novel, it is perhaps better described as an expression of the author's creativity. I was wowed continually as I read it and I can't recommend it enough.
As I said earlier Bitter Spirits is an intelligent read. It sparks the readers curiosity about more than just the setting. Paranormal Romance is a genre that is receiving so much attention right now, that it is difficult to sort the wheat from the chafe, so to speak. There are a lot of writers who are just jumping on the band wagon and throwing books out there, that do not possess anything close the quality of this one. Bennett makes the reader believe in ghosts to such a degree that they will find themselves wanting to know more about them in the real world. Her writing is intuitive and riveting. Readers will not want to put this one down!
The 1920's Prohibition Era is not a setting that I would expect for paranormal romance. Actually, it is not one I was very familiar with until reading this book. I had of course the cursory knowledge we get from history class, but that doesn't begin to convey the rich history and beauty that is described so effortlessly in this book. Bitter Spirits takes setting to a whole new level. Bennett's descriptions of everything from the dress and customs to the speakeasies and the bootlegging industry was flawless. The reader is basically immersed in the culture of the time and almost becomes a part of the setting themselves. It was amazing!
I loved the main characters for a lot of reasons. Winter is a man with a commanding presence, an alpha in every sense of the word, but he is flawed and real. Bennett brings his character to life with such resonance that readers will be hard put not to fall for him instantly. Who could blame Aida, he is quite a man. Aida was a sharp and world wise character who had a gift for spirits. She was sassy and resourceful and definitely held her own next to a strong male lead. Both characters were so much more than two dimensional. They had fire and depth and showed strength and vulnerability. Bennett has created a couple who were meant for each other and the reader will know it from their very first meeting, though the author is cognizant to keep readers off balance and wondering until the end.
Absolutely nothing! I loved this book!
If you have even the barest inkling that you might be interested in reading a paranormal romance, this is a great book to start with. Fans of the genre will be in awe as well. Bitter Spirits is a smart, highly detailed novel, with an unusual setting and a fiery romance. This book sizzles in more ways than one. You gotta read it!
Bitter Spirits is available NOW from your favorite bookseller.
Jenn Bennett is an award-winning artist and author of the Arcadia Bell urban fantasy series. Born in Germany, she's lived and traveled extensively throughout Europe, the U.S., and the Far East. She currently lives near Atlanta with one husband and two evil pugs.
The publisher is sponsoring a giveaway for one copy of Bitter Spirits by Jenn Bennett.
1. Please leave a comment describing your thoughts on the roaring twenties and having a paranormal series set in that time period.
Blog Tour Stop: The Silence of the Library by Miranda James
Please join me in welcoming the author behind Miranda James, Dean James to Debbie's Book Bag today. Dean is here promoting his book, The Silence of the Library. Enjoy this guest post about his inspiration for his latest book in the Cat in the Stacks series. Don't forget to check out my review following the guest post and the publisher is sponsoring a giveaway for one copy of The Silence of the Library, see details at the end.
I blame Nancy Drew...
I blame Nancy Drew for my life of crime. Reading it and writing it, that is.
I was ten when I borrowed The Secret of Shadow Ranch from a cousin. It was the first mystery I ever read, and I was hooked. Then, to my delight, I discovered this was only one of a long series of adventures in which Nancy solved mystery after mystery. Just as exciting, I soon found other amateur mystery-solvers: the Hardy Boys, the Dana Girls, Judy Bolton, Trixie Belden, and many more. By the time I began reading adult mysteries, my love of the amateur detective was completely entrenched.
When I decided I wanted to write a mystery myself, I knew my main character would be an amateur. After all, I'm not a policeman, or a lawyer, or a private detective. But I do have a healthy dose of curiosity about the world around me and the people in it. Charlie Harris, the sleuth in my "Cat in the Stacks" series, is just like me in that respect. He's also about my age (fiftyish, if you must know), he's a librarian, he grew up in Mississippi, and he has a Maine coon cat. That's as far as it goes, however. (I have two cats, by the way, neither of which is a Maine coon. Also, Diesel is much better behaved than my two.)
I wanted to incorporate my knowledge of, and love for, these juvenile series books into one of my own books, and in the new book, The Silence of the Library, I have done so. I created a series character in the mold of Nancy Drew and the other girl detectives and called her Veronica Thane. The author of the series, Electra Barnes Cartwright, was inspired by Mildred Wirt Benson (the first writer, aka "Carolyn Keene", of the Nancy Drew series, Margaret Sutton, author of the Judy Bolton series, and Julie Campbell Tatham, the original author of Trixie Belden. In fact the book is dedicated to their memories.
The most fun part of writing this book for me was the "excerpts" from the first Veronica Thane book. I reread some of my favorite girl detective stories from the 1930s to get a sense of the style in my head, and off I went. I hope readers will get a kick out of this aspect of the story. Of course, The Silence of the Library includes murder – a subject that usually didn't come up in the classic juvenile series books. But when you get a number of rabid book collectors together, with hints of a rare and highly collectible volume, something deadly is sure to happen.
The Silence of the Library (Cat in the Stacks - Book 5) by Miranda James
Imprint: Berkley Prime Crime
Miranda James on the WEB: website, facebook, goodreads
Murder Past Due (2010), Classified as Murder (2011), File M for Murder (2012), Out of Circulation (2013), The Silence of the Library (2014)
Everyone in Athena, Mississippi, knows Charlie Harris, the librarian with a rescued Maine coon cat named Diesel. He's returned to his hometown to immerse himself in books, but a celebrated author's visit draws an unruly swarm of fanatic mystery buffs…and one devious killer.
It's National Library Week, and the Athena Public Library is planning an exhibit to honor the centenary of famous novelist Electra Barnes Cartwright—creator of the beloved Veronica Thane series.
Charlie has a soft spot for Cartwright's girl detective stories (not to mention an extensive collection of her books!). When the author agrees to make a rare public appearance, the news of her whereabouts goes viral overnight, and series devotees and book collectors converge on Athena.
After all, it's rumored that Cartwright penned Veronica Thane stories that remain under wraps, and one rabid fan will stop at nothing—not even murder—to get hold of the rare books…
The Silence of the Library is the fifth book in the Cat in the Stacks series by author Miranda James. Long time fans of the series always enjoy a visit with librarian, Charlie Harris and his Maine Coon Cat, Diesel. This title is a nod to the girl detective series like Nancy Drew and Trixie Beldon and will make cozy readers nostalgic for the reads of their childhood. James always delivers a mystery that is entertaining and full of surprises and this one is no different. A great addition to the series!
As always with this series, I love the fact that we have a hero, rather than a heroine. Too many cozy mystery writers believe that the protagonist has to be female for cozy readers to enjoy. Miranda/Dean James proves that theory wrong. Charlie Harris is a well beloved character who has a different perspective on amateur sleuthing perhaps because of his gender. I thought James did a wonderful job of developing this character and continues to delight readers with his antics.
Diesel the Maine Coon Cat is always a favorite with this series. He doesn't have magical abilities or anything like that. He is just a companion, a partner in crime, if you will and he adds a certain flavor to book that is unmistakable. Charlie practically takes Diesel everywhere with him and that includes sleuthing. Diesel is as big as a small dog and would be hard to go unnoticed. I just think Diesel adds something to story that is hard to describe but easy to understand.
The main reason I was so taken with this particular book in the series, was the girl detective theme that James uses. I was a huge fan of girl detective stories when I was young and it led me to books by authors and like Agatha Christie and then to the cozies I enjoy as an adult like The Silence of the Library. That homage or nod to the authors like Mildred Wirt Benson and Julie Campbell Tatham was simply magical. I loved the nostalgic feel this book gives off and I know many cozy readers will as well.
The mystery aspects of this novel were very well written. I always enjoy the fact that James allows Charlie to have his own way with the investigation. He doesn't have to resort to going behind the backs of authorities and he doesn't sleuth where sleuthing isn't warranted. He knows the right questions to ask to the right suspects or witnesses or even killers. This was one was a little hard to figure out which is a good thing and I didn't have it until the last couple of chapters. The suspect list was believable as well. Rare book collectors are definitely a breed of their own and James did a wonderful job of bringing that world to life for the reader.
There weren't any huge glaring mistakes with this one. James as usual provides a very succinct mystery that is beautiful in it's simplicity and still complex enough to keep the reader glued to the pages.
If you are a fan of this series, The Silence of the Library is easily my favorite so far. James delivers a cozy that is entertaining and harkens back to the girl detectives of old. His style and prose make readers want to continue reading about Charlie and Diesel. Definitely, a keeper!
The Silence of the Library is available NOW from your favorite bookseller.
Miranda James is a pseudonym for Dean James, the Agatha Award-winning author of several works of mystery nonfiction as well as four mystery series including the New York Times bestselling Cat in the Stacks series.
The publisher is sponsoring a giveaway for one copy of The Silence of the Library by Miranda James.
1. Please leave a comment describing any book signings or special author events you have been to.
Review: With Autumn's Return
With Autumn's Return (Westward Winds - Book 3) by Amanda Cabot
Imprint: Revell Books
Genre: Christian/Inspirational Historical Romance
(Received for an honest review from Revell Books)
Purchase: Amazon, Barnes & Noble, Book Depository, ChristianBook, Indiebound
Amanda Cabot on the WEB: website, blog, facebook, goodreads
Summer of Promise (2012), Waiting for Spring (2013), With Autumn's Return (2014)
Excerpt from, With Autumn's Return, courtesy of the author's website.
Elizabeth Harding arrives in Cheyenne, Wyoming, to establish her medical practice thanks to the wooing of her two older sisters who extolled the beauty of the land. She's certain she'll have a line of patients eager for her expertise and gentle bedside manner. However, she soon discovers the town and its older doctor may not welcome a new physician. Even more frustrating, the handsome young attorney next door may not be ready for the idea of a woman doctor. For his part, Jason Nordling has nothing against women, but he's promised himself that the woman he marries will be a full-time mother.
Despite their firm principles, Elizabeth and Jason find that mutual attraction--and disdain from the community--is drawing them ever closer. And when the two find themselves working to save the life and tattered reputation of a local woman, they'll have to decide how far they're willing to go to find justice--and true love.
The third and final book in the Westward Winds trilogy by Amanda Cabot entitled With Autumn's Return is set in the late 1800's Cheyenne, Wyoming. It has enough Western imagery and heartfelt emotion to draw the reader in quickly and keep them flipping those pages. Cabot's knack for characterization and keen sense of integrating faith into her books is one of the things that keeps her readers returning with each new book she writes. A great example of Christian/Inspirational writing at it's best!
First let me say that I have not read the first two books in this series, though I have read several others by this author. It always amazes me how subtle and intuitive she is with writing about issues of faith. She weaves the ideas she has and the beliefs she follows into every aspect of the book, so that it becomes a part of the story and not just added to get the title of Christian or Inspirational fiction. Cabot knows how to allow her characters to show their faith in their daily lives as we should and that makes all the difference.
I loved the heroine in this one for several reasons. Elizabeth is the final Harding sister and she is certainly a go-getter. She knows what she wants and how to get it. She is a female doctor, which was kind of unheard of at the time. It reminded me of Jane Seymour's television show, Dr. Quinn Medicine Woman in some ways, which was a good thing, since I loved that show. She was determined to do everything she could for her patients even in the face of adversity, gossip and overall disdain of her abilities. I liked her tenacity and grasp of what was important.
I liked Jason as well, as a new attorney he was having a bit of self-doubt. He wasn't sure he could hack it in this cut throat business, but in the end he was able to find his inner strength and show that he had the stuff to make it. I liked the way his relationship with Elizabeth grew as they both faced their fears and worked to become accepted and valued in the community.
There was actually a murder in this book, several secondary characters that were well written and helped move the plot along, more than one tender romance and plenty of wonderful characters. There was quite a bit going on, but it all seemed to flow seamlessly and gave me more than one reason to smile.
Not much to report here. This is probably one of my favorites by Cabot, so I can't offer any insights on how it would have been better. It's already good.
If you enjoy good Christian/Inspirational fiction you can't go wrong with an Amanda Cabot novel. With Autumn's Return is the final novel in the series but can easily be read as a stand alone. The characters are engaging and full of life and the research into the time period is flawless. Pick it up, you'll be glad you did!
Available January 2014 at your favorite bookseller from Revell, a division of Baker Publishing Group.
Amanda Cabot is the bestselling author of the Texas Dreams series, Christmas Roses, Summer of Promise, and Waiting for Spring. Her books have been finalists for the ACFW Carol Awards and RWA's Booksellers Best. She lives in Wyoming. Find out more at www.amandacabot.com.
Blog Tour Stop: The Silence of the Library by Mira...
Blog Tour Stop: The Dancing Master by Julie Klassen
Guest Post and Giveaway: Tammy Faulkner
Review and Giveaway: Tapestry of Lies
Review and Giveaway: River of Dreams
Review and Giveaway: A Chorus Lineup
Review: After the Storm
Review and Giveaway: Murder Sends a Postcard
Review: The Valentine's Arrangement
Review: Dare to Love Again
Review: Come To Me Quietly
Review and Giveaway: The Last Man on Earth
Review: A Hint of Seduction
Review: The Unwelcomed Child
Review and Giveaway: The Home of the Braised
Review: The Calling
Review: The Lion and the Rose
Review and Giveaway: Pecan Pies and Homicides
Review: Before Jamaica Lane
Review and Giveaway: Cursed by Destiny
Review: The Experts Guide to Driving a Man Wild
Review: Mrs. Lincoln's Rival
Interview, Review and Giveaway: Samantha Grace
Review and Giveaway: Playing With Fire
Review: Boots Under Her Bed
Review; Everyday Confetti
Interview, Review and Giveaway: M.L. Rowland
Review and Giveaway: Eggs in a Casket
Review and Giveaway: Desperate and Deceptive
Winners Post!
Review: No One To Trust
Review: The Magic Between Us
Review and Giveaway: The Lone Warrior
Guest Post, Review and Giveaway: Victoria Laurie
Review and Giveaway: Teacup Turbulence
Review: How to Master Your Marquis
Review and Giveaway: Paws for Murder
Early Review: The Splendour Falls
Review and Giveaway: Merry Market Murder
Review: Full Throttle
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,011
|
Peńskie [ˈpɛɲskʲɛ] est un village polonais de la gmina de Krypno dans le powiat de Mońki et dans la voïvodie de Podlachie. Il se situe à environ 10 kilomètres au sud de Mońki et à 33 kilomètres au nord-ouest de Białystok.
Selon le recenssement de la commune de 1921, ont habité dans le village 476 personnes, dont 467 étaient catholiques, 2 orthodoxes, et 7 judaïques. Parallèlement, tous habitants ont déclaré avoir la nationalité polonaise. Dans le village, il y avait 69 bâtiments habitables.
Notes et références
Village dans la voïvodie de Podlachie
Powiat de Mońki
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 576
|
{-# LANGUAGE TemplateHaskell #-}
module Debug where
import Language.Haskell.TH
import Language.Haskell.TH.Syntax
import Language.Haskell.TH.Ppr
{-|
Debugging Template Haskell stuff at the GHCi REPL is hard because everything
ends up in the "Q" monad and there is no way to print the "Q" monad. These functions
call pprint and show but return a "Q Exp" so that the following expressions work
to print a given value (e.g. "x"):
$(pprintQ x)
$(showQ x)
NOTE: GHCi likes to run the contents of splices twice or more, so the results
may be printed multiple times.
-}
pprintQ :: (Ppr a) => Q a -> Q Exp
pprintQ x = x >>= (qRunIO . putStrLn . pprint) >> [|return ()|]
showQ :: (Show a) => Q a -> Q Exp
showQ x = x >>= (qRunIO . putStrLn . show) >> [|return ()|]
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,568
|
Break free from the blankets. Say see-ya to boots, mittens and scarves. It's finally time to open the windows and breathe in a bit of spring. But before you gallop into the sunshine and bid farewell to months spent hibernating, spruce-up your world to put a spring in your gangs' steps.
Be a Night Owl: Since it stays lighter for longer into the evening hours, spend a few minutes before bed each night prepping lunches, packing book bags, and choosing the kids' clothing. Then, spare yourself a few seconds, too, by playing barista. Grind your beans and set the timer on your coffee pot, so you can spend more time sipping that warm, steamy latte come sunrise. Just add milk in the morning, voila!
Keep Breakfast on the Brain: On Sundays, plot a breakfast menu for the week. Each day affords your family the opportunity to try a new breakfast food or savor some favorites. One rule: every one eats the same breakfast together at the table.
Makeover the Fridge: Assign shelves and drawers to specific food groups, and pledge to stick with it. This will help you easily track your food quantity and plan outings to the store, so you spend less time running up and down the aisles and more time enjoying eating (and being) as a family!
Spring-ify the Family Car: Nothing says spring fun like a car wash! First, eat breakfast that includes milk's high-quality protein to get the brood fueled – then grab a bucket of water and suds, and get ready to rid the signs of winter from your four wheels. Shelve the ice scraper and winter gear tucked between the kids' seats to make room for some April musts, including an umbrella and sunglasses!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 92
|
package quasar.fs.mount
import scala.Predef.$conforms
import slamdata.Predef._
import quasar.Variables
import quasar.contrib.pathy.{ADir, AFile, APath}
import quasar.fs.{PathError, FileSystemType}
import quasar.sql, sql.{ScopedExpr, Sql}
import matryoshka._
import matryoshka.data.Fix
import matryoshka.implicits._
import monocle.function.Field1
import monocle.std.{disjunction => D}
import org.specs2.execute._
import pathy.Path._
import scalaz._, Scalaz._
import scalaz.concurrent.Task
abstract class MountingSpec[S[_]](
implicit
S0: Mounting :<: S,
S1: MountingFailure :<: S,
S2: PathMismatchFailure :<: S
) extends quasar.Qspec {
import MountConfig.{viewConfig, fileSystemConfig}
def interpName: String
def interpret: S ~> Task
val mnt = Mounting.Ops[S]
val mntErr = MountingFailure.Ops[S]
val mmErr = PathMismatchFailure.Ops[S]
// NB: Without the explicit imports, scalac complains of an import cycle
import mnt.{FreeS, havingPrefix, lookupConfig, lookupType, mountView, mountFileSystem, mountModule, remount, replace, unmount, viewsHavingPrefix, modulesHavingPrefix}
implicit class StrOps(s: String) {
def >>*[A: AsResult](a: => FreeS[A]) =
s >> a.foldMap(interpret).unsafePerformSync
}
val noVars = Variables.fromMap(Map.empty)
val exprA = ScopedExpr(sql.stringLiteral[Fix[Sql]]("A").embed, Nil)
val exprB = ScopedExpr(sql.stringLiteral[Fix[Sql]]("B").embed, Nil)
val viewCfgA = viewConfig(exprA, noVars)
val viewCfgB = viewConfig(exprB, noVars)
val dbType = FileSystemType("db")
val uriA = ConnectionUri("db://example.com/A")
val uriB = ConnectionUri("db://example.com/B")
val fsCfgA = fileSystemConfig(dbType, uriA)
val fsCfgB = fileSystemConfig(dbType, uriB)
val invalidPath = MountingError.pathError composePrism
PathError.invalidPath composeLens
Field1.first
val notFound = MountingError.pathError composePrism
PathError.pathNotFound
val pathExists = MountingError.pathError composePrism
PathError.pathExists
def maybeNotFound[A](dj: MountingError \/ A): Option[APath] =
D.left composePrism notFound getOption dj
def maybeExists[A](dj: MountingError \/ A): Option[APath] =
D.left composePrism pathExists getOption dj
def mountViewNoVars(loc: AFile, scopedExpr: ScopedExpr[Fix[Sql]]): FreeS[Unit] =
mountView(loc, scopedExpr, noVars)
s"$interpName mounting interpreter" should {
"havingPrefix" >> {
"returns all prefixed locations and types" >>* {
val f1 = rootDir </> dir("d1") </> dir("d1.1") </> file("f1")
val f2 = rootDir </> dir("d1") </> file("f2")
val dA = rootDir </> dir("d1") </> dir("A")
val dB = rootDir </> dir("d2") </> dir("B")
val mnts = Map[APath, MountType](
f1 -> MountType.viewMount(),
f2 -> MountType.viewMount(),
dA -> MountType.fileSystemMount(dbType))
val setup =
mountViewNoVars(f1, exprA) *>
mountViewNoVars(f2, exprA) *>
mountFileSystem(dA, dbType, uriA) *>
mountFileSystem(dB, dbType, uriB)
(setup *> havingPrefix(rootDir </> dir("d1")))
.map(_ must_=== mnts ∘ (_.right))
}
"returns nothing when no mounts have the given prefix" >>* {
havingPrefix(rootDir </> dir("dne")) map (_ must beEmpty)
}
"does not include a mount at the prefix" >>* {
val d = rootDir </> dir("d3") </> dir("someMount")
(mountFileSystem(d, dbType, uriA) *> havingPrefix(d))
.map(_ must beEmpty)
}
}
"lookupConfig" >> {
"returns a view config when asked for an existing view path" >>* {
val f = rootDir </> dir("d1") </> file("f1")
(mountViewNoVars(f, exprA) *> lookupConfig(f).run.run)
.map(_ must beSome(viewCfgA.right[MountingError]))
}
"returns a filesystem config when asked for an existing fs path" >>* {
val d = rootDir </> dir("d1")
(mountFileSystem(d, dbType, uriA) *> lookupConfig(d).run.run)
.map(_ must beSome(fsCfgA.right[MountingError]))
}
"returns none when nothing mounted at the requested path" >>* {
val f = rootDir </> dir("d2") </> file("f2")
val d = rootDir </> dir("d3")
lookupConfig(f).run.run.tuple(lookupConfig(d).run.run)
.map(_ must_=== ((None, None)))
}
"viewsHavingPrefix" >>* {
val v = rootDir </> dir("d1") </> file("f1")
val m = rootDir </> dir("d1") </> dir("d4")
val f = rootDir </> dir("d1") </> dir("d5")
(mountViewNoVars(v, exprA) *> mountModule(m, Nil) *> mountFileSystem(f, dbType, uriA) *> viewsHavingPrefix(rootDir))
.map(_ must_= Set(v))
}
"modulesHavingPrefix" >>* {
val v = rootDir </> dir("d1") </> file("f1")
val m = rootDir </> dir("d1") </> dir("d4")
val f = rootDir </> dir("d1") </> dir("d5")
(mountViewNoVars(v, exprA) *> mountModule(m, Nil) *> mountFileSystem(f, dbType, uriA) *> modulesHavingPrefix(rootDir))
.map(_ must_= Set(m))
}
}
"lookupType" >> {
"returns the view type when asked for an existing view path" >>* {
val f = rootDir </> dir("d1") </> file("f1")
(mountViewNoVars(f, exprA) *> lookupType(f).run.run)
.map(_ must beSome(MountType.viewMount().right[MountingError]))
}
"returns a filesystem type when asked for an existing fs path" >>* {
val d = rootDir </> dir("d1")
(mountFileSystem(d, dbType, uriA) *> lookupType(d).run.run)
.map(_ must beSome(MountType.fileSystemMount(dbType).right[MountingError]))
}
"returns none when nothing mounted at the requested path" >>* {
val f = rootDir </> dir("d2") </> file("f2")
val d = rootDir </> dir("d3")
lookupType(f).run.run.tuple(lookupType(d).run.run)
.map(_ must_=== ((None, None)))
}
}
"mountViewNoVars" >> {
"allow mounting a view at a file path" >>* {
val f = rootDir </> file("f1")
(mountViewNoVars(f, exprA) *> lookupConfig(f).run.run)
.map(_ must beSome(viewCfgA.right[MountingError]))
}
"allow mounting a view above another view" >>* {
val f1 = rootDir </> dir("d1") </> dir("d2") </> file("f1")
val f2 = rootDir </> dir("d1") </> file("d2")
val r = (
mountViewNoVars(f1, exprA) *>
mountViewNoVars(f2, exprB)
) *> lookupConfig(f2).run.run
r map (_ must beSome(viewCfgB.right[MountingError]))
}
"allow mounting a view below another view" >>* {
val f1 = rootDir </> dir("d1") </> file("d2")
val f2 = rootDir </> dir("d1") </> dir("d2") </> file("f2")
val r = (
mountViewNoVars(f1, exprA) *>
mountViewNoVars(f2, exprB)
) *> lookupConfig(f2).run.run
r map (_ must beSome(viewCfgB.right[MountingError]))
}
"allow mounting a view above a filesystem" >>* {
val d = rootDir </> dir("d1") </> dir("db")
val f = rootDir </> file("d1")
val r = (
mountFileSystem(d, dbType, uriA) *>
mountViewNoVars(f, exprA)
) *> lookupConfig(f).run.run
r map (_ must beSome(viewCfgA.right[MountingError]))
}
"allow mounting a view at a file with the same name as an fs mount" >>* {
val d = rootDir </> dir("d1")
val f = rootDir </> file("d1")
val r = (
mountFileSystem(d, dbType, uriA) *>
mountViewNoVars(f, exprA)
) *> lookupConfig(f).run.run
r map (_ must beSome(viewCfgA.right[MountingError]))
}
"allow mounting a view below a filesystem" >>* {
val d = rootDir </> dir("d1")
val f = rootDir </> dir("d1") </> file("f1")
val r = (
mountFileSystem(d, dbType, uriA) *>
mountViewNoVars(f, exprA)
) *> lookupConfig(f).run.run
r map (_ must beSome(viewCfgA.right[MountingError]))
}
"fail when a view is already mounted at the file path" >>* {
val f = rootDir </> dir("d1") </> file("f1")
mntErr.attempt(mountViewNoVars(f, exprA) *> mountViewNoVars(f, exprB)) map { r =>
maybeExists(r) must beSome(f)
}
}
}
"mountFileSystem" >> {
def mountFF(d1: ADir, d2: ADir): FreeS[Unit] =
mountFileSystem(d1, dbType, uriA) *> mountFileSystem(d2, dbType, uriB)
def mountVF(f: AFile, d: ADir): FreeS[Option[MountConfig]] =
mountViewNoVars(f, exprA) *>
mountFileSystem(d, dbType, uriA) *>
lookupConfig(d).run.run ∘ (_ >>= (_.toOption))
"mounts a filesystem at a directory" >>* {
val d = rootDir </> dir("d1")
(mountFileSystem(d, dbType, uriA) *> lookupConfig(d).run.run)
.map(_ must beSome(fsCfgA.right[MountingError]))
}
"succeed mounting above an existing fs mount" >>* {
val d1 = rootDir </> dir("d1")
val d2 = d1 </> dir("d2")
(mountFF(d2, d1) *> lookupConfig(d1).run.run) map (_ must beSome(fsCfgB.right[MountingError]))
}
"succeed mounting below an existing fs mount" >>* {
val d1 = rootDir </> dir("d1")
val d2 = d1 </> dir("d2")
(mountFF(d1, d2) *> lookupConfig(d2).run.run) map (_ must beSome(fsCfgB.right[MountingError]))
}
"fail when mounting at an existing fs mount" >>* {
val d = rootDir </> dir("exists")
mntErr.attempt(mountFF(d, d)).tuple(lookupConfig(d).run.run) map { case (dj, cfg) =>
maybeExists(dj).tuple(cfg) must beSome((d, fsCfgA.right[MountingError]))
}
}
"succeed when mounting above an existing view mount" >>* {
val d = rootDir </> dir("d1")
val f = d </> file("view")
mountVF(f, d) map (_ must beSome(fsCfgA))
}
"succeed when mounting at a dir with same name as existing view mount" >>* {
val f = rootDir </> dir("d2") </> file("view")
val d = rootDir </> dir("d2") </> dir("view")
mountVF(f, d) map (_ must beSome(fsCfgA))
}
"succeed when mounting below an existing view mount" >>* {
val f = rootDir </> dir("d2") </> file("view")
val d = rootDir </> dir("d2") </> dir("view") </> dir("db")
mountVF(f, d) map (_ must beSome(fsCfgA))
}
}
"mount" >> {
"succeeds when loc and config agree" >>* {
val f = rootDir </> file("view")
(mnt.mount(f, viewCfgA) *> lookupConfig(f).run.run)
.map(_ must beSome(viewCfgA.right[MountingError]))
}
"fails when using a dir for a view" >>* {
mmErr.attempt(mnt.mount(rootDir, viewCfgA))
.map(_ must be_-\/(Mounting.PathTypeMismatch(rootDir)))
}
"fails when using a file for a filesystem" >>* {
val f = rootDir </> file("foo")
mmErr.attempt(mnt.mount(f, fsCfgA))
.map(_ must be_-\/(Mounting.PathTypeMismatch(f)))
}
}
"remount" >> {
"moves the mount at src to dst" >>* {
val d1 = rootDir </> dir("d1")
val d2 = rootDir </> dir("d2")
val r =
(mnt.mount(d1, fsCfgA) *> remount(d1, d2)) *>
(lookupConfig(d1).run.run.tuple(lookupConfig(d2).run.run))
r map (_ must_=== ((None, Some(fsCfgA.right[MountingError]))))
}
"moves the mount at src to dst (file)" >>* {
val d1 = rootDir </> file("d1")
val d2 = rootDir </> file("d2")
val r =
(mnt.mount(d1, viewCfgA) *> remount(d1, d2)) *>
(lookupConfig(d1).run.run.tuple(lookupConfig(d2).run.run))
r map (_ must_=== ((None, Some(viewCfgA.right[MountingError]))))
}
"moves mount nested under src to dst" >>* {
val d1 = rootDir </> dir("d1")
val d2 = rootDir </> dir("d2")
val f = dir("d3") </> file("foo")
val r =
(mnt.mount(d1, fsCfgA) *> mnt.mount(d1 </> f, viewCfgA) *> remount(d1, d2)) *>
(lookupConfig(d1 </> f).run.run.tuple(lookupConfig(d2 </> f).run.run))
r map (_ must_=== ((None, Some(viewCfgA.right[MountingError]))))
}
"succeeds when src == dst" >>* {
val d = rootDir </> dir("srcdst")
val r =
mnt.mount(d, fsCfgB) *>
remount(d, d) *>
lookupConfig(d).run.run
r map (_ must beSome(fsCfgB.right[MountingError]))
}
"fails if there is no mount at src" >>* {
val d = rootDir </> dir("dne")
mntErr.attempt(remount(d, rootDir)) map (dj =>
maybeNotFound(dj) must beSome(d))
}
"restores the mount at src if mounting fails at dst" >>* {
val f1 = rootDir </> file("f1")
val f2 = rootDir </> file("f2")
val r =
mountViewNoVars(f1, exprA) *>
mountViewNoVars(f2, exprB) *>
remount(f1, f2)
mntErr.attempt(r).tuple(lookupConfig(f1).run.run) map { case (dj, cfg) =>
maybeExists(dj).tuple(cfg) must beSome((f2, viewCfgA.right[MountingError]))
}
}
}
"replace" >> {
"replaces the mount at the location with a new one" >>* {
val d = rootDir </> dir("replace")
val r =
mnt.mount(d, fsCfgA) *>
replace(d, fsCfgB) *>
lookupConfig(d).run.run
r map (_ must beSome(fsCfgB.right[MountingError]))
}
"fails if there is no mount at the given src location" >>* {
val f = rootDir </> dir("dne") </> file("f1")
mntErr.attempt(replace(f, viewCfgA)) map (dj =>
maybeNotFound(dj) must beSome(f))
}
"restores the previous mount if mounting the new config fails" >>* {
val f = rootDir </> file("f1")
val r = mountViewNoVars(f, exprA) *> replace(f, fsCfgB)
mmErr.attempt(r).tuple(lookupConfig(f).run.run)
.map(_ must_=== ((-\/(Mounting.PathTypeMismatch(f)), Some(viewCfgA.right[MountingError]))))
}
}
"unmount" >> {
"should remove an existing view mount" >>* {
val f = rootDir </> file("tounmount")
val r =
mountViewNoVars(f, exprA) *>
unmount(f) *>
lookupConfig(f).run.run
r map (_ must beNone)
}
"should remove an existing fs mount" >>* {
val d = rootDir </> dir("tounmount")
val r =
mountFileSystem(d, dbType, uriB) *>
unmount(d) *>
lookupConfig(d).run.run
r map (_ must beNone)
}
"should remove a nested view mount" >>* {
val d = rootDir </> dir("tounmount")
val f = d </> file("nested")
val r =
mountFileSystem(d, dbType, uriB) *>
mountViewNoVars(f, exprA) *>
unmount(d) *>
lookupConfig(f).run.run
r map (_ must beNone)
}
"should fail when nothing mounted at path" >>* {
val f = rootDir </> dir("nothing") </> file("there")
mntErr.attempt(unmount(f)) map (dj => maybeNotFound(dj) must beSome(f))
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,071
|
Q: How to pass 'next' parameter to Django Auth logout_then_login view? I am using Django 1.6, django.contrib.auth.views. I am using the login and logout url definitions below. Right now, when a user clicks my logout link, the 'logout_then_login' view gets triggered like it should, but when the user logs back in, the 'next' parameter is undefined and the page is forwarded to the default 'accounts/profile' path. What I would rather have happen is the logout_then_login view forward a 'next' value to the login view.
For example, the user is on page /foo/bar when they click logout. I would like them to be sent to the login screen with '/foo/bar' set as the next paramter, so when they login right away again, they are back on the same page they were at. I want it to be dynamic, based on the request.path when the logout link is clicked. I am aware of the static solution of overriding the default value for next when it is undefined.
Current urls.py:
from django.contrib.auth.views import login, logout_then_login
urlpatterns = patterns('base.views',
url(r'^accounts/login/$', login, name='login'),
url(r'^accounts/logout/$', logout_then_login, name='logout'),
)
I have tried using the extra_context argument for logout_then_login, but not even sure how it should look using a named url in a template:
<a href="{% url 'logout' extra_context={'next': request.path } %}">Log Out</a>
Am I on the right track - or is there a better way to do this?
A: Ok, so this turned out to be way easy. I was over thinking it when I was trying to cram the next parameter into the url template tag.
I simply needed to tack on the next parameter after the url template tag:
<a href="{% url 'logout' %}?next={{request.path}}"> Log Out</a>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,355
|
\section{Introduction}\label{}
Optimization problem is an important topic in vast area of science.
The most typical example is searching the ground state of random spin system such as the spin glass.
The free energy landscape of random spin system is very complex.
Therefore, we encounter the difficulty that the system does not reach
the ground state but stays in the metastable one.
There are two types of slowing down. One is the case where the state is
trapped at a energetically metastable state.
An energy barrier prevents the state from escaping to the equilibrium state.
This is '{\it energetic slowing down}'.
On the other hand, though there is no energy barrier,
the system can not reach the equilibrium state in a short time which has been found
in the regularly decorated bond system \cite{Tanaka1},\cite{Tanaka2}.
We called this situation '{\it entropic slowing down}'.
In the latter case, we have found that effective time scale $\tau_0$ of the ordering process
becomes extremely long and the relaxation process is practically frozen.
In this case it is difficult to find the energy minimam stracture using the thermal annealing process.
Kadowaki and Nishimori \cite{Kadowaki} have proposed
the quantum annealing method for the energetic slowing down system.
Afterward, the efficiency of the quantum annealing method has been confirmed
in the energetic slowing down system \cite{Santoro},\cite{Suzuki}.
Our aim of this study is to demonstrate the efficiency of the quantum annealing method
in the entropic slowing down system.
\section{Model}
We study the decorated bond system depicted in Fig. 1 as an example of systems with
"entropically slowing down".
The Hamiltonian of this system is
\[
\mathcal{H}_0 = -J' \sigma_1^z \sigma_2^z -J \sigma_1^z \sum_{i=1}^N S_i^z
-J \sigma_2^z \left( \sum_{i=1}^{\frac{N}{2}} S_i^z - \sum_{i=\frac{N}{2}+1}^N S_i^z\right),
\]
where $N$ is the number of the decorated spin $\left\{ S_i \right\}$.
The effective coupling $J_{\mathrm{eff}}$
between $\sigma_1^z$ and $\sigma_2^z$ is defined
$\left\langle \sigma_1^z \sigma_2^z \right\rangle = \tanh \beta J_{\mathrm{eff}}$.
In the case of Fig. 1, the contributions through the decoration spins are canceled out
each other and $J_{\mathrm{eff}} = J'$ due to the direct interaction.
However, the dynamics is not simple as the case of the single bond.
If the system is trapped at the state depicted in Fig. 1-(b),
it does not easily reach the equilibrium state due to the entropy effect \cite{Tanaka1}.
In this case, the thermal annealing is not efficient and the relaxation time is about
$2^{N/2}$ for this system to reach the ground state \cite{Tanaka2}.
Here, we introduce quantum annealing using the time dependent transverse field
$\mathcal{H}_t\left( t \right) = -\Gamma \left( t \right) \left( \sigma_1^x + \sigma_2^x + \sum_i S_i^x \right)$,
and we change the transverse field as $\Gamma \left( t\right)=\Gamma_0 \left(1-t/\tau \right)$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=1]{tanaka-2-6-000434-f1.eps}
\end{center}
\caption{
The thick and thin lines denote the ferromagnetic coupling $J'>0$ and $J>0$, respectively.
The dotted lines denote the antiferromagnetic coupling $-J<0$.
The black triangles denote $+$ spin and the gray spins are not fixed ($+$ or $-$) spin.
(a) The ground state with the correlation function $\sigma_1^z \sigma_2^z = +1$.
(b) The '{\it entropically metastable state}' with $\sigma_1^z \sigma_2^z = -1$.
In order to relax to (a) from this configuration,
all the gray spins must be align to cancel the interaction from the black spins.
But the probability of happening of this situation is very small.
This is entropic slowing down.
}
\label{fig-1}
\end{figure}
\section{Real Time Dynamics of the Decorated Bond System by Schr\"odinger Equation}
We study the real time dynamics of the decorated bond system with the time dependent transverse field.
Here, the time evolution of the state vector $\left| \Phi \left( t \right) \right\rangle$ is
determined by the Schr\"odinger equation.
The initial state $\left| \Phi \left( 0 \right) \right\rangle$
is the ground state of the initial Hamiltonian
$\mathcal{H}\left( 0 \right)$.
Figure 2 shows the energy as a function of the transverse field, where
$J=1$, $J'=0.1$, $\Gamma_0 = 1.5$, and $N=6$.
It should be noted that there are two-fold evident degeneracy because of the up-down symmetry of spin.
In Fig. 2, we find that the ground state at $\Gamma = \Gamma_0$ is connected to that of $\Gamma=0$.
Thus the adiabatic motion leads the system to the true ground state of $\Gamma=0$.
In this sense, the success of quantum annealing is insured.
Now we study properties of the quantum annealing. First, the $\tau$ dependence of
the probability $P_0$ to reach the ground state of $\mathcal{H}_0$.
Fig. 3 shows $P_0$ as a function of $\tau$.
There we find that the slower the speed of the sweep, the larger $P_0$.
In the quantum mechanical system, we have several method to obtain the ground state
such as the power method, Lanczos method {\it etc}.
These methods are also possible processes to obtain the ground state.
We will compare these methods as the annealing process and be reported elsewhere.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.37]{tanaka-2-6-000434-f2.eps}
\end{center}
\caption{
The energy diagram of decorated bond system ($N=6$) as a function of transverse field.
}
\label{fig-1}
\end{figure}
\section{Conclusion}
Here we assume ferromagnetic interaction between $\sigma_1$ and $\sigma_2$.
thus the Fig. 1(a) gives the ground state. However if $\sigma_1$ and $\sigma_2$ are
antiparallel initially, the decorated spins denoted by rightward triangles align upwards and
the decorated spins denoted as the right half of the triangles remain disorder.
In order to change $\sigma_1$ and $\sigma_2$ antiparallel to parallel,
all of the left half of the triangle must align, which is difficult.
This is called "entropically slowing down".
Here we treated an 'easy' example to find the ground state
and to demonstrate the efficiency of the quantum annealing.
The decorated bond systems with larger number of $N$ ({\it e.g.} $N=20$)
have been studied to research the temperature dependent structure \cite{Tanaka1}.
When $N$ increases the energy levels are much closer and the annealing becomes more
difficult, which will be reported elsewhere \cite{Tanaka2}.
The present work is partially supported by Grand-in-Aid from the Ministry of Education,
Culture, Sports, Science, and Technology, and also by NAREGI Nanoscience Project, Ministry of
Education Culture, Sports, Science, and Technology, Japan.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.35]{tanaka-2-6-000434-f3.eps}
\end{center}
\caption{
The probability of reaching the ground state of $\Gamma = 0$
as a function of $\tau$.
}
\label{fig-1}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,247
|
Q: After upgrading flutter to 1.20 textfield gives error on lower Android API devices final textController = TextEditingController();
TextField(
controller:textController,
decoration:InputDecoration(...)
)
Error log:
W/System.err( 8349): java.lang.NullPointerException: Attempt to invoke virtual method 'int android.content.ClipData.getItemCount()' on a null object reference
W/System.err( 8349): at android.os.Parcel.readException(Parcel.java:1626)
W/System.err( 8349): at android.os.Parcel.readException(Parcel.java:1573)
W/System.err( 8349): at android.content.IClipboard$Stub$Proxy.getPrimaryClip(IClipboard.java:197)
W/System.err( 8349): at android.content.ClipboardManager.getPrimaryClip(ClipboardManager.java:258)
W/System.err( 8349): at io.flutter.plugin.platform.PlatformPlugin.getClipboardData(PlatformPlugin.java:274)
On lower Android API devices, it is giving me an error. I am using a Samsung j7 having Android 6. Help me out.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,261
|
James T. Williams papers, 1836-1947
Bookmark: James T. Williams papers, 1836-1947
Collection is open for research. Researchers must register and agree to copyright and privacy laws before using this collection. All or portions of this collection may be housed off-site in Duke...
Williams, James T., 1881-1969
48 Linear Feet
The Williams Papers span the period 1836 to 1947 with the bulk dating from 1904 to 1942. The collection contains the following series: Diaries and Reminiscences; Correspondence; Subject Files; Legal Papers; Financial Papers; Writings and Speeches; Miscellaneous; Clippings; Printed Material; and Pictures. Correspondence comprises the majority of the collection and particularly focuses on Williams's professional career during the period from 1910 to 1925 when he was editor of the Tucson Citizen and the Boston Evening Transcript. While the collection documents aspects of Williams's personal and professional life from his college days through the early 1940s, the last twenty years of his life are not included. There is as well very little information about the Teapot Dome Affair in the correspondence, which occurred during the period covered by the collection.
Williams wrote, spoke, and accumulated material about a variety of topics and concerns which are represented in different parts of the collection. Among the most prominent are Aviation and the Presidential Elections of 1916, 1920, and 1924 which are found in the Correspondence, Subject Files, Writings and Speeches, Clippings, Printed Material and Pictures Series; Military preparedness before the entry of the United States into World War I in the Correspondence, Subject Files, Writings and Speeches, and Pictures Series; Arizona's efforts to achieve statehood in the Correspondence, Legal Papers, and Writings and Speeches Series; Massachusetts politics in the Diaries and Reminiscences, Correspondence, Writings and Speeches, Clippings, and Printed Material Series; and Peace and disarmament in the Correspondence, Subject Files, Clippings and Printed Material Series. Prominent politicians such as Warren G. Harding and Herbert Hoover are represented in the Correspondence, Writings and Speeches, and Clippings Series. The collection would be of interest to researchers studying the League of Nations, the Republican Party during the first quarter of the 20th century, and the political and social climate in Greenville, S.C..
The Correspondence Series illustrates that as a leading spokesman for the Republican Party, Williams corresponded with many public figures concerning the topics above. After moving to Tucson, Williams became involved in Arizona's efforts to become a state. He represented the positions taken by President Taft and expressed these viewpoints in numerous editorials related to political matters. Many letters criticize Woodrow Wilson and Josephus Daniels for their policies relating to military preparedness and foreign relations. Of particular note are Williams's strong opposition to the League of Nations and his correspondence in the collection with leading opponents of the League, including Henry Cabot Lodge (1850-1924), William Edgar Borah, Hiram Warren Johnson, and Frank Bosworth Brandegee.
Also included in the Correspondence Series is extensive family correspondence containing material about the social life and political affairs in Greenville, S.C., where Williams's father was mayor, and about his mother's family, the McBees of Lincolnton, N.C. Numerous letters were written by his uncles, Silas McBee, a noted Episcopal clergyman and editor in New York; William Ephraim Mikell, Dean of the Law School at the University of Pennsylvania; and William Alexander Guerry, an Episcopal bishop in South Carolina. There are also letters from cousins, Mary Vardrine McBee, who founded Ashley Hall, a school for girls in Charleston, South Carolina, and Alexander Guerry, who served in various positions at the University of Chatanooga and at The University of the South. Other correspondents in the series include William Howard Taft, Leonard Wood, Nicholas Murray Butler, Albert J. Beveridge, Calvin Coolidge, Frank H. Hitchcock, Charles Nagel, Theodore Roosevelt, and John Wingate Weeks.
Related collections include the Vardry Alexander McBee Papers at Duke University, the Silas McBee and the McBee Family collections at the University of North Carolina at Chapel Hill, the James Thomas Williams (1845-1936) Papers at the University of South Carolina, and an interview with Williams in the Biographical Oral History Collection at Columbia University.
1881, Aug. 10 Born, Lincolnton, N.C.
1897-1898 Attended Furman College in Greenville, S.C.
1898-1900 Attended the University of the South, Sewanee, Tenn.
1901 Received B.A. from Columbia University
1901-1902 Employed at The State newspaper in Columbia, S.C.
1902-1906 Washington correspondent for Associated Press (AP)
1904 AP representative at Democratic and Republican national conventions
1905 AP representative at Portsmouth Peace Conference
1906-1908 Washington correspondent for the Boston Evening Transcript
1907 Elected member of U.S. Naval Institute
1908 Worked for Republican Party and on campaign staff of William Howard Taft
1909 U.S. Civil Service Commissioner. Contracted tuberculosis and admitted to Fort Bayard Army Hospital in New Mexico.
1910-1912 Editor and part owner of the Tucson Citizen newspaper
1912 Delegate-at-large from Arizona at Republican National Convention
1912-1925 Editor of the Boston Evening Transcript, a Republican newspaper
1917 Applied for service in U.S. Army and was rejected because of physical disability
1920 Decorated Knight of the Order of Leopold (Belgium) and Commander of the Order of the Crown (Italy)
1924 Received honorary (D.C.L.) degree from the University of the South; Appointed to Board of Visitors to the U.S. Naval Academy by President Coolidge
1925-1927 Columnist for the Boston American, a Hearst paper
1925-1937 Contributing editor for Hearst's National Syndicate in Washington, D.C.
1928 Received LL.D. from Norwich University
1937-1938 Washington representative for the Chicago Daily News foreign service
1939-1947 Employed at Sperry Gyroscope Corporation for public relations
1967 Returned to Greenville, S.C. to live with sister
1969, Dec. 26 Died in Greenville, S.C.
After 1937 and for the remainder of his career, he lectured across the country and was a free lance writer. Member of Sigma Alpha Epsilon fraternity, Society of the Cincinnati, and Metropolitan, Cosmos, Army and Navy clubs.
The Williams Papers were purchased by Duke University Library in 1970.
Processed by: Janie C. Morris
Encoded by Stephen Miller
Completed October 25, 1991
Presidents -- United States -- Election -- 1916
Air mail service -- United States -- Photographs
Aeronautics -- United States
Republican Party (U.S. : 1854- )
Beveridge, Albert Jeremiah, 1862-1947
Brandegee, Frank Bosworth, 1864-1924
Butler, Nicholas Murray, 1862-1947
Coolidge, Calvin, 1872-1933
Daniels, Josephus, 1862-1948
Guerry, Alexander
Harding, Warren G. (Warren Gamaliel), 1865-1923
Hitchcock, Frank H. (Frank Harris), 1867-1935
Hoover, Herbert, 1874-1964
Johnson, Hiram, 1866-1945
Nagel, Charles, 1849-1940
Weeks, John W. (John Wingate), 1860-1926
Taft, William H. (William Howard), 1857-1930
Roosevelt, Theodore
Wilson, Woodrow
Lodge, Henry Cabot
United States -- Military policy
Arizona -- Politics and government -- To 1950
Greenville (S.C.)
Massachusetts -- Politics and government 1865-1950
All or portions of this collection may be housed off-site in Duke University's Library Service Center. The library may require up to 48 hours to retrieve these materials for research use.
copyright interests in these papers have not been transferred to the University. For more information consult the section on copyright in the Regulations and Procedures of the David M. Rubenstein Rare Book & Manuscript Library.
[Identification of item], James T. Williams Papers, David M. Rubenstein Rare Book & Manuscript Library, Duke University.
EAD ID williams
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,574
|
Rhyolite and dacite domes and flows and small hypabyssal intrusive bodies
Mostly light-gray to red, dense, flow-banded, nonporphyritic and porphyritic rhyolite and dacite in nested domes, small intrusive bodies, and related flows. Includes some near-vent breccias, pumice-lapilli tuffs, and coarse pumicites. Commonly associated with mercury mineralization. Includes several small hypabyssal intrusions of diorite, granodiorite, and quartz monzonite exposed in Paisley Hills of Lake County (Muntzert, 1969; Muntzert and Field, 1968). In many places represents vents for lava flows and tuff of unit Tsf
Late Eocene to Miocene
Igneous > Volcanic > Felsic-volcanic > Dacite (Dome, Flow, Pyroclastic-air fall)
Igneous > Volcanic > Felsic-volcanic > Rhyolite (Dome, Flow, Pyroclastic-air fall)Also containsd acite, diorite, granodiorite, and quartz monzonite
Igneous > Hypabyssal > Felsic-hypabyssal > Hypabyssal-dacite (Stock or pipe)
Igneous > Hypabyssal > Felsic-hypabyssal (Pluton)
Igneous > Hypabyssal > Felsic-hypabyssal > Hypabyssal-rhyolite (Stock or pipe)
Ferns, M.L., Brooks, H.C., Evans, J.G., and Cummings, M.L., 1993b, Geologic map of the Vale 30 x 60 minute quadrangle, Malheur County Oregon: Oregon Department of Geology and Mineral Industries Geological Map Series GMS-77, scale 1:100,000.
Swanson, D.A., 1969a, Reconnaissance geologic map of the east half of the Bend quadrangle, Crook, Wheeler, Jefferson, Wasco, , and Deschutes Counties, Oregon: U.S. Geological Survey Miscellaneous Geologic Investigations Map I-568, scale 1:250,000.
Robinson, P.T., 1975, Reconnaissance geologic map of the John Day Formation in the southwestern part of the Blue Mountains and adjacent areas, north-central Oregon: U.S. Geological Survey Miscellaneous Geologic Investigations Map I-872, scale 1:125,000.
Muntzert, J.K., 1969, Geology and mineral deposits of the Bratain district, Lake County, Oregon: Corvallis, Oregon State University, master's thesis, 70 p.
Muntzert, J.K. and Field, C.W., 1968, Geology and mineral deposits of the Brattain district, Lake County, Oregon [abs.]: Geological Society of America Special Paper 121, p. 616-617.
Evans, J.G., 1996, Geologic map of the Monument Peak quadrangle, Malheur County, Oregon: U.S. Geological Survey Miscellaneous Field Studies Map MF-2317, scale 1:24,000.
https://pubs.er.usgs.gov/publication/mf2317
Ferns, M.L. and Gilbert, Deborah, 1992, Preliminary geologic map of the Rockville quadrangle, Malheur County, Oregon: Oregon Department of Geology and Mineral Industries Open-File Map O-92-12, scale 1:24,000.
Vander Meulen, D.B., Rytuba, J.J., Grubensky, M.J., and Goeldner, C.A., 1987, Geologic map of the Bannock Ridge 7.5' quadrangle, Malheur County, Oregon: U.S. Geological Survey Miscellaneous Field Studies Map MF-1903A, scale 1:24,000.
Vander Meulen, B.B., Rytuba, J.J., Grubensky, M.J., Vercoutere, T.L., and Minor, S.A., 1987, Geologic map of the Pelican Point quadrangle, Malheur County, Oregon: U.S. Geological Survey Miscellaneous Field Studies Map MF-1904, scale 1:24,000.
Vander Meulen, D.B., Rytuba, J.J., Minor, S.A., and Harwood, C.S., 1989, Preliminary geologic map of the Three Fingers Rock quadrangle, Malheur County, Oregon: U.S. Geological Survey Open-File Report 89-344, scale 1:24,000.
https://pubs.er.usgs.gov/publication/ofr89344
Vander Meulen, D.B., Rytuba, J.J., Vercoutere, T.L., and Minor, S.A., 1987, Geologic map of the Rooster Comb 7.5' quadrangle, Malheur County, Oregon: U.S. Geological Survey Miscellaneous Field Studies Map MF-1902C, scale 1:24,000.
Baker - Crook - Grant - Harney - Jefferson - Lake - Malheur - Wasco - Wheeler
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,341
|
Q: Swift MapView -> start "didSelectAnnotation" programmatically isn't the same like to tap with a finger on annotation My princip is easy: When you tap on an annotation (didSelectAnnotation) a infoscreen appears.
When you tap on the Map the infoscreen (didDeselectAnnotation) disappears.
Now it should be possible to let the infoscreen appear, when the user tap a button, too.
So I start/call mapView(mapView: MKMapView, didSelectAnnotationView view: MKAnnotationView) in my code. This works fine. But when I click on my map, it doesn't disappear. I have first to click on the annotation (didSelectAnnoation) and after that the map listen for the didDeselectAnnoation.
What kind of problem is that and how can I fix that?
I can't find anything at google but maybe I'm searching wrong. Thank you for help and sorry for bad english (I'm still learning :) )
A: Use this method:
func selectAnnotation(_ annotation: MKAnnotation,
animated animated: Bool)
It selects the specified annotation and displays a callout view for it.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,785
|
Бартоломео Бімбі (; 15 травня, 1648, Сеттіньяно — 14 січня, 1729, Флоренція) — італійський художник зламу 17—18 ст., майстер різноманітних натюрмортів, переважно ботанічної тематики.
Життєпис
Народився в місті Сеттіньяно. Художню освіту опановував у майстерні Лоренцо Ліппі. Починав зі створення картин біблійних, як і його вчитель Лоренцо Ліппі (1606—1665). Ще одним вчителем для нього став художник Оноріо Марінарі (1627—1715).
Стажувався в Римі в майстерні художника Маріо де Фйорі (1603—1673). Перейшов до створення різноманітних за сюжетами натюрмортів та зображення екзотичних для Італії птахів чи тварин. Бароко відбилось в його картинах зацікавленістю до незвичного чи екзотичного. Звідси зображення рослин з американського континенту, всіляких ботанічних рідкісних речей, окремих овочів, звертання до анімалістичного жанру.
Його покровителем став сам герцог Тосканський Козімо ІІІ Медічі, а після його смерті донька Козімо ІІІ — Марія Луїза Медічі.
Художня манера митця відрізнялась точністю у відтворенні характерних рис рослин, побутових речей чи тварин. Низка його картин — це точна фіксація рослин з ботанічних збірок Медічі. Низка його картин із зображеннями рослин перейшла до університету, де слугувала навчальними експонатами. Згодом частка його творів передана до музею натюрмортів у місті Флоренція, що є філією комплексу музеїв Палаццо Пітті.
Вибрані твори (неповний перелік)
«Мушлі на арабському килимі»
«Цитрини і лимони»
«Грона винограду»
«Турецька зброя»
«Цитруси Медічі»
«Гарбуз на пагорбі»
«Кольорова капуста»
«Плід і квітка цитрину»
«Ботанічна рідкість (pianta di fave prodigiosa)»
«Мисливські трофеї, заяць і дичина»
«Теля з двома головами»
«Самиця опоссума з щенятами»
«Папуга»
«Декоративний соняшник»
«Короп звичайний»
«Фламінго і собака»
Вибрані твори (галерея)
Див. також
Мистецтво Італії
Живопис бароко
Натюрморт
Кабінет курйозів
Анімалістичний жанр
Музей натюрмортів, Флоренція
Палаццо Пітті
Томмазо Саліні
Франческо Нолетті
Квітковий натюрморт 17 століття
Джерела
Silvia Meloni Trkulja: BIMBI, Bartolomeo. In: Alberto M. Ghisalberti (Hrsg.): Dizionario Biografico degli Italiani (DBI). Band 10 (Biagio–Boccaccio), Istituto della Enciclopedia Italiana, Rom 1968.
Mina Gregori, Le Musée des Offices et le Palais Pitti: La Peinture à Florence, Editions Place des Victoires, 2000 (ISBN 2-84459-006-3), p. 422—427
Посилання
Італійські барокові живописці
Італійські натюрмортисти
Художники XVII століття
Художники, чиї твори перебувають у суспільному надбанні
Уродженці Флоренції
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 586
|
\section*{Abstract}
We present a new approach for online handwritten signature classification and verification based on
descriptors stemming from Information Theory.
The proposal uses the Shannon Entropy, the Statistical Complexity, and the Fisher Information evaluated over the Bandt and Pompe symbolization of the horizontal and vertical coordinates of signatures.
These six features are easy and fast to compute, and they are the input to an One-Class Support Vector Machine classifier.
The results produced surpass state-of-the-art techniques that employ higher-dimensional feature spaces which often require specialized software and hardware.
We assess the consistency of our proposal with respect to the size of the training sample, and we also use it to classify the signatures into meaningful groups.
\section*{Introduction}
The word {\it biometrics\/} is associated to human traits or behaviors which can be measured
and used for individual recognition.
In fact, the biometry recognition, as a personal authentication signal processing, can
be used in applications where users need to be security identified\cite{OrtegaGarcia2004}.
Clearly, these kind of systems can either verify or identify.
Two types of biometrics can be defined according to the personal traits considered:
physical/physiological or behavioral.
Physical/physiological biometrics is about catering the biological traits of users, like fingerprints,
iris, face, hand, etc.
Behavioral biometrics takes into account dynamic traits of users, such as, voice, handwritten and
signature expressions.
One of the main advantages of biometric systems is that users do not have
to remember passwords or carry access keys.
Another important advantage lies in the difficulty to steal, imitate or generate genuine
biometric data, leading to enhanced security\cite{OrtegaGarcia2004}.
As mentioned, behavioral biometrics is based on measurements extracted from an activity
performed by the user, in conscious or unconscious way, that are inherent to his/her
own personality or learned behavior.
In this aspect, behavioral biometrics has interesting pros, like user acceptance and
cancelability, but it still lacks of some level of the uniqueness physiological biometrics has.
Among the pure behavioral biometric traits, the handwritten
signature and the way we sign is the one with widest social and legal acceptance\cite{Plamondon1989,Leclerc1994,Gupta1997,Impedovo2008,Ahmed2013}.
Identity verification by signature analysis requires no invasive measurements and people
are familiar with the use of signatures in their daily life.
Also, it is the modality confronted with the highest level of attacks.
A signature is a handwritten depiction of someone's name or some other mark of
identification written on documents and devices as proof of identification.
The formation of signature varies from person to person, or even from the same person due
to the psychophysical state of the signer and the conditions under which the signature
apposition process occurs.
Hilton\cite{Hilton1992} studied how signatures are produced, and found that the signature has at least three attributes:
form, movement and variation; being movement the most important, because signatures
are produced by moving a writing device.
The study also noted that a person's signature does evolve over time and, with the vast majority
of users, once the signature style has been established the modifications are usually slight.
The movement is produced by muscles of fingers, hand, wrist, and in some writers the arm;
these muscles are controlled by nerve impulses.
When one person is signing these nerve impulses are controlled by the brain without any
particular attention to detail.
The signing processes can be described then, at high level, as how the central nervous
system (the brain) recovers information from long term memory in which parameters such as
size, shape, timing etc. are specified.
At the peripheral level, commands are generated for muscles.
In consequence, the signing process is believed to be a reflex action
(ballistic action\footnote{Ballistic movement can be defined as muscle contractions that exhibit maximum velocities and accelerations over a very short period of time.
They exhibit high firing rates, high force production, and very brief contraction times\cite{ballistic}.})
rather than a deliberate action.
Then, the production of genuine signatures is associated to a ballistic handwriting, which
is characterized by a spurt of activity, without positional feedback, whereas the
production of forgery signature is associated to a deliberate handwriting which is
characterized by a conscious attempt to produce a visual pattern with the aid of positional
feedback\cite{DerGon1965,Nalwa1997}.
Handwritten signature verification is a problem in which the input signature
(a test signature) is classified as genuine or forged.
This process is usually performed in three main phases:\cite{Plamondon1989,Leclerc1994,Gupta1997,Impedovo2008,Ahmed2013}
\begin{itemize}
\item [$\bullet$]{\bf Data acquisition and pre-processing.}
Two different categories of systems can be identified, depending on whether there is electronic
access to the handwritten process or not.
{\it a)~Online or dynamic recognition\/}, in which the pen's instantaneous information trajectories,
and also information like pressure, speed or pen-up movements can be captured.
{\it b)~Offline or static recognition\/}: those that record signatures as images
on paper which can be later digitized by means of a scanner, and processed.
In the latter, the pre-processing phase involves filtering, noise reduction and
smoothing.
Online signature verification offers reliable identity protection, as it employs dynamic information not available on the signature image itself but in the process of signing. As a consequence,
online signature verification systems usually achieve better accuracy than offline systems.
\item [$\bullet$]{\bf Feature extraction.\/}
Two types of features can be used.
{\it a)~Function features of the signature\/}: time
functions whose values constitute the feature set.
{\it b)~Parameter features\/}: the signature is characterized as a vector of
elements, each one representative of the value of the feature.
Usually, the last one yields better performance, but it is also time-consuming.
\item [$\bullet$]{\bf Classification.\/}
In the verification process, the authenticity of the test signature is evaluated by matching
it against those stored in the knowledge base developed during the enrollment stage.
This process produces a single response that attests to the authenticity of the test signature.
When template matching techniques are considered, a questioned sample is matched against
templates of authentic/forgery signatures.
Distance-based classifiers, mostly when parameters are used as features, are usually developed
with statistical techniques, e.g. with Mahalanobis and Euclidean distances.
The performance of a signature verification system is commonly assessed in terms
of the percentage Equal Error Rate.
\end{itemize}
On the one hand, template matching attempts at finding similarities between the input signature and those in a data base.
Most approaches use Dynamic Time Warping to perform this match\cite{Impedovo2008,Ahmed2013}.
On the other hand, distance-based classifiers rely on the use of features derived from the signatures.
Two opposite mechanisms describing the signing process can be found in the literature.
The nonlinear character and chaotic behavior of several physiological complex processes are well
established\cite{Goldberger1990,West2013}.
In particular, Longstaff and Heath\cite{Longstaff1999}
found evidence of chaotic behavior on the underlying dynamics
of time series related to velocity profiles of handwritten texts.
Taking into account the inherent behavioral nature of the online signing process, the input
information could be associated to deterministic (nonlinear low dimensional chaotic) signals, and
the handwritten signature variations as a consequence of chaos (sensibility to initial conditions).
In opposition, most of the research in the field of signal verification considers the input
information as well described by a random process\cite{Plamondon1989,Leclerc1994,Gupta1997,Impedovo2008,Ahmed2013}.
Then, the dynamic input information acquired through a time sampling procedure must be
consequently considered as discrete time random sequence.
In any case, the signature analysis taken as a time-based sequence characterization process is
strongly related to the way in which a reference model is established.
From the stochastic point of view, Hidden Markov Models are among the most commonly used in the
literature, and the ones with the best performance in signature
verification\cite{Plamondon1989,Leclerc1994,Gupta1997,Impedovo2008,Ahmed2013}.
Our proposal relies on the use of time causal quantifiers based on Information Theory for the
characterization of online handwritten signatures:
normalized permutation Shannon entropy, permutation statistical complexity and permutation Fisher
information measure.
These quantifiers have proved to be useful in the identification of chaotic and stochastic
dynamics throughout the associated time series\cite{Rosso2007,Rosso2015}.
Their evaluation is simple and fast, making them apt for the signature verification problem.
We apply our proposal to the well know MCYT online signature data base\cite{MCYT2003}.
Next section describes the database used in this study, followed by a section where we detail
the quantifiers employed and by their application to the data.
In addition to the usual data flow, we present an exploratory data analysis (EDA) of the features that enhances their appropriateness for this problem.
The expressiveness and usefulness of these descriptors for the problem of online signature
classification and verification follows in the sequence: we experiment their application to the test-bed.
\section*{Handwritten signatures database}
\label{Sec:HandwrittenData}
The present study is carried out on the freely available and widely used handwritten signatures
database MCYT-100 subset of 100 persons\cite{MCYT2003}.
The acquisition of each on-line signature is accomplished dynamically using a graphics tablet.
The signatures are acquired on a
WACOM$^\copyright$ graphic tablet, model~INTUOS~A6~USB.
The tablet resolution is \SIunits{2540}~{lines\per in} (\SIunits{100}~{lines\per\milli\meter}),
and the precision is \SIunits{$\pm$0.25}~{\milli\meter}.
The maximum detection height is \SIunits{10}~{\milli\meter}
(so also pen-up movements are considered), and the capture area is
\SIunits{127}~{\milli\meter} (width) $\times$ \SIunits{97}~{\milli\meter} (height).
This tablet provides the following discrete-time sequences:
{\it a)\/} position $x_t$ in the $x$-axis,
{\it b)\/} position $y_t$ in the $y$-axis, and
{\it c)\/}
also the time series corresponding to the pressure $p_t$ applied by the pen, as well as the
azimuth $\gamma_t$ and altitude $\varphi_t$ angles of the pen with respect to the tablet, not used in
the present work.
The sampling frequency is set to \SIunits{100}~{\hertz}.
Taking into account the Nyquist sampling criterion and the fact that the maximum frequencies of the
related biomechanical sequences are always under \SIunits{20-30}~{\hertz}\cite{Baron1989},
this sampling frequency leads to a precise discrete-time signature representation.
The signature corpus comprises genuine and shape-based highly skilled forgeries with natural
dynamics\cite{MCYT2003,Salicetti2009}.
In order to obtain the forgeries, each contributor is requested to imitate other signers by writing
naturally.
For this task, they were given the printed signature to imitate and were asked not only to imitate the
shape but also to generate the imitation without artifacts such as breaks or slow-downs (see
\cite{MCYT2003,Salicetti2009} for more details of the acquisition procedure).
Each signer contributes with $25$ genuine signatures in five groups of five signatures each,
and is forged $25$ times by five different imitators.
Figure~\ref{fig:MCYT-firmas} presents examples for six different subjects,
being the first two columns genuine and the third column forgery signatures.
\begin{figure}[hbt]
\centering
\includegraphics[width = \linewidth]{3firmas-22-H1A-new}
\includegraphics[width = \linewidth]{3firmas-39-H1B-new}
\includegraphics[width = \linewidth]{3firmas-60-H2A-new}
\includegraphics[width = \linewidth]{3firmas-6-H2B-new}
\includegraphics[width = \linewidth]{3firmas-98-H3A-new}
\includegraphics[width = \linewidth]{3firmas-46-H3B-new}
\caption{Six different subjects signatures from the MCYT database.
Two genuine signatures (left, blue) and a skilled forgery (right, red).
The two first signatures were classified as H1A and H1B,
the following two to types H2A and H2B, and the last two to types H3A and H3B; cf.\ Sec.\ Signatures classification.}
\label{fig:MCYT-firmas}
\end{figure}
Since signers are concentrated in a different writing task between genuine signature sets, the
variability between client signatures from different acquisition sets is expected to be higher than
the variability of signatures within the same set.
The total number of contributors in the MCYT is $330$, and the total number of signatures present in
the signature database is $16,500$, half of them genuine signatures and the rest
forgeries\cite{MCYT2003,Salicetti2009}.
As previously mentioned, we used a subset of the database, denominated MCYT-100,
which includes $100$ subjects and for each one, $25$ genuine and $25$
skilled
forged signatures, and only the
corresponding time series corresponding to the $x$- and $y$-coordinates of each signature will be analyzed.
In particular, one must note that the time series' lengths are quite variable.
In order to facilitate our Information Theory analysis, we pre-processed each time series as follows:
{\it a)\/}~the coordinates were re-scaled into the unit square $[0,1]\times[0,1]$;
{\it b)\/}~taken as base these scaled values, the original total number of data for
each time series is expanded to $M = 2000$ points using a cubic Hermite polynomial.
In this way, for each subject $k$ ($k = 1, \dots, 100$) and associated signatures $j$ ($j=1, \dots, 25$)
we will analyze two time series, denoted by
$ {\mathbf X}^{(k;\alpha)}_j = \{ 0 \leq {\tilde{x}}^{(k;\alpha)}_{j;i} \leq 1,~i = 1, \ldots, M \}$ and
$ {\mathbf Y}^{(k;\alpha)}_j = \{ 0 \leq {\tilde{y}}^{(k;\alpha)}_{j;i} \leq 1,~i = 1, \ldots, M \}$,
in which the supra-index $\alpha = G,~F$ denotes genuine and forgery signature, and
${\tilde{x}}$ and ${\tilde{y}}$ are the interpolated values, respectively.
\section*{Information Theory quantifiers}
\label{Sec:Quantifiers}
The basic elements for the study of a system dynamics, either natural or man-made, are sequences of measurements or observations whose evolution can be tracked through time.
Then, given
an observable of such system, a natural question that arises is: how much information is this
observable encoding about the dynamics of the underlying system?
The information contents of a system are typically evaluated via a probability distribution function (PDF) $P$
obtained from such observable.
We can define Information Theory quantifiers as measures able to characterize relevant properties of the
PDF
associated with these time series, and in this way we should judiciously
extract information on the dynamical system under study.
\subsection*{Shannon entropy, Fisher Information Measure, and Statistical Complexity}
\label{Sec:HFC}
Entropy is a basic quantity with multiple field-specific interpretations; for instance, it has been associated
with disorder, state-space volume, and lack of information\cite{Brissaud2005}.
When dealing with information content, the Shannon entropy is often considered the foundational and most
natural one\cite{Shannon1948,Shannon1949}.
Given a continuous probability distribution function (PDF) $f(x)$ with $x \in \Omega \subset {\mathbb R}$ and
$\int_{\Omega} f(x)~dx = 1$, its associated {\it Shannon Entropy\/} $S$ \cite{Shannon1948,Shannon1949} is
\begin{equation}
\label{shannon}
{\mathrm S}[f] = -\int_{\Omega} f(x) \ln f(x) dx .
\end{equation}
It is a global measure, that is, it is not too sensitive to strong changes in the distribution taking
place on a small-sized region of $\Omega$.
Such is not the case with {\it Fisher's Information Measure\/} (FIM) $\mathcal F$\cite{Fisher1922,Frieden2004},
which constitutes a measure of the gradient content of the distribution $f$, thus being quite sensitive even
to tiny localized perturbations.
It reads
\begin{equation}
\label{fisher}
{\mathcal F}[f] = \int_{\Omega} { \frac{1}{f(x)} } \left[ { \frac{df(x)}{dx} }\right]^2 dx
= 4 \int_{\Omega} \left[ { \frac{d \psi(x)} {dx} }\right]^2 ,\quad
\text{where } \psi(x) = \sqrt{f(x)}.
\end{equation}
The Fisher Information Measure can be variously interpreted as a measure of the ability to estimate a parameter,
as the amount of information that can be extracted from a set of measurements, and also as a measure of the state
of disorder of a system or phenomenon\cite{Frieden2004}, its most important property being the so-called
Cramer-Rao bound.
It is important to remark that the gradient operator significantly influences the contribution of minute local
$f$-variations to the Fisher information value, accordingly, this quantifier is called ``local"\cite{Frieden2004}.
Note that the Shannon entropy decreases with the distribution skewness, while the Fisher information increases.
Local sensitivity is useful in scenarios whose description necessitates an appeal to a notion of ``order''.
In the previous definition of FIM (Eq.~(\ref{fisher})) the division by $f(x)$ is not convenient if
$f(x) \rightarrow 0$ at certain points of the support $\Omega$.
We avoid this if we work with real probability amplitudes, by means of the alternative expression that employs
$\psi(x)$\cite{Fisher1922,Frieden2004}.
This form requires no divisions, and shows that $\mathcal F$ simply measures the gradient content in
$\psi(x)$.
Let now $P=\{p_i; i=1,\ldots, N\}$ be a discrete probability distribution, with $N$ the number of possible
states of the system under study.
The Shannon's logarithmic information measure reads
\begin{equation}
\label{shannon-disc}
{\mathrm S}[P] = -\sum_{i=1}^{N} p_i \ln p_i.
\end{equation}
This can be regarded to as a measure of the uncertainty associated (information) to the physical process described by $P$.
For instance, if ${\mathrm S}[P] = {\mathrm S}_{\min} = 0$, we are in position to predict with complete certainty
which of the possible outcomes $i$, whose probabilities are given by $p_i$, will actually take place.
Our knowledge of the underlying process described by the probability distribution is maximal in this instance.
In contrast, our knowledge is minimal for a uniform distribution $P_e = \{ p_i = 1/N, \forall i=1, \ldots , N \}$
since every outcome exhibits the same probability of occurrence, and the uncertainty is maximal, i.e.,
${\mathrm S}[P_e] = {\mathrm S}_{\max} = \ln N$.
In the discrete case, we define a ``normalized" Shannon entropy, $0 \leq {\mathcal H} \leq 1$, as
\begin{equation}
\label{shannon-disc-normalizada}
{\mathcal H} [P] = {\mathrm S}[P] / {\mathrm S}_{\max} \ .
\end{equation}
The concomitant problem of loss of information due to the discretization has been thoroughly studied (see, for
instance, \cite{Zografos1986,Pardo1994} and references therein) and, in particular, it entails the
loss of Fisher's shift-invariance, which is of no importance for our present purposes.
For the FIM we take the expression in terms of real probability amplitudes as starting point, then a discrete
normalized FIM, $0 \leq {\mathcal F} \leq 1$, convenient for our present purposes, is given by
\begin{equation}
\label{Fisher-disc}
{\mathcal F}[P]=F_0\sum_{i=1}^{N-1} \big[\sqrt{p_{i+1}} - \sqrt{p_{i}}\big]^2 .
\end{equation}
It has been extensively discussed that this discretization is the best behaved in a discrete environment\cite{Dehesa2009}.
Here the normalization constant $F_0$ reads
\begin{equation}
\label{F0}
F_0=\left\{
\begin{array}{rl}
1, & \text{if } p_{i^*} = 1 \text{ for }
i^* = 1 \text{ or } i^* = N \text{ and } p_{i} = 0, \forall i \neq i^*, \\
1/2, & \text{otherwise.}
\end{array}
\right.
\end{equation}
The perfect crystal and the isolated ideal gas are two typical examples of systems with minimum and
maximum entropy, respectively.
However, they are also examples of simple models and therefore of systems with zero complexity, as the
structure of the perfect crystal is completely described by minimal information (i.e., distances and
symmetries that define the elementary cell) and the probability distribution for the accessible states
is centered around a prevailing state of perfect symmetry.
On the other hand, all the accessible states of the ideal gas occur with the same probability and can be
described by a ``simple" uniform distribution.
According to L\'opez-Ruiz {\it et al.}\cite{LMC1995}, and using an oxymoron, an object, a procedure,
or system is said to be complex when it does not exhibit patterns regarded as simple.
It follows that a suitable complexity measure should vanish both for completely ordered and for completely
random systems and cannot only rely on the concept of information (which is maximal and minimal for the
above mentioned systems).
A suitable measure of complexity can be defined as the product of a measure of information and a measure of
disequilibrium, i.e. some kind of distance from the equiprobable distribution of the accessible states of
a system.
In this respect, Rosso and coworkers\cite{Lamberti2004} introduced an effective {\it Statistical Complexity
Measure\/} (SCM) ${\mathcal C}$, that is able to detect essential details of the dynamical processes
underlying the dataset.
Based on the seminal notion advanced by L\'opez-Ruiz {\it et al.}\cite{LMC1995}, this statistical complexity
measure\cite{Lamberti2004} is defined through the product
\begin{equation}
{\mathcal C}[P] = {\mathcal Q}_{J}[P,P_e] \cdot {\mathcal H}[P]
\label{complexity}
\end{equation}
of the normalized Shannon entropy ${\mathcal H}$, see Eq.~\eqref{shannon-disc-normalizada}, and the disequilibrium
${\mathcal Q}_{J}$ defined in terms of the Jensen-Shannon divergence ${\mathcal J}[ P, P_e]$.
That is,
\begin{equation}
\label{disequilibrium}
{\mathcal Q}_{J} [ P, P_e] = Q_{0} {\mathcal J}[ P, P_e] =
Q_{0} \{ {\mathrm S}[(P + P_e)/2 ] - {\mathrm S}[ P ]/2 - {\mathrm S}[P_e]/2\},
\end{equation}
the above-mentioned Jensen-Shannon divergence and $Q_0$, a normalization constant
such that $0 \leq {\mathcal Q}_{J} \leq 1$:
\begin{equation}
Q_0 = -2 \left\{ {\frac{N+1}{N}} \ln (N+1) - \ln (2N) + \ln N \right\}^{-1} \ ,
\label{q0-jensen-1}
\end{equation}
are equal to the inverse of the maximum possible value of ${\mathcal J} [P,P_e]$.
This value is obtained when one of the components of $P$, say $p_m$, is equal to one and the remaining $p_j$
are zero.
The Jensen-Shannon divergence, which quantifies the difference between probability distributions,
is especially useful to compare the symbolic composition between different sequences\cite{Grosse2002}.
Note that the above introduced
SCM
depends on two different probability distributions: one associated with the system under analysis, $P$, and the other the uniform distribution, $P_e$.
Furthermore, it was shown that for a given value of ${\mathcal H}$, the range of possible ${\mathcal C}$
values varies between a minimum ${\mathcal C}_{min}$ and a maximum ${\mathcal C}_{max}$,
restricting the possible values of the
SCM\cite{Martin2006}.
Thus, it is clear that important additional information related to the correlational structure between the
components of the physical system is provided by evaluating the statistical complexity measure.
In this way, the information plane ${\mathcal H} \times {\mathcal C}$ constitute a nice tool to visualizate
and characterize different dynamical systems.
If our system lies in a very ordered state, which occurs when almost all the $p_{i}$--values are zeros
except for a particular state $k \neq i$ with $p_{k} \cong 1$, both
the normalized Shannon entropy and statistical complexity are close to zero (${\mathcal H} \approx 0$ and
${\mathcal C} \approx 0$), and the normalized Fisher's information measure is close to one (${\mathcal F} \approx 1$).
On the other hand, when the system under study is represented by a very disordered state, that is when all the
$p_{i}$--values oscillate around the same value, we have ${\mathcal H} \approx 1$ while
${\mathcal C} \approx 0$ and ${\mathcal F} \approx 0$.
One can state that the general FIM--behavior of the present discrete version
(Eq.~(\ref{Fisher-disc})),
is opposite to that of the Shannon entropy, except for periodic motions.
The local sensitivity of FIM for discrete--PDFs is reflected in the fact that the specific ``$i-$ordering"
of the discrete values $p_{i}$ must be seriously taken into account in evaluating the sum in
Eq.~(\ref{Fisher-disc}).
This point was extensively discussed by Rosso
and co-workers\cite{Olivares2012A,Olivares2012B}.
The summands can be regarded to as a kind of ``distance" between two contiguous probabilities.
Thus, a different ordering of the pertinent summands would lead to a different FIM-value, hereby its local nature.
In the present work, we follow the Lehmer lexicographic order\cite{Lehmer} in the generation
of Bandt and Pompe PDF (see next section).
Given the local character of FIM, when combined with a global quantifier as the normalized Shannon entropy,
conforms the Shannon--Fisher plane, ${\mathcal H} \times {\mathcal F}$, introduced by Vignat and Bercher\cite{Vignat2003}.
These authors showed that this plane is able to characterize the non-stationary behavior of a complex signal.
\subsection*{The Bandt and Pompe approach to the PDF determination}
\label{Sec:Bandt-Pompe}
The evaluation of the Information Theory derived quantifiers, like those previously introduced (Shannon entropy,
Fisher information and statistical complexity), suppose some prior knowledge about the system; specifically,
a probability distribution associated to the time series under analysis should be provided beforehand.
The determination of the most adequate PDF is a fundamental problem because the PDF $P$ and the sample space
$\Omega$ are inextricably linked.
Usual methodologies assign to each time point of the series ${\mathcal X}$ a symbol from a finite alphabet
$\mathfrak{A}$, thus creating a {\it symbolic sequence} that can be regarded to as a {\it non causal coarse grained}
description of the time series under consideration.
As a consequence, order relations and the time scales of the dynamics are lost.
The usual histogram technique corresponds to this kind of assignment.
{\it Causal information\/} may be duly incorporated if information about the past dynamics of the system is
included in the symbolic sequence, i.e., symbols of alphabet $\mathfrak{A}$ are assigned to a portion of the
phase-space or trajectory.
Many methods have been proposed for a proper selection of the probability space $(\Omega, P)$.
Bandt and Pompe (BP)\cite{Bandt2002} introduced a simple and robust symbolic methodology that takes into account
time causality of the time series (causal coarse grained methodology) by comparing neighboring values in a
time series.
The symbolic data are:
{\it (i)\/}~created by ranking the values of the series; and
{\it (ii)\/}~defined by reordering the embedded data in ascending order, which is tantamount to a phase space
reconstruction with embedding dimension (pattern length) $D$ and time lag $\tau$.
In this way, it is possible to quantify the diversity of the ordering symbols (patterns) derived from a scalar
time series.
Note that the appropriate symbol sequence arises naturally from the time series, and no model-based assumptions
are needed.
In fact, the necessary ``partitions'' are devised by comparing the order of neighboring relative values rather
than by apportioning amplitudes according to different levels.
This technique, as opposed to most of those in current practice, takes into account the temporal structure of the time series generated by the physical process under study.
As such, it allows us to uncover important details concerning the ordinal structure of the time
series\cite{Rosso2007,Rosso2012} and can also yield information about temporal correlation\cite{Rosso2009A,Rosso2009B}.
It is clear that this type of analysis of a time series entails losing details of the original series'
amplitude information.
Nevertheless, by just referring to the series' intrinsic structure, a meaningful difficulty reduction has
indeed been achieved by BP with regard to the description of complex systems.
The symbolic representation of time series by recourse to a comparison of consecutive ($\tau = 1$) or
nonconsecutive ($\tau > 1$) values allows for an accurate empirical reconstruction of the underlying phase-space,
even in the presence of weak (observational and dynamic) noise\cite{Bandt2002}.
Furthermore, the ordinal patterns associated with the PDF are invariant with respect to nonlinear monotonous
transformations.
Accordingly, nonlinear drifts or scaling artificially introduced by a measurement device will not modify the
estimation of quantifiers, a nice property if one deals with experimental data (see, e.g.,\cite{Saco2010}).
These advantages make the BP methodology more convenient than conventional methods based on range
partitioning, i.e., a PDF based on histograms.
To use the BP methodology\cite{Bandt2002} for evaluating the PDF, $P$, associated with the time
series (dynamical system) under study, one starts by considering partitions of the $D$-dimensional space that
will hopefully ``reveal'' relevant details of the ordinal structure of a given one-dimensional time series
${\mathcal X}(t) = \{ x_t; t = 1, \ldots, M\}$ with embedding dimension $D > 1$ ($D \in {\mathbb N}$)
and time lag $\tau$ ($\tau \in {\mathbb N}$).
We are interested in ``ordinal patterns'' of order (length) $D$ generated by
\begin{equation}
\label{asignation1}
(s)\mapsto \left(x_{s-(D-1)\tau},x_{s-(D-2)\tau},\ldots, x_{s-\tau},x_{s}\right) ,
\end{equation}
which assign to each time $s$ the $D$-dimensional vector of values at times $s, s-\tau,\ldots,s-(D-1)\tau$.
Clearly, the greater $D$, the more information on the past is incorporated into our vectors.
By ``ordinal pattern'' related to the time $(s)$, we mean the permutation $\pi=(r_0,r_1, \ldots,r_{D-1})$
of $[0,1,\ldots,D-1]$ defined by
\begin{equation}
\label{asignation2}
x_{s-r_{D-1}\tau} \le~x_{s-r_{D-2}\tau} \le \cdots \le~x_{s-r_{1}\tau} \le x_{s-r_0\tau} .
\end{equation}
We set $r_i < r_{i-1}$ if $x_{s-r_{i}} = x_{s-r_{i-1}}$ for uniqueness, although ties in samples from continuous
distributions have null probability.
For all the $D!$ possible orderings (permutations) $\pi_i$ when embedding dimension is $D$,
and time-lag $\tau$,
their relative
frequencies can be naturally computed according to the number of times this particular order sequence is found
in the time series, divided by the total number of sequences,
\begin{equation}
\label{eq:frequ}
p(\pi_i)= \frac{\# \{s|s\leq N-(D-1)\tau ; (s) \text{ is of type } \pi_i \}}{N-(D-1)\tau} ,
\end{equation}
where $\#$ denotes cardinality.
Thus, an ordinal pattern probability distribution $P = \{ p(\pi_i), i = 1, \dots, D! \}$ is obtained from the
time series.
Figure~\ref{fig:patrones} illustrates the construction principle of the ordinal patterns of length
$D=2$, $3$ and $4$ with $\tau = 1$\cite{Parlitz2012}.
Consider the sequence of observations $\{x_0, x_1, x_2, x_3\}$.
For $D=2$, there are only two possible directions from $x_0$ to $x_1$: up and down.
For $D=3$, starting from $x_1$ (up) the third part of the pattern can be above $x_1$, below $x_0$, or between
$x_0$ and $x_1$.
A similar situation can be found starting from $x_1$ (down).
For $D=4$, for each one of the six possible positions for $x_2$, there are four possible localizations for $x_3$,
yielding $D!=4!=24$ different possible ordinal patterns.
In Fig.~\ref{fig:patrones}, full circles and continuous lines represent the sequence values
$x_0 < x_1 > x_2 > x_3$, which leads to the pattern $\pi=[0321]$.
A graphical representation of all possible patterns corresponding to $D = 3, 4$ and $5$
can be found in Fig.~2 of Parlitz \textit{et al.}\cite{Parlitz2012}.
\begin{figure}[hbt]
\centering
\includegraphics[width = \linewidth]{esquema-permuta-0321}
\caption{Illustration of the construction principle for ordinal patterns of length $D$ \cite{Parlitz2012}.
If $D=4$ and $\tau=1$, full circles and continuous lines represent the sequence of values $x_0 < x_1 > x_2 > x_3$
which lead to the pattern $\pi=[0321]$.}
\label{fig:patrones}
\end{figure}
The embedding dimension $D$ plays an important role in the evaluation of the appropriate probability
distribution, because $D$ determines the number of accessible states $D!$ and also conditions the minimum acceptable
length $M \gg D!$ of the time series that one needs in order to work with reliable statistics\cite{Kowalski2007}.
Regarding the selection of the parameters, Bandt and Pompe suggested working with $4 \leq D \leq 6$, and specifically
considered a time lag $\tau = 1$ in their cornerstone paper\cite{Bandt2002}.
Nevertheless, it is clear that other values of $\tau$ could provide additional information.
It has been recently shown that this parameter is strongly related, if it is relevant, to the intrinsic time scales
of the system under analysis\cite{Zunino2010B,Soriano2011,Zunino2012}.
Additional advantages of the method reside in
{\it i)\/} its simplicity (it requires few parameters: the pattern length/embedding dimension $D$ and the time lag
$\tau$), and
{\it ii)\/} the extremely fast nature of the calculation process.
The BP methodology can be applied not only to time series representative of low dimensional dynamical systems,
but also to any type of time series (regular, chaotic, noisy, or reality based).
In fact, the existence of an attractor in the $D$-dimensional phase space in not assumed.
The only condition for the applicability of the BP method is a very weak stationary assumption: for $k \leq D$,
the probability for $x_t < x_{t+k}$ should not depend on $t$.
For a review of BP's methodology and its applications to physics, biomedical and econophysics signals see
Zanin {\it et al.\/}\cite{Zanin2012}.
Moreover,
Rosso {\it et al.\/}\cite{Rosso2007} show that the above mentioned quantifiers produce better descriptions
of the process associated dynamics when the PDF is computed using BP rather than using the usual histogram methodology.
The BP proposal for associating probability distributions to time series (of an underlying symbolic
nature) constitutes a significant advance in the study of nonlinear dynamical systems\cite{Bandt2002}.
The method provides univocal prescription for ordinary, global entropic quantifiers of the Shannon-kind.
However, as was shown by Rosso and coworkers\cite{Olivares2012A,Olivares2012B}, ambiguities arise in applying the
BP technique with reference to the permutation of ordinal patterns.
This happens if one wishes to employ the BP-probability density to construct local entropic quantifiers,
like the Fisher information measure, which would characterize time series generated by nonlinear dynamical systems.
The local sensitivity of the Fisher information measure for discrete PDFs is reflected in the fact that the specific
``$i$-ordering'' of the discrete values $p_i$ must be seriously taken into account in evaluating Eq.~(\ref{Fisher-disc}).
The numerator can be regarded to as a kind of ``distance'' between two contiguous probabilities.
Thus, a different ordering of the summands will lead, in most cases, to a different Fisher information value.
In fact, if we have a discrete PDF given by $P = \{ p_i, i = 1, \ldots , N\}$, we will have $N!$ possibilities
{for the $i$-ordering.}
The question is, which is the arrangement that one could regard as the ``proper'' ordering?
The answer is straightforward in some cases, the histogram-based PDF constituting a conspicuous example.
For such a procedure, one first divides the interval $[a, b]$ (with $a$ and $b$ the minimum and maximum amplitude
values in the time series) into a finite number on non-overlapping sub-intervals (bins).
Thus, the division procedure of the interval $[a, b]$ provides the natural order sequence for the evaluation of
the PDF gradient involved in the Fisher information measure.
In our current paper, we chose the lexicographic ordering given by the algorithm of Lehmer\cite{Lehmer},
among other possibilities, due to its better distinction of different
dynamics in the Shannon--Fisher plane, ${\mathcal H} \times {\mathcal F}$ (see\cite{Olivares2012A,Olivares2012B}).
\section*{Signature features and exploratory data analysis}
\label{Sec:Results}
Online handwritten classification and verification is an interesting and challenging classification problem.
On the one hand, intra-personal variation information can be large.
Some people provide signatures with poor consistency.
The speed, pressure and inclination, for example, pertaining to the signatures made by the same person can
differ greatly on regularity which makes it quite challenging to extract consistent features.
On the other hand, we can only obtain few samples from one person and no forgeries in practice.
This makes it very difficult to determine the reliability of extracted features.
The main idea is to construct an efficient classification scheme for data acquisition, or the reduction of
often unmanageable large datasets to a parsimonious form, without mislay important statistical information.
We aim at discovering relevant characteristic statistical structures which could be exploited if the key
information can be efficiently condensed into a suitable low-dimensional object.
The features we employ in this work are the Information Theory quantifiers already presented.
For each of the $k$ subjects ($k=1,\ldots,100$) in the database and its $j$ associated signatures
($25$ genuine and $25$
skilled
forgery), two associated time series ${\mathbf X}^{(k;\alpha)}_j$ and
${\mathbf Y}^{(k;\alpha)}_j$ are extracted and transformed into BP's PDFs with pattern length
(embedding dimension) $D = 5$ and time lag $\tau = 1$.
Note that the condition $M \gg D!$ its satisfied.
We denoted these PDFs as:
\begin{align*}
P_{X;j}^{(k;\alpha)}&= \text{ Bandt and Pompe's PDF~of } {\mathbf X}^{(k;\alpha)}_j |_{D,\tau}, \text{ and}\\
P_{Y;j}^{(k;\alpha)}&= \text{ Bandt and Pompe's PDF of } {\mathbf Y}^{(k;\alpha)}_j |_{D,\tau},
\end{align*}
in which $j=1, \ldots, 25$, and $\alpha = G, F$ identify genuine and skilled forgery signatures, respectively.
We computed the normalized permutation Shannon entropy ${\mathcal H}$,
the permutation statistical complexity ${\mathcal C}$,
and the permutation Fisher information measure
${\mathcal F}$ from these PDFs, and the obtained values are denoted as:
\begin{align*}
{\mathcal H}_{X;j}^{(k;\alpha)} &= {\mathcal H}[P_{X;j}^{(k;\alpha)}], &{\mathcal H}_{Y;j}^{(k;\alpha)} &= {\mathcal H}[P_{Y;j}^{(k;\alpha)}];\\
{\mathcal C}_{X;j}^{(k;\alpha)} &= \ {\mathcal C}[P_{X;j}^{(k;\alpha)}], &{\mathcal C}_{Y;j}^{(k;\alpha)} &= {\mathcal C}[P_{Y;j}^{(k;\alpha)}];\\
{\mathcal F}_{X;j}^{(k;\alpha)} &= {\mathcal F}[P_{X;j}^{(k;\alpha)}], &{\mathcal F}_{Y;j}^{(k;\alpha)} &= {\mathcal F}[P_{Y;j}^{(k;\alpha)}].
\end{align*}
We perform Exploratory Data Analysis (EDA)
on the Information Theory quantifiers
looking for simple descriptions of the data.
Apart from simple descriptive univariate measures, we use the Pearson correlation to measure the association between features.
This analysis was performed using the {\tt R} language and
platform version~3.2.1 (\url{http:\\www.R-project.org}).
Figure~\ref{Fig:HistEntropy} shows a scatterplot of the Entropy for both the genuine and skilled forgery signatures.
The $5000$ points correspond to $25$ genuine signatures (in blue) and $25$ forgery signatures
(in red) for each of the $100$ subjects.
Both types of signatures show similar association (Correlation):
${\rm Corr}({\mathcal H}_{X;j}^{(k;G)}, {\mathcal H}_{Y;j}^{(k;G)}) = 0.9665$ and
${\rm Corr}({\mathcal H}_{X;j}^{(k;F)}, {\mathcal H}_{Y;j}^{(k;F)}) = 0.9770$.
The
entropies of both types of signatures are overlapped and scattered elliptically.
However, the bivariate mean and dispersion values differ.
Entropies are less dispersed in the genuine than in the skilled forgery signatures,
a signal of the separability between them.
Marginal density plots show the distribution of entropy for each coordinate of both types of signatures.
These plots, however limited due to its marginal nature, reveal several modes, and suggest both wide and narrow
structures in the data.
\begin{figure}[hbt]
\centering
\includegraphics[width=0.9\linewidth]{HistEntropy}
\caption{Scatter plot with marginal kernel density estimates of entropy quantifiers in both trajectory coordinates
time series
${\mathbf X}$ and ${\mathbf Y}$.
Genuine (blue) and skilled forgery signatures (red points), 100 subjects.
Marginal kernel densities depict the distribution of entropy quantifiers along both axes.
}\label{Fig:HistEntropy}
\end{figure}
Figure~\ref{Fig:ContourEntropy} shows the contour plots of bivariate kernel density estimates for the
entropy in genuine and forgery signatures.
A number of features are immediately noticeable.
The dispersion in the former group is much smaller than in the latter (less than $0.4$).
The kernel density estimates reveal skewness and a mild multimodality in the joint distribution of the data.
There are also quite many points that are far from these curves and cluster centers.
These points correspond to abnormal local estimates obtained in heterogeneous blocks, possibly induced by the presence of clusters.
The modes in genuine signatures are smaller than in forgery signatures, and this may be used as discriminatory measure.
Similar results are obtained for the Complexity and the Fisher information;
these are
reported in the Supplementary Information, see Figs.~S1 to~S4, respectively.
\begin{figure}[hbt]
\centering
\includegraphics[width=.95\linewidth]{ContourEntropy}
\caption{Contour plot superimposed on the scatterplot of entropy quantifiers for genuine (right panel)
and skilled forgery signatures (left panel)}
\label{Fig:ContourEntropy}
\end{figure}
\section*{Signatures classification}
\label{Sec-Classification}
As pointed out by Boul\'etreal {\it et al.\/}\cite{Bouletreau1998}, a signature is characterized
by two aspects:
{\it a)\/}~a conscious one associated to the pattern signature; and
{\it b)\/}~an unconscious one which leads spontaneous movements constituting the drawing.
These two factors produce high variability, being the amount of signature variability
strongly writer-dependent.
In fact, the signature {\it variability\/} or, conversely, the signature {\it stability\/} can be considered
an important indicator for writer characterization\cite{Houmani2014}.
Houmani and Garcia-Salicetti\cite{Houmani2014} argue that signature stability is required in genuine signatures
in order to characterize a writer: the less stable a signature is, the more likely it is that forgery
will be dangerously close to genuine signatures for any classifier.
Also, complex enough signatures are required in order to guarantee a certain level of security, in the sense
that the more complex a signature is, the more difficult it will be to forge it\cite{Houmani2014}.
Boul\'etreal and collaborators\cite{Bouletreau1998,Vincent2000} propose a signature complexity measure related
to signature legibility and based on fractal dimension.
They classify writer styles into: highly cursive, very legible, separated, badly formed and small
writings, using only genuine signatures.
Unfortunately, such resulting categories were not confronted to classifiers for performance analysis.
We classify the genuine signatures based on causal Information Theory quantifiers:
Normalized Permutation Shannon Entropy,
Permutation Statistical Complexity and
Permutation Fisher Information Measure of both $\mathbf X$ and $\mathbf Y$ trajectories
on each of the one hundred writers in the MCYT data base, and their $25$ original signatures.
The mean and standard deviation values were clustered using the neighbor-joining method and an automatic Hierarchical Clustering
with the Euclidean distance-based dissimilarity matrix.
Each feature was treated independently, and the results are shown as circular dendrograms.
Figure~\ref{Fig:DendroEntropy} shows the results of clustering the Entropy.
We distinguish three classes of genuine signatures denoted by H1, H2, and H3.
\begin{figure}[hbt]
\centering
\includegraphics[width=.9\linewidth, angle=0]{DendogramEntropy}
\caption{
Neighbor-joining, rooted, circular dendrogram clustering of genuine signatures by Entropy:
H1, H2, and H3, in red, blue, and green, respectively.
}\label{Fig:DendroEntropy}
\end{figure}
The H1 group is the first group to form, i.e., the one comprised of the most similar individuals.
It is formed below the $25\%$ level, and it is composed by two subgroups: H1A and H1B.
The H1A group is formed exclusively by oversimplified signatures made by simple loops without identifiable letters.
It encompasses the following subjects: 1, 16, 17, 22, 23, 27, 29, 37, 83.
The same group is formed when the other features are used.
The H1B group is comprised of the following subjects: 2, 5, 8, 10, 19, 21, 24, 28, 32, 35, 36, 39, 43, 48,
49, 51, 55, 58, 59, 64, 69, 70, 74, 77, 89.
Although these are simplified signatures, traces of letters and/or more complex curves appear and differentiate them
from the members of the H1A group.
The H2 group is formed approximately at the $32\%$ level, and, again, it is comprised of two distinct groups:
H2A and H2B.
The subjects that make the H2A group are: 4, 7, 12, 15, 18, 20, 30, 31, 34, 38, 40, 41, 42, 52, 57, 60, 62,
66, 67, 68, 71, 73, 75, 79, 80, 81, 86, 87, 91, 96, 100.
It is composed by signatures with traces that resemble letters, but that are not perfectly identifiable,
and that include circling traces of large or moderate size.
Signatures in this group are kind of framed by large loops.
The H2B group is similar to the previous one, i.e., it is formed by signatures with large and medium size
circling traces, but with more identifiable letters than in the previous groups.
Names and surnames are more readable in this group than in previous ones.
It is formed by the following signatures: 6, 9, 13, 25, 33, 45, 50, 63, 65, 76, 78, 82, 84, 85, 88, 92, 94, 95, 97, 99.
The H3 group is formed at, approximately, the $43\%$ level by the fusion of two other highly unbalanced subgroups:
one, H3A, with only two subjects (44, 46) and the other, H3B, with thirteen subjects (3, 11, 14, 26, 47, 53, 54, 56, 61,
72, 90, 93, 98).
These two clusters form at approximately the same level.
The former is composed of calligraphic signatures where vertical traces predominate over horizontal ones.
The latter is composed of highly cursive signatures, where separation between the surname and the family name
predominates.
The same results of clustering was obtained with the Manhattan (norm ${\cal L}_1$) and Maximum distances ($\mathcal L_\infty$ norm), showing that Entropy is an expressive and stable quantifier.
Similar analyses were carried with the Permutation Statistical Complexity and Permutation Fisher Information
(presented in figures Figs.~S5 and~S6 in the Supplementary Information).
Complexity produces the same clusters identified by Entropy, so it adds no new information.
The Fisher information measure forms the same H1A group that was identified by the Entropy, but with less
cohesion, at about $15\%$.
In other words, these nine subjects are more similar locally than globally.
As with Entropy, three main groups form at similar levels.
The members of these clusters are slight variations of those identified using Entropy, with very similar structure.
Table~\ref{tab:tab-Measure-subject} presents
the mean and standard deviation of the three quantifiers over the $25$ genuine and $25$ skilled forgery signatures
(${\mathbf X}$ and ${\mathbf Y}$ time series) for each of the typical subjects, split in the three aforementioned types H1, H2 and~H3.
There are interesting tendencies in these data.
Genuine signatures present quantifiers values lower than those corresponding to forgery signatures,
and the latter also exhibit larger standard deviation.
This could be explained by the imitative character of these signatures, however it deserves closer studies.
\begin{sidewaystable}[hbt] \centering
\centering
\begin{tabular}{@{\extracolsep{5pt}} ccccccc|cc|cc}
\\ \cmidrule(r){6-11}
& & & & & \multicolumn{2}{c|}{Entropy}&\multicolumn{2}{c|}{Complexity}&\multicolumn{2}{c}{Fisher Information} \\ \midrule
$Type$ & $Sub-Type$ & Subject & Coordinate & Class & Mean & S.D. & Mean & S.D. & Mean & S.D. \\
\midrule
\multirow{8}{*}{H1} & \multirow{4}{*}{H1A} & \multirow{4}{*}{22} & \multirow{2}{*}{${\mathbf X}$} & F & $0.1568$ & $0.0052$ & $0.1490$ & $0.0039$ & $0.4688$ & $0.0070$ \\
& & & & G & $0.1519$ & $0.0019$ & $0.1457$ & $0.0015$ & $0.4766$ & $0.0035$ \\ \cmidrule(r){4-11}
& & & \multirow{2}{*}{${\mathbf Y}$} & F & $0.1595$ & $0.0071$ & $0.1511$ & $0.0052$ & $0.4665$ & $0.0097$ \\
& & & & G & $0.1512$ & $0.0042$ & $0.1447$ & $0.0037$ & $0.4734$ & $0.0046$ \\
\cmidrule(r){3-11}
& \multirow{4}{*}{H1B} &\multirow{4}{*}{39} & \multirow{2}{*}{${\mathbf X}$} & F & $0.2212$ & $0.0384$ & $0.1941$ & $0.0257$ & $0.4286$ & $0.0147$ \\
& & & & G & $0.1749$ & $0.0037$ & $0.1620$ & $0.0028$ & $0.4497$ & $0.0029$ \\ \cmidrule(r){4-11}
& & & \multirow{2}{*}{${\mathbf Y}$} & F & $0.2270$ & $0.0449$ & $0.1980$ & $0.0296$ & $0.4277$ & $0.0153$ \\
& & & & G & $0.1776$ & $0.0043$ & $0.1644$ & $0.0031$ & $0.4491$ & $0.0035$ \\
\midrule
\multirow{8}{*}{H2} & \multirow{4}{*}{H2A} &\multirow{4}{*}{60} & \multirow{2}{*}{${\mathbf X}$} & F & $0.2482$ & $0.0593$ & $0.2112$ & $0.0365$ & $0.4212$ & $0.0107$ \\
& & & & G & $0.2010$ & $0.0056$ & $0.1803$ & $0.0040$ & $0.4331$ & $0.0031$ \\ \cmidrule(r){4-11}
& & & \multirow{2}{*}{${\mathbf Y}$} & F & $0.2442$ & $0.0544$ & $0.2090$ & $0.0339$ & $0.4219$ & $0.0134$ \\
& & & & G & $0.2079$ & $0.0043$ & $0.1861$ & $0.0030$ & $0.4315$ & $0.0024$ \\
\cmidrule(r){3-11}
& \multirow{4}{*}{H2B}&\multirow{4}{*}{6} & \multirow{2}{*}{${\mathbf X}$} & F & $0.2621$ & $0.0584$ & $0.2194$ & $0.0334$ & $0.4143$ & $0.0137$ \\
& & & & G & $0.2337$ & $0.0149$ & $0.2032$ & $0.0095$ & $0.4205$ & $0.0066$ \\ \cmidrule(r){4-11}
& & & \multirow{2}{*}{${\mathbf Y}$} & F & $0.2648$ & $0.0538$ & $0.2218$ & $0.0304$ & $0.4136$ & $0.0134$ \\
& & & & G & $0.2314$ & $0.0102$ & $0.2018$ & $0.0067$ & $0.4211$ & $0.0050$ \\
\midrule
\multirow{8}{*}{H3} & \multirow{4}{*}{H3A} &\multirow{4}{*}{98} & \multirow{2}{*}{${\mathbf X}$} & F & $0.3236$ & $0.0646$ & $0.2529$ & $0.0320$ & $0.3937$ & $0.0208$ \\
& & & & G & $0.2707$ & $0.0101$ & $0.2268$ & $0.0064$ & $0.4106$ & $0.0032$ \\ \cmidrule(r){4-11}
& & & \multirow{2}{*}{${\mathbf Y}$} & F & $0.3204$ & $0.0794$ & $0.2497$ & $0.0388$ & $0.3970$ & $0.0208$ \\
& & & & G & $0.2664$ & $0.0124$ & $0.2243$ & $0.0077$ & $0.4105$ & $0.0034$ \\
\cmidrule(r){3-11}
& \multirow{4}{*}{H3B}&\multirow{4}{*}{46} & \multirow{2}{*}{${\mathbf X}$} & F & $0.3514$ & $0.0641$ & $0.2691$ & $0.0294$ & $0.3940$ & $0.0156$ \\
& & & & G & $0.3480$ & $0.0282$ & $0.2720$ & $0.0156$ & $0.4019$ & $0.0047$ \\ \cmidrule(r){4-11}
& & & \multirow{2}{*}{${\mathbf Y}$} & F & $0.3419$ & $0.0681$ & $0.2639$ & $0.0323$ & $0.3940$ & $0.0163$ \\
& & & & G & $0.3270$ & $0.0263$ & $0.2599$ & $0.0148$ & $0.4008$ & $0.0052$ \\
\bottomrule
\end{tabular}
\caption{Sample mean and standard deviation (S.D.) of the time series quantifiers for the 25 genuine (G)
and 25
skilled
forged (F) signatures, for each of the typical subjects: H1A, H1B, H2A, H2B, H3A, and H3B (same order as in Fig.~\ref{fig:MCYT-firmas}).}
\label{tab:tab-Measure-subject}
\end{sidewaystable}
The classification into subclasses of genuine signatures was also carried by the parallelepiped
algorithm\cite{Richards1999}, arguably the simplest model-free classification procedure.
Entropy leads to clusters with nice interpretability.
Figure~\ref{Fig:EntropyClasificationBoxesEntropy} shows the regions that define the three classes
identified by the dendrogram based on Entropy presented in Fig~\ref{Fig:DendroEntropy}.
All subclasses are well separated by disjoint boxes, with the only exception of H1B and H2A that
overlap slightly but without compromising the discrimination.
The classes are preserved using this classification superimposed with Complexity and Fisher Information features; see Figs.~S7 and~S8 in the Supplementary Information.
\begin{figure}[hbt]
\centering
\includegraphics[width=.9\linewidth, angle=0]{ClassificationWithSignatureTrue}
\caption{Classification by the rule of the parallelepiped of genuine signatures using Entropy
(one signature example from each of the three groups is shown).
Each subject is identified by its ID.
}\label{Fig:EntropyClasificationBoxesEntropy}
\end{figure}
\section*{Online signature verification}
\label{Sec-Verification}
The problem we have at hand consists in identifying suspicious signatures, given that we only have examples from genuine signatures.
This is due to the fact that, in practice, it is too expensive, too hard or even impossible to obtain a significant number of good quality forgery signatures for every possible individual in the data base.
This, thus, configures a One-Class classification problem.
Among the many ways of tackling such problems, Support Vector Machines (SVMs) are suitable for solving machine learning problems even in large dimensional feature spaces\cite{Campbell2011,Boser1992,Vapnik1995}.
SVMs were introduced by Vapnik and co-workers\cite{Boser1992,Vapnik1998},
and extended by a number of other researchers.
Their remarkably robust performance with respect to sparse and noisy data makes them the choice in several applications.
A SVM is primarily a method that performs classification tasks by
constructing hyperplanes in a multidimensional space that separates cases of different class labels.
SVMs perform both regression and classification tasks and can handle multiple continuous and categorical
variables.
To construct an optimal hyperplane, a SVM employs an iterative training algorithm, which is used
to minimize an error function.
One-Class Support Vector Machines (OC-SVMs) are a natural extension of SVMs\cite{Scholkopf2001, Scholkopf2002}.
The solution consists in estimating a distribution that encompasses most of the observations, and then labeling as ``suspicious'' those that lie far from it with respect to a suitable metric.
An OC-SVM solution is built estimating a probability distribution function which makes most of the observed data more likely than the rest, and a decision rule that separates these observation by the largest possible margin.
The computational complexity of the learning phase is intensive because the training of an OC-SVM involves a quadratic programming problem\cite{Boser1992}, but once the decision function is determined, it can be used to predict the class label of new test data effortlessly.
In our case, the observations are six-dimensional vectors: Entropy, Complexity and Fisher Information in each of the two directions, horizontal and vertical, and
we train the OC-SVM with genuine signatures.
Let ${\cal Z} = \{ z_1, z_2, \dots , z_N \}$ be the six-dimensional training examples of genuine signatures.
Let $\Phi \colon {\cal Z} \rightarrow {\cal G}$ be a kernel map which transforms the
training examples to another space.
Then, to separate the data set from the origin, one needs to solve the following quadratic programming problem:
\begin{equation}
\label{eq:oc-svm-1}
\min_{{\mathbf{w}\in {\cal G},~\xi_i , b \in \mathbb{R}}} \qquad
\left\{\frac{1}{2} \|\mathbf{w}\|^2 + \frac{1}{\nu N} \sum_{i=1}^N \xi_i - b\right\}
\end{equation}
subject to
\begin{align}
& \nu \in (0,1], \ \xi_i \ge 0, \ \forall i=1,\dots, N \label{eq:oc-svm-2}, \text{ and}\\
& (\mathbf{w} \cdot \Phi(z_i) ) \geq b- \xi_i, \ \forall i=1,\dots, N ,
\label{eq:oc-svm-3}
\end{align}
where $\xi_i$ are nonzero slack variables which allow the procedure to incur in errors.
The parameter $\nu$ characterizes the solution as
{\it a)\/} it sets an upper bound on the fraction of outliers (training examples regarded out-of-class) and,
{\it b)\/} it is a lower bound on the number of training examples used as Support Vectors.
We used $\nu=0.1$ in our proposal.
Using Lagrange techniques and a kernel function $K(z,z_i) = \Phi(z)^T \Phi(z_i)$, for the dot-product
calculations, the decision function $f(z)$ becomes:
\begin{equation}
\label{eq:oc-svm-primal-3}
f(z) = \text{sign}\left\{(\mathbf{w}\cdot \Phi(z)) - b \right\} =
\text{sign}\left\{ \sum_{i=1}^{N}~\alpha_i ~K(z,z_i) - b \right\}.
\end{equation}
This method thus creates a hyperplane characterized by $\mathbf{w}$ and $b$ which has maximal distance
from the origin in the feature space $\cal G$ and separates all the data points from the origin.
Here $\alpha_i$ are the Lagrange multipliers; every $\alpha_i >0 $ is {\it weighted in\/} the decision function and thus ``supports" the machine; hence the name Support Vector Machine.
Since SVMs are considered to be sparse, there will be relatively few Lagrange multipliers with a nonzero value.
Our choice for the kernel is the Gaussian Radial Base function:
\begin{equation}
\label{eq:oc-svm-primal-4}
K(z_i,z_j) = \exp \Big(-\frac{1}{2\sigma^2} \|z_i - z_j\|^2
\Big) ,
\end{equation}
where $\sigma \in {\mathbb{R}}$ is a kernel parameter and ${\|z_i - z_j\|^2}$ is the
dissimilarity measure; we used Euclidean distance.
The parameter $\sigma^2 = {10}$ was selected by 5-fold-cross validation, that its, the dataset is divided
into five disjoint subsets, and the method is repeated five times.
Each time, one of the subsets is used as the test set and the other four subsets are put together to form the training set.
Then the average error across all trials is computed.
Every observation belongs to a test set exactly once, and belongs to a training set four times.
Accuracy (ACC), Area Under the ROC Curve (AUC) and Equal Error Rate (EER) are used as performance measures~\cite{Kohavi1998}.
In the context of signature verification one-class classification problems, a false positive occurs when a genuine signature is erroneously classified as being atypical.
The probability of false positive misclassification is the false positive rate, which is controlled by the parameters $\nu$ in the aforementioned OC-SVM formulation.
The parameter $\nu$ can be fixed a {\it priori} and it corresponds to the percentage of observations of the typical data which will be assigned as the Type~I Error.
We used the LIBSVM (version 2.0) tool, linked with the R software, that supports vector classification and regression, including OC-SVM.\cite{Chang2001}
We used the standard parameters of the algorithm.
In order to assess the consistency of our procedure, and to promote the comparison with other methods reported in the literature, we evaluate the performance of the proposed verification system for different training samples:
random samples of size $n$ ($n=5, 10, 14, 18, 22$) of
genuine signatures were selected for each user.
Table~\ref{tab:ntrain} presents the average value of all performance metrics using $\sigma^2=10$.
ACC suggests that the larger the training sample, the better the performance is.
AUC presents a similar tendency, and its average is larger than $0.88$, indicating that our verification system produces an excellent classification.
\begin{table}[hbt]
\centering
\begin{tabular}{rccc}
\toprule
$n$ & ACC ($\uparrow$) & AUC ($\uparrow$) & EER ($\downarrow$)(\%)\\
\midrule
5 & 0.6940 & 0.8816 & 0.1890 \\
10 & 0.7678 & 0.8940 & 0.1711 \\
14 & 0.8144 & 0.8975 & 0.1634 \\
18 & 0.8250 & 0.8866 & 0.1731 \\
22 & 0.8389 & 0.8909 & 0.1632 \\
\bottomrule
\end{tabular}
\vspace{0.25in}
\caption{Performance of the system trained with varying number $n$
of samples of genuine signatures.
$\uparrow$ and $\downarrow$ denote measures of quality (the higher the better) and of error (the smaller the better), respectively.
}
\label{tab:ntrain}
\end{table}
As mentioned in the introduction, the two methodologies with best results are those based on Dynamic Time Warping (DTW) and Hidden Markov Models (HMM).
In the following we compare our proposal with these two recent state-of-the-art methods using the ERR(\%) over the same data base:
\begin{itemize}
\item Fierrez-Aguilar {\it et al.\/}\cite{Fierrez2005}, ERR(\%) = 2.12 (five training signatures; Global (Parzen WC)
and local (HMM) experts function);
\item Fierrez-Aguilar {\it et al.\/}\cite{Fierrez2007}, ERR(\%) = 0.74 (ten training signatures; HMM based algorithm);
\item Pascual-Gaspar {\it et al.\/}\cite{Pascual2009}, ERR(\%) = 1.23 (five training signatures; DTW-bases algorithm,
result with scenario-dependent optimal features.
\end{itemize}
The results of our proposal using five (ten, respectively) training samples, are ERR(\%) = 0.19 (0.17, respectively).
Clearly, our system provides better performance using similar number of training signatures (see Table~\ref{tab:ntrain} for more details).
In the following we analyze the performance of the proposed procedure applied selectively to the pre-classified samples.
Table~\ref{tab:class} presents the performance of the system when applied to genuine pre-classified signatures.
For all classes we observe that the larger the training sample, also the larger the average ACC is.
The best average AUC are observed for the class H2, followed by H1 and H3.
This indicate that H2 signatures are easily identifiable.
Note that the mean values of ERR(\%) for H2 are smaller than H1 and H3.
The ERR(\%) values in H3 indicate that identifying forgeries in this class is hard.
\begin{table}[hbt]
\centering
\begin{tabular}{crccc} \toprule
Class & $n$ & ACC ($\uparrow$) & AUC ($\uparrow$) & EER(\%) ($\downarrow$) \\
\midrule
\multirow{5}{*}{H1}
& 5 & 0.6758 & 0.8692 & 0.1976 \\
&10 & 0.7566 & 0.8828 & 0.1812 \\
&14 & 0.8039 & 0.8857 & 0.1717 \\
&18 & 0.8217 & 0.8894 & 0.1662 \\
&22 & 0.8277 & 0.8788 & 0.1631 \\
\midrule
\multirow{5}{*}{H2}
& 5 & 0.7059 & 0.8945 & 0.1784 \\
&10 & 0.7819 & 0.9079 & 0.1548 \\
&14 & 0.8284 & 0.9096 & 0.1509 \\
&18 & 0.8327 & 0.8900 & 0.1734 \\
&22 & 0.8515 & 0.8996 & 0.1608 \\
\midrule
\multirow{5}{*}{H3}
& 5 & 0.6948 & 0.8653 & 0.2053 \\
&10 & 0.7450 & 0.8720 & 0.2036 \\
&14 & 0.7907 & 0.8832 & 0.1874 \\
&18 & 0.8062 & 0.8686 & 0.1874 \\
&22 & 0.8214 & 0.8889 & 0.1716 \\
\bottomrule
\end{tabular}
\caption{Performance of the classification of pre-classified samples varying the number $n$ of
samples of genuine signatures used for training; same coding as in Tab.~\ref{tab:ntrain}.}
\label{tab:class}
\end{table}
\section*{Conclusions}
\label{Sec:Conclusions}
We proposed a procedure for identifying skilled forgery online handwritten signatures using time causal Information Theory quantifiers and One-Class Support Vector Machines.
This is a competitive proposal from the computational viewpoint as it uses only the signatures coordinates, and it produces better results than state-of-the-art techniques.
The technique also produces meaningful classification of the input data, as it is able to separate different types of signatures.
To the best of our knowledge, this is the first time Information Theory quantifiers have been used for this problem.
The central contribution is the use of the Bandt and Pompe (BP) PDF symbolization which is invariant to a number of transformations of the input data.
In fact, the original time series are pre-processed only to facilitate the signal sampling, and this scaling has no effect on the BP PDFs.
This representation, which is sensitive to the time causality, is able to capture essential dynamical characteristics of the signatures that lead to excellent discrimination between skilled forgery and genuine online handwritten signatures, despite the high variability the data possess.
Additionally, obtaining the BP PDFs is computationally simple and efficient.
Only six Information Theory features are required for the classification, three from each horizontal and vertical direction:
Shannon Entropy, Statistical Complexity and Fisher Information.
This contrasts many state-of-the-art works that require features in high-dimensional spaces, e.g. forty or even more.
As said, our proposal does not require highly specialized hardware able to capture signature speed, pressure, orientation etc.
The classification was performed by a One-Class Support Vector Machine trained with genuine signatures.
The learned rule is consistent with respect to the number of training samples, and with as few as five examples it surpasses the performance of recent successful techniques.
We assessed the performance of our proposal using the same data base employed in the current literature, with also the same measures of quality and error.
\section*{Acknowledgments}
The authors are grateful to CONICET, CNPq and FACEPE for partial funding of this research.
The Biometrics Research Lab (ATVS), Universidad Aut\'onoma de Madrid, provided the MCYT-100 signature corpus employed in this work.
\section*{Authors Contributions}
OAR, RO and ACF conceived and designed the research.
OAR performed the numerical data analysis.
RO and ACF performed the statistical analysis.
OAR and RO prepared figures.
OAR and ACF wrote the manuscript.
All authors reviewed and approved the manuscript
\section*{Competing interests}
The authors declare no competing financial interests.
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{"url":"https:\/\/web2.0calc.com\/questions\/hard-algebra_22","text":"+0\n\n# hard algebra\n\n0\n34\n1\n\nIf $g(x)=\\sqrt[3]{\\frac{x+3}{4}}$, for what value of x\u00a0will g(2x)=g(x)? Express your answer in simplest form.\n\nJun 26, 2022\n\n#1\n+282\n+1\n\n$$\\displaystyle g(x)=\\sqrt[3]{\\frac{x+3}{4}}$$, for what value of x will g(2x) = g(x)? x=0\n\nJun 27, 2022\n\n$$\\displaystyle g(x)=\\sqrt[3]{\\frac{x+3}{4}}$$, for what value of x will g(2x) = g(x)? x=0","date":"2022-08-07 19:20:45","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\": 0.9308417439460754, \"perplexity\": 5442.090789889278}, \"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-2022-33\/segments\/1659882570692.22\/warc\/CC-MAIN-20220807181008-20220807211008-00736.warc.gz\"}"}
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{"url":"http:\/\/mathhelpforum.com\/differential-geometry\/129690-rolles-theorem-print.html","text":"# Rolle's Theorem\n\n\u2022 Feb 19th 2010, 08:10 PM\nABigSmile\nRolle's Theorem\nThis is not exactly Rolle's Theorem but it's a problem that was given to me that is similar that is bugging me. It states \"if $f$ is differentiable on $(a,b)$, and $f(a) = f(b) = 0$, then $f$ is uniformly continuous on $[a,b]$\" I understand since $f$ is differentiable on $(a,b)$ that it must also be continuous on $(a,b)$.\nBut continuity doesn't imply uniform continuity. So is there any way to show uniform continuity? Because if there is, I haven't been able to figure it out so far. Is this statement always false since our interval is open? Would providing a simple counter example suffice if that's the case?\n\u2022 Feb 19th 2010, 08:17 PM\nDrexel28\nQuote:\n\nOriginally Posted by ABigSmile\nThis is not exactly Rolle's Theorem but it's a problem that was given to me that is similar that is bugging me. It states \"if $f$ is differentiable on $(a,b)$, and $f(a) = f(b) = 0$, then $f$ is uniformly continuous on $[a,b]$\" I understand since $f$ is differentiable on $(a,b)$ that it must also be continuous on $(a,b)$.\nBut continuity doesn't imply uniform continuity. So is there any way to show uniform continuity? Because if there is, I haven't been able to figure it out so far. Is this statement always false since our interval is open? Would providing a simple counter example suffice if that's the case?\n\nI don't understand the question? Does $f$ have to be continuous on $[a,b]$ otherwise this follows directly since any continuous function on a compact space is uniformly continuous.\n\u2022 Feb 19th 2010, 08:27 PM\nABigSmile\nQuote:\n\nOriginally Posted by ABigSmile\n\"if $f$ is differentiable on $(a,b)$, and $f(a) = f(b) = 0$, then $f$ is uniformly continuous on $[a,b]$\"\n\nThe problem I am trying to solve is this. It doesn't say prove it so I am assuming it could be a true or false statement. Also I am pretty sure $f$ does not have to be continuous on $[a,b]$ .Does that clear anything up? Sorry for the confusion.\n\u2022 Feb 19th 2010, 08:31 PM\nDrexel28\nQuote:\n\nOriginally Posted by ABigSmile\nThe problem I am trying to solve is this. It doesn't say prove it so I am assuming it could be a true or false statement. Also I am pretty sure $f$ does not have to be continuous on $[a,b]$ .Does that clear anything up? Sorry for the confusion.\n\nOk. So why doesn't the function $f(x)=\\begin{cases} 0 & \\mbox{if} \\quad x=0,1\\\\ 1 & \\mbox{if} \\quad 0 serve as a counter example?. Clearly $f(a)=f(b)=0$ and $f$ is differentiable on $(0,1)$ but it isn't continuous, let alone uniformly continuous, on $[0,1]$.","date":"2017-01-19 09:04:42","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\": 32, \"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\": 0.9121299386024475, \"perplexity\": 150.34451696026485}, \"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-2017-04\/segments\/1484560280504.74\/warc\/CC-MAIN-20170116095120-00503-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
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Review: Naughty Bear
By Brittany Vincent On July 20, 2010 In Review, Xbox 360 With Comments Off on Review: Naughty Bear Permalink
Naughty Bear is a sociopath. When he isn't invited to a fellow bear's soiree (Daddles' birthday party; the event of the year, no doubt) he deals with this rejection in what is perhaps the most unhealthy manner one could think of: the mass murder of a quaint little community of bears. Armed with a variety of weapons (axes, pistols, baseball bats, you name it) he's on a mission to punish each and every fluffy cuddle buddy on Perfection Island who ever dared to cross him. But he can't do it alone. That's where you come in. In 505 Games' latest stab at an original release, Naughty Bear, takes up the mantle of the "naughtiest" bear of them all. He's crass, angry, and, well, naughty. He's suffered humiliation and anguish at the hands of his so-called friends and neighbors, and it's your job to ensure justice is served…in that psychotic, no-regard-for-others kind of way.
The game advertises quite the rollicking premise: beat the stuffing out of your mortal enemies in a "cuter," more tongue-in-cheek way than, say, games like Rampage or Manhunt. Unfortunately, like in childhood, that's where the sidewalk ends. Naughty Bear, despite valiant attempts at accomplishing otherwise, is a veritable exercise in mediocrity. Only two levels into my review run-through, it became the less favorable option to going to work, mowing the lawn, or organizing my sock drawer. That's when I knew there was a problem.
Naughty Bear is all about beating your fellow teddy bears to death, burying machetes in their stomachs, setting them ablaze, and eventually driving them to suicide. You know, kid stuff. I was shocked when I learned the game is only rated T by the ESRB, but then realized that eviscerating such cuddly victims mustn't merit the same hard 'M' that games celebrating more realistic violence and mayhem are slapped with. The game spans seven main levels, each with their own subset of missions with unique goals to accomplish. You might need to track down a certain bear ripe for "punishment," complete an area without taking damage (difficult to do with collision detection this horrible), or refrain from killing anything at all – a frustrating exercise in self-restraint.
Each level opens with a sprinkling of a "story" to guide to the game's wanton destruction. First, he wasn't invited to a birthday party. Next, he gets tangled up in some dirty politics. It's really all downhill from there. If you make it through all seven episodes, you'll find aliens, ninjas, and zombies to contend with in the exact same manner you yawned through previously. There's little variation in the fresh meat, er, bears save for color or decor such as hats, police garb, etc. Get used to the island and its inhabitants. You're going to become good friends.
To enable Naughty Bear's terrifying behavior, you'll pick up several different items scattered across the island: baseball bats, sticks, axes, and even pistols. Pick up a weapon (or use your bear paws) and proceed to bash skulls in. You can button mash a bear to death and quickly use the right trigger to perform a kill with your weapon of choice, or utilize several points littered across the maps to pull off environmental kills, none of which are particularly entertaining but certainly predictable. It's a thrill to watch a bear bury an axe in another's skull…the first few times. Anything after that begins to grate on the nerves. Yeah, it was thrilling to watch dressphere transformations in Final Fantasy X-2 while the spectacle was fairly new. Started to skip them after the first thousand times though, didn't you?
The game gives the illusion of variety, when in reality each and every mission requires much of the same thing of you, in the same locations, using the same weapons. For example, technically you can play Naughty Bear one of three ways. You can can go on an all-out rampage to take out as many bears as fast as you can to feed your combo meter, chaining senseless violence together with an absolutely lazy, NOT scary "BOO!" in an attempt to scare the plush toys out of their wits for extra points. Smash balloons, statues, and household fixtures. Burn precious gifts found around the island. Decimate the place and maximize the damage before moving on to the next area.
Option two? Take your time, creeping through the grass where Naughty Bear will automatically attempt to cover himself with foliage. Sabotage the bears' belongings and catch them unaware either to pull of a particularly heinous kill or scare them silly, perhaps enough so that they pull the trigger on their own little fluffy heads. In short, channel your inner Garrett. Taff your way through the homes of your half-wit victims and ensure they walk into their own demise.
The third way? It's less an option on its own than just straight advice: play a combination of both tactics because Naughty Bear is all about maximizing points via multipliers and combo chains. In essence, anything that sees you taking the time to attempt creativity in killing off the bears is discouraged. I found that I made much quicker progress simply by planting an axe in each bear I found, quickly triggering a kill, then rinsing and repeating. I applaud the direction the game touts as the "best" way to play — premeditation, cunning, and imaginative ways to toy with your prey — but it becomes apparent early on that running around like a furry serial killer is what's necessary to really get by, especially since you need to participate in sub-challenges in order to unlock the main episodes. They don't unlock through sheer will alone and those party hats and Naughty Bear costumes won't magically appear unless you put in a little work. Work that you aren't going to want to do.
If it weren't enough that the game practically forces you to play in a manner that isn't entertaining, it's full of frustrating glitches and game-breaking freezes. Not once, twice, or even three times was I plagued by my entire game freezing upon journeying to the next part of the island or Naughty Bear's lair, but a whopping eight times before I just gave up. The first few were misdemeanors – I chalked them up to my aging Xbox 360, but research confirmed these issues are present in the PlayStation 3 version as well, I was increasingly ready to begin using the disc as a coaster. It's this kind of issue that consistently breaks any sort of feeling of involvement I had with the game as a "real" adventure or even an enjoyable little beat-'em-up. Let's not forget the countless environmental kills where the stodgy camera decided to focus on the trees behind Naughty Bear's head rather than the carnage going on in front of the player.
All of these factors simply blended together to convince me that my time was better spent elsewhere. And should you encounter them, I have a feeling your mind will be made up for you as well. It makes you want to grab your own teddy bear and shed a few tears, doesn't it? I am deeply disappointed, as I initially and quite gleefully followed the many trailers and niceties like classic horror costumes for Naughty Bear with a song in my heart.
The shallow gameplay mechanics, shoddy graphics, and miserable excuse for a commentator were the cherry on top of a cake filled with unexpected, bitter jelly. I find it absolutely laughable that it is expected you will pay full price for an inane adventure that would be better served as a throwaway XBLA or PSN title. When even dark humor and over-the-top violence can't save your game, perhaps it's a good idea to downsize an intended full retail release. Oh ho ho! The bears "bleed" stuffing and "cute" cop bears come to the island to try and bring order to the havoc. One cop, mind you. It's hilarious because these cute things should not be violent! No. It's not. It could have been. It could have been a blast! Unfortunately, all the could haves and would haves can't save this product from what it's destined for: a tomb at the bottom of the bargain bin.
With a title and a premise like Naughty Bear, it seems impossible to walk away from the game with the feeling that you wasted your money. But as Juliet lamented, "what's in a name?" In this case, everything. In fact, that's all this game really has going for it. Unfortunately, that's not enough to save it from absolute, soul-sucking mediocrity. And it pains me to say this as I'm one who routinely looks for releases that go above and beyond the call of duty (both literally and figuratively) to bring something new to the table. This excellent premise is wasted with clunky, repetitive gameplay and a host of other irreparable issues. And it's such a shame that what could have been a delightful, tongue-in-cheek jab at games such as Manhunt or even Grand Theft Auto is really, like its cast of characters, so full of fluff. Perhaps it's time for Naughty Bear to be locked away for good.
← Review: Trinity Universe
Impressions: Blacklight: Tango Down →
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CES 2016| Forget Rearview Mirrors! Upcoming Cars Will Have 3D Cameras Instead
Connie Nguyen
on January 11, 2016 at 9:29 am
You can't change the world from the rearview mirrors. — Anita Roddick
They may not change the world, but the new style of mirrors will definitely give you a new perspective!
Soon rearview mirrors will be just a memory thanks to the Department of Transportation's mandate for rear-view cameras in all cars built after May 1, 2018. Many automakers have been ready for this change, including Cadillac, Audi, Tesla, and the latest name in that list is BMW.
At CES 2016 in Las Vegas, Nevada, BMW introduced the i8 Mirrorless concept, which is designed with three cameras to replace the mirrors. A brilliant idea to keep you safe on the road! Using these three cameras will offer a much-wider-angle image of what's behind you by just a glance. A display in place of the conventional rearview mirror gives you a full view of cars around you, including vehicles in your blind spot.
The i8 Mirrorless uses wing-like side cameras, which serve as turn signal indicators, giving the car a more aerodynamic look. Check them out in these images:
Connie loves to hear your feedback, so feel free to email her or add her on LinkedIn.
AutomotiveFast & FuriousFeatured PostsGreen + Sexy
BMWbmw i8car conceptcesfuture of fastgreen and sexytechdrive
Connie has adored words since she learned how to rhyme at age three. She is currently a Freelance Copywriter/Editor in the San Francisco Bay Area. Before that, Connie had 6+ years working for WPP & Dentsu Group in Vietnam. She yearns to be a part of this creative juggernaut that will echo throughout the generations, getting people to think about new technology and venture outside their comfort zones. She is also a soldier in the fight against mediocrity. In addition to making ads, Connie plays piano, swing dances, and mountain bikes. She studied Communications at RMIT in Melbourne, Australia, where she finds it horribly inconvenient not to shout at parties.
This Colorful E-Citroen Will Soon Be Yours
CES 2016| After The Big Scandal, VW Introduced Its Flagship Smart Car Concept
Tech-In-Motion | An Ingenious Sidecar To A Stroller
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{"url":"https:\/\/physics.stackexchange.com\/questions\/439853\/amount-of-steam-generated-using-gas-burner-and-induction-cooker","text":"# Amount of steam generated using gas burner and Induction cooker\n\nI did an experiment by boiling water using two different heat sources. At first, I boiled water using a gas stove, then I repeated by using an induction cooker. I noticed that the amount of steam (visible vapour) formed is not the same. When using induction cooker, I could see a massive amount of steam but very little when I boil water using gas burner. For both cases, Im using the same pot size and the room condition is the same. I'm interested to know why there is different in steam formed.\n\n\u2022 Lots of standard induction cooktops can easily put out much more power than a typical gas cooktop. Could the difference in the amount of steam you're seeing simply be explained by the fact that the induction cooktop may be boiling off water at a significantly faster rate than the gas cooktop? \u2013\u00a0Samuel Weir Nov 9 '18 at 17:58\n\u2022 Hi Samuel, the induction cooktops that im using is only 7kw. But the capacity of the gas burner that im using in this test is about 40kw, commercial high pressure gas burner. The purpose im carry out this test is i thought that it could generate more visible steam using gas burner because the power is much higher. But it turned out another way. \u2013\u00a0cktan Nov 12 '18 at 0:46","date":"2019-10-19 03:10:39","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\": 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\": 0.5710131525993347, \"perplexity\": 770.7493515552778}, \"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-2019-43\/segments\/1570986688674.52\/warc\/CC-MAIN-20191019013909-20191019041409-00438.warc.gz\"}"}
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Tree Felling professionals trained in planting, cultivating and maintaining trees. You could call them tree doctors, trained to look after the needs of individual trees instead of entire forests. Because they are professionals, you can expect that they have undergone rigorous training and some can be further trained into specializations. Tree experts are generally hired by landscaping companies, commercial tree companies and tree nurseries. Benefits of Hiring Tree Fellers, Tree Loppers, as mentioned previously, are tree cutting and removal professional contractors. They know how best trim or remove your trees without issues. Specialist tree felling experts are further equipped with skills needed to do the manual labor of tree care such as climbing and cutting so you won't have to worry about hiring additional labour. Tree felling specialists also remove dead trees and use special equipment involved in the process. Tree cutting is a dangerous job that should be left to professionals like them. Tree felling is usually performed for general households as well as for commercial use, such as councils or large companies that require assistance with problem trees which require to be cut down and removed.
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\section{The SBF method in a nutshell}
Employing the surface brightness fluctuation signal of unresolved stars in distant galaxies is an effective
and inexpensive new way to measure accurate distances to early-type (dwarf) galaxies. Unlike other extragalactic distance indicators (e.g. TRGB, RR Lyrae stars), this method does {\it not} require resolved stars therefore allowing distance measurements for early-type galaxies far beyond the practical limits of any of the classical distance indicators ($\sim$5\,Mpc). With Fourier analysis techniques, the SBF method quantifies the mean stellar flux per CCD pixel and rms variation due to Poisson noise across a designated area in a dwarf galaxy.
Initially the SBF method was almost exclusively applied on nearby giant ellipticals and MW globular clusters (e.g. Tonry et al.~1989, 1994) but was found to work equally well with dwarf elliptical (dE) galaxies (e.g. Jerjen et al.~1998, 2000, 2001, 2004, and Rekola et al. 2005). As dE galaxies are by far
the most numerous galaxy type at the current cosmological epoch, the SBF method in combination
with wide-field CCD imaging offers the opportunity for the first time to spatially locate dEs in vast numbers and thereby to map in 3D the densest environments of the local Universe (for first results see contributions by
C\^ot\'e et al., Jerjen, Jordan et al., and Rekola et al.~in this volume). First SBF distances are published for dEs as distant as 15Mpc (using 2m ground-based telescopes) and 25Mpc (using 8m VLT+FORS and HST \& ACS).
\section{Analysis prerequisites}
The {\it minimal requirements} for the SBF analysis of an early-type galaxy are:
\smallskip
\begin{itemize}
\item Galaxy morphology: the light distribution of the stellar system must be radially symmetric and have
minimal structure. An overall elliptical shape of the galaxy is crucial as this is modelled
and subtracted as part of the SBF analysis.
\smallskip
\item{Photometry: calibrated CCD images are required in two photometric bands, e.g. ($B,R$) or ($g_{475}$, $z_{850}$), as the fluctuation magnitude shows a colour dependency.}
\smallskip
\item{Image quality: FWHM $ \leq r_{\rm eff}['']/20$, where $r_{\rm eff}$ is the half-light radius of the galaxy.}
\smallskip
\item{Integration time: $t=$S/N$\cdot 10^{0.4\cdot(\mu_{\rm gal}-\mu_{\rm sky}+DM+\overline{M}-m_1)}$,
where $\mu_{\rm gal}$ is the mean surface brightness of the galaxy, $\mu_{\rm sky}$ the surface brightness of the sky background, $DM$ the estimates distance modulus of the galaxy, $\overline{M}$ the fluctuation luminosity of the underlying stellar population, and $m_1$ the magnitude of a star
providing 1 count/sec on the CCD detector at the telescope. }
\smallskip
\end{itemize}
\begin{figure}
\begin{centering}
\includegraphics[height=0.47\textheight]{Dunn.fig1.eps}
\caption{An illustration how the signal-to-noise in the SBF power spectrum increases
with length of exposure time and galaxy surface brightness at the distance of the Fornax
Cluster.}
\label{fig1}
\end{centering}
\end{figure}
To give a general idea of these constraints, Fig.~\ref{fig1} illustrates the depth required for
an image of a dE at the distance of the Fornax cluster observed with VLT+FORS1.
The SBF amplitude above the shot noise level (signal-to-noise) in the power spectrum is
shown as a function of integration time and mean effective surface brightness of the galaxy.
A SBF distance can be determined when the S/N is approximately 0.5, (see Fig.~8 in Rekola et al.~2005), but that depends largely on the image quality i.e. seeing. For example, to
achieve a S/N$\sim$2 in the galaxy power spectrum, the minimum exposure time required for a
dE with a mean surface brightness of 25 mag\,arcsec$^{-2}$ is 1600s.
It is interesting to note
that this exposure time is by a factor of 20 shorter than the 32,000s of HST time spent by Harris et al.~(1998)
to measure the TRGB distance of a dwarf elliptical at a similar distance.
\section{SBF Reduction Pipeline}
Previous SBF work has entailed individuals hand selecting regions in galaxy images for the analysis.
To make the results as impartial as possible and data reduction more efficient we are developing
a rapid, semi-automatic SBF analysis package named SAPAC that can process large numbers of galaxies.
SAPAC is a software package that carries out a semi-automatic SBF analysis of any early-type
galaxy for which CCD data meets the requirements as discussed above.
For a detailed description of the fluctuation magnitude calibration and the individual reduction
steps such as the modelling of the galaxy, foreground star removal, selection of SBF fields
etc.~we refer the reader to Jerjen (2003).
SAPAC consists of Perl scripts using and IRAF module and uses a sophisticated graphical user interface, also written in
Perl. The average processing time for 10 SBF fields in a galaxy and measuring a distance is approximately 20 minutes. Initially we have concentrated the pipeline on $B$, $R$ images, but the
implementation of calibration information for a wider range of commonly used filter sets for
SBF work like $J,H,K$ of the SDSS $g, z$ filters is in process.
\medskip
Potential users of SAPAC who are interested in testing this package for calculating accurate
distances of early-type dwarfs are welcome to contact Laura Dunn.
This software package will be made available to the astronomical community soon.
\begin{acknowledgments}
L.P.D would like to acknowledge partial financial support from the Astronomical Society of
Australia, the International Astronomical Union, and Alex Rodgers Travel Scholarship.
\end{acknowledgments}
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A meteorological app (chrome extension).
# API for Met (chrome extension)
Entry for - http://www.venturesity.com/challenge/id/264</br>
Api-server - https://met-iamkdev.rhcloud.com/ (Not actually needed but I created as the challenge mention about it. Also it protect the api key.) *powered by PHP and Silex*
Weather data from - http://openweathermap.org/</br>
Card design form - http://codepen.io/ajerez/pen/KwYNWZ</br>
Icon form - http://www.flaticon.com/free-icon/umbrella_133385</br>
You can also view the application at - https://met-iamkdev.rhcloud.com/
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International experience, Phil Matthews, Executive Director, Nuleaf
Nuleaf's Executive Director looks at the similarities, and differences, between nuclear communities around the world.
Engaging with Nuleaf – a personal journey! Sean Morris, Secretary, NFLA
Sean Morris, outgoing Secretary of NFLA looks back on his engagement with Nuleaf.
Hearing the voice of local people, Dr Sam King, RWM Ltd
Dr Sam King, Head of Community Engagement and Site Evaluation at RWM Ltd talks about the importance of listening to the local community.
Interview with Cllr Marion Fitzgerald, Allerdale Borough Council
Councillor Marion Fitzgerald from Allerdale Borough Council explains why the council got involved in the Allerdale GDF working Group.
Mike Starkie, Mayor of Copeland, shares his views on geological disposal
The work that is under way to assess whether there is a willing host community and suitable place in our locality to accommodate a Geological Disposal Facility is of significant national importance and, quite rightly, is attracting substantial interest and opinion. But what is not up for debate is that the Copeland borough has a significant role to play in this process, regardless of the final location of a GDF in England or Wales.
Local government is always working on two fronts: to deliver effective services today and to yield better future outcomes for their community. In the middle of this unprecedented crisis, the need for action on public health and the economy, while laying the foundations of a future recovery, is more urgent than ever.
Welcome to the Nuleaf Blog
This is a time of huge change in nuclear decommissioning and waste management.
The Nuclear Decommissioning Authority (NDA) is being restructured and moving towards an approach based on Integrated Waste Management.
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Q: if whole data makes only one batch does it makes sense to shuffle data? if whole data makes only one batch does it makes sense to shuffle data? from my point of view it's not necessary because you will not have any bias per batch
A: When training your model with mini batch gradient descent alike algorithm, you want the batches to differ from each other, but not too much. When the batches differ, learning from some slightly different sample has the regularizing effect, since the model needs to be flexible enough to adapt to those different batches. When the batches are too different, it may have problems with converging, since from batch to batch it could need to make drastic changes in the parameters. To achieve good results, we shuffle the data before splitting into batches, so that splitting the shuffled data leads to getting random samples from the whole dataset.
When you learn on whole data, there is not point in shuffling. At each step, you would apply the same operations to whole dataset, so it wouldn't matter. You would be multiplying all the samples by same weights, adding same biases, transforming using same activation function etc., so the order of the samples would not matter. In the end you would use a cost function that usually is a sum of losses over all samples, and it doesn't matter in what order you take the sum.
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'XCOM 2' Review: Good Luck, Commander
2K Games/Fireaxis Games
Nate Church
XCOM 2, the sequel to 2012's XCOM: Enemy Unknown, can proudly stand beside both the old and new in the X-COM series — no small feat — but it also surpasses them with its brilliance.
The alien forces of ADVENT are relaying information through some sort of glowing obelisk, in the sanctuary of a hastily-abandoned church. We've dropped into the crumbling neighborhood under the cover of night, the four best that the human resistance had on hand.
Our ranger "Ghost" sweeps forward with an affirmative murmur, the eye-black face paint and camouflaged hood she wears evoking her namesake as she haunts the graveyard just to the east of the chapel. "Shakes," our appropriately named surgeon, squawks confirmation into his headset as he moves to support her from a safe distance. "Knockout" James Meyer takes a typically blunt approach — the demolitionist barrels toward one of the stained-glass windows on the side of the building and unceremoniously fires a grenade straight through.
The window shatters inward. The hoarse thud of the detonation shakes the chamber, and the alien relay is left shaken, sparking, and damaged. Two of its unfortunate guardians lie in a heap amidst the splinters of what used to be a row of pews, but the last one scrambles for cover behind the podium. He starts to snarl and point an accusatory finger toward Knockout, looking like nothing so much as the twisted parody of an apocalyptic evangelist. To his left, Ghost leaps through a shower of multicolored glass, dashing forward to bisect him with an elegant sweep of her blade.
It's from this vantage that she can spy the full extent of Knockout's damage to the church — the entire western wall is wide open, and fire is creeping along the edges. She can also see that there are no less than six more ADVENT troops waiting for us in what I can only assume is a rectory next door. One of them wields a glowing red weapon, a cross between a chainsaw and a cattle prod, and charges toward her with an echoing shriek.
That's when "Showtime" makes her appearance. Situated atop a roof nearly a block away, she shoulders a tiger-striped rifle. Time slows, and she fires. The shock trooper charging Ghost instead rolls to a heap at her feet, quite dead. The remaining enemies duck into cover, and my squad does the same.
This isn't a carefully orchestrated set-piece moment in XCOM 2; it's a couple of turns in a routine sort of side event. It's described exactly as it appears on screen. Even the nicknames of the soldiers have been chosen by the game itself, almost eerily appropriate to the roles and behaviors of each character. Perhaps that's why I'm so attached to them already; they sprang fully-formed from the nebulous generation of the game's procedural systems and then quickly evolved over the course of several missions. Already I'm more invested in their fates than I am in the entire cast of most games.
I've dubbed my squad the Devil Dogs, in honor of the only soldier to survive from the beginning until the very end of my campaign in XCOM: Enemy Unknown. He too was a cobbled together mass of random generation and personality, but he'd pale in comparison to the intricate characterization of his successors in XCOM 2. I'm not sure of the exact sum of customizations available to your troops, but it begins in the dozens and expands from there.
Walking simulators are all well and good for telling a story, but XCOM 2 shows the power of storytelling through gameplay. In this world I am the author, protagonist, and the reader as well. I am both the observer and and intrinsic part of this experience, despite the fact that I'm simply moving units across a tiled, turn-based battlefield. What's more, so much of the experience is procedurally generated that no two campaigns play out the same way. Even reloading a mission can cause the AI to adopt vastly different strategies. You never know exactly what to expect, in the best way possible.
The mechanical skeleton of the game is much the same as its predecessor. Instead of digging out rooms to expand an underground base, you'll clear debris from chambers on a ship co-opted from the alien menace that now controls the globe in order to upgrade your headquarters. You'll research both new and adapted alien technologies, upgrade and outfit your troops, and receive constant updates from the various heads of resistance staff. In a small bar just below the dropship hangar in the stern of the ship, there is a wall gradually filling with the portraits and epitaphs of fallen heroes.
XCOM 2's graphics don't stand against the best of today's releases, but they've traded cutting edge visuals for a level of organic personality that very few titles of any generation can achieve. They're well above serviceable, though performance takes needless random dips that Firaxis are already hard at work investigating. It says something about the overall experience that the frame rate hiccups and sometimes over-long loading screens did absolutely nothing to diminish my enjoyment.
However the game looks, it plays out in a manner immediately familiar to series veterans. Combat, while enhanced and deepened by the introduction of several new features, is generally the same as Enemy Unknown. You'll pump a victorious fist as many times as you'll cringe at the fates of your soldiers, dictated in ways large and small by the infamous RNG.
The introduction of stealth can mitigate this to some extent — it's easy and fun to set up elaborate ambushes, in which one unit purposefully alerts the enemy with the lob of a grenade into the midst of their patrol, followed by a hail of gunfire from the squadmate's hidden compatriots. A host of enemy variations blends almost every mission into a heady mix of terror and triumph. It's rare that a turn-based experience can generate the bursts of adrenaline normally associated with the very best action games and movies, but XCOM 2 knows exactly how to play with your mind.
Should your soldiers prevail (read: survive), you'll have even more choices to make. Each of the five classes comes with divergent skills that can very easily cause a couple of otherwise identical soldiers to operate entirely differently on the battlefield. Add that to the weapon and equipment modification, the elaborate character customization, and the very limited resources you can allocate to any of it, and you are all but forced to invest yourself in the life of each resistance fighter.
As you lumber across the globe in your flying fortress, you'll make constant choices about priority. ADVENT will kill those hostages if you don't reach them soon, but will you be able to save them without the supplies lurking just a little too far out of your way? The game's clock never stops ticking, and every click is a hard choice that the game uses liberally both to reward and to punish your decisions as Commander.
The game is merciless at times on the default difficulty, and outright sadistic at any point above it. It's an experience that will launch a million braggarts onto the internet, and for good reason. Even competing the game on Legend is an achievement that ranks among the most difficult gaming experiences available on any platform. I stuck with Veteran, though I wouldn't fault anyone from dialing it down to Rookie. Do so, however, and you could potentially miss one of the highlights of the game — its inherent white-knuckled risk assessment.
Like the best horror games, the most persistent feeling that XCOM 2 gives you is that you are never, ever safe. Whether it's the doomsday clock working its way ever downward as you scramble to develop the resistance, or a clockwork mission that turns disastrous in a single turn, the developers at Firaxis have taken a dark sort of delight in punishing any misstep. It's rare that a game so gleefully antagonistic can remain so engaging and fun.
It's only February, but it's hard to believe that XCOM 2 won't stand fast as one of the best gaming experiences of the year. I wholeheartedly recommend it to anyone who enjoys a bit of suffering with their entertainment. Once it gets some of its technical inconsistencies under control, it will be as close to flawless as I've seen a game come in a long, long time.
By the time the Devil Dogs complete their extraction, the church is a ruin of broken beams and rubble. It consigns its final twisted parishioners to the earth, but I'm not thinking about poetic imagery at the time. Civilian resistance in Mexico is suffering a retaliatory strike because of our operation, and we have just enough time to reach them.
Follow Nate Church @Get2Church on Twitter for the latest news in gaming and technology, and snarky opinions on both.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,048
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Q: My view follows the standing version of my character but not the walking version I am making a platformer game with 4 character objects: Ninja_Standing_Right, Ninja_Walking_Right,Ninja_Walking_Left, and Ninja_Standing_Left. I have applied a view that follows Ninja_Walking_Right but whenever the ninja changes to a different object, such as Ninja_Walking_Left, the view doesn't follow it. This is obviously because Ninja_Walking_Left and Ninja_Walking_Right are different objects but how can I make it so that the view follows both Ninja_Walking_Right and Ninja_Walking_Left or whatever other method there is, thanks!
A: I would recommend to rather change the sprite of the object than changing the whole object itself. This would both fix your problem and avoid having the same code for all player direction objects.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"http:\/\/lists.gnu.org\/archive\/html\/bug-texinfo\/2007-06\/msg00030.html","text":"bug-texinfo\n[Top][All Lists]\n\n## Re: pdfeTeX error when compiling lilypond.texi with accents in node name\n\n From: Oleg Katsitadze Subject: Re: pdfeTeX error when compiling lilypond.texi with accents in node names Date: Wed, 27 Jun 2007 00:07:37 +0300 User-agent: Mutt\/1.5.13 (2006-08-11)\n\nOn Tue, Jun 26, 2007 at 12:23:33AM +0200, John Mandereau wrote:\n> Here's attached an almost minimal example showing the problem. If you\n> replace the umlaut-U with U, texi2pdf successfully compiles it.\n\nThanks again for the short example, now it makes sense. The problem\nlies in the fact that @macro expands (\\xdef's) the body without\ntaking care to avoid expansion of the (active) non-ASCII characters.\nI believe the problem is not limited to UTF-8, but to any non-ASCII\nencoding.\n\nThe patch I attach avoids this expansion, but I'm not sure this\ndoesn't break something else. So I'm afraid we'll have to wait for\nKarl's opinion, whenever he has time to look into this.\n\nBest,\nOleg\n\nP.S. I'm not sure if \\setnonasciicharscatcode really needs to set\ncatcodes globally. Can it ever be called from inside a group? If\nnot, we can simply remove \\global from it, and thus avoid the need for\n\\setnonasciicharscatcodenonglobal. Or maybe just copy the loop from\n\\setnonasciicharscatcodenonglobal inside \\scanctxt.\n\npatch\nDescription: Text document","date":"2017-01-19 13:04:57","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\": 0.874883770942688, \"perplexity\": 10693.48779696448}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2017-04\/segments\/1484560280668.34\/warc\/CC-MAIN-20170116095120-00369-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
| null | null |
var model = require('../domain/model.js');
var mongoose = require('mongoose');
var push = require("../routes/push");
exports.Status = {
New: 'new',
Unread: 'unread',
Read: 'read'
};
exports.notifyNewCollaborators = function(recipe, collaborators) {
for ( var i=0; i<collaborators.length; i++ ) {
var c = collaborators[i];
var c_id = c._id || c;
var data = "Ha sido agregado como colaborador de la receta <b>";
data += recipe.NAME + "</b>";
var link = "#/recipe/edit/" + encodeURIComponent(recipe._id);
notify(c_id,data,link);
}
};
exports.notifyOnPublish = function(recipe_name,recipe_id,user_name,user_id) {
return Promise.resolve();
// avoid notification on publish, creo qeu nadie las lee.
// model.User.find().exec(function(err,users) {
// for ( var i=0; i<users.length; i++) {
// if ( users[i]._id != user_id) {
// var data = "El usuario <b>{{user_name}}</b> ha publicado la receta <b>{{recipe_name}}</b>";
// data = data.replace('{{user_name}}',user_name);
// data = data.replace('{{recipe_name}}',recipe_name);
// var link = "/share.html#/" + encodeURIComponent(recipe_id);
// notify(users[i]._id, data, link);
// }
// }
// });
};
/**
* Notifica al owner q han agregado a favoritos su receta.
*
* @param owner_id id de usuario que es el dueño original de la receta
* @param recipe
* @param user_id id de usuario q clono la receta
* @param user_name nombre de usuario q clono la receta
*/
exports.notifyAddFavorite = function(owner_id, recipe, user_id, user_name) {
var data = "<b>{{user_name}}</b> ha agregado a favoritos tu receta <b>{{recipe.NAME}}</b>";
data = data.replace('{{user_name}}',user_name);
data = data.replace('{{recipe.NAME}}',recipe.NAME);
var link = "#/recipe/edit/" + encodeURIComponent(recipe._id);
notify(owner_id, data, link);
};
/**
* Notifica al owner q han clonado su receta.
*
* @param owner_id id de usuario que es el dueño original de la receta
* @param recipe
* @param user_id id de usuario q clono la receta
* @param user_name nombre de usuario q clono la receta
* @param recipe_name nombre de la receta original
*/
exports.notifyRecipeCloned = function(owner_id, recipe,user_id,user_name,recipe_name) {
if (owner_id != user_id) {
var data = "<b>{{user_name}}</b> ha clonado tu receta <b>{{recipe.NAME}}</b>";
data = data.replace('{{user_name}}',user_name);
data = data.replace('{{recipe.NAME}}',recipe_name);
var link = "/share.html#/" + encodeURIComponent(recipe._id);
notify(owner_id, data, link);
}
};
/**
* Notifica modificacion de receta a los q la tienen como favorita.
* @param recipe receta en la que se hizo update
*/
exports.notifyUpdateFavorite = function(recipe) {
for (var i=0; i<recipe.starredBy.length; i++) {
var data = "Han actualizado tu receta favorita <b>{{recipe.NAME}}</b>";
data = data.replace('{{recipe.NAME}}',recipe.NAME);
var link = "/share.html#/" + encodeURIComponent(recipe._id);
notify(recipe.starredBy[i]._id, data, link);
}
};
/**
* Notifica modificacion de receta a los q son colaboradores.
* @param recipe receta en la que se hizo update
*/
exports.notifyUpdateCollaborators = function(recipe, user_id, user_name) {
console.log("OWNER", recipe.owner);
if ( recipe.owner._id != user_id ) {
var data = "<b>{{user_name}}</b> ha actualizado tu receta <b>{{recipe.NAME}}</b>";
data = data.replace('{{recipe.NAME}}',recipe.NAME);
data = data.replace('{{user_name}}',user_name);
var link = "#/recipe/edit/" + encodeURIComponent(recipe._id);
notify(recipe.owner._id, data, link);
}
for (var i=0; i<recipe.collaborators.length; i++) {
var col_id = recipe.collaborators[i]._id || recipe.collaborators[i];
if ( col_id != user_id ) {
var data = "<b>{{user_name}}</b> ha actualizado la receta <b>{{recipe.NAME}}</b> en la que eres colaborador";
data = data.replace('{{recipe.NAME}}',recipe.NAME);
data = data.replace('{{user_name}}',user_name);
var link = "#/recipe/edit/" + encodeURIComponent(recipe._id);
notify(col_id, data, link);
}
}
};
/**
* Cuando se hace una modificacion en una receta favorita. Esto le llegaria a varios.
* Tambien notifica a los que hicieron algun comentario
* @param recipe la receta en la cual se hizo el comentario
* @param user_id usuario el cual comenta
* @param user_name nombre del usuario que comento la receta.
*/
exports.notifyCommentOnFavorite = function(recipe, user_id , user_name) {
var notified = [];
for (var i=0; i<recipe.starredBy.length; i++) {
if ( recipe.starredBy[i]._id != user_id) {
var data = "<b>{{user_name}}</b> ha comentado tu receta favorita <b>{{recipe.NAME}}</b>";
data = data.replace('{{user_name}}',user_name);
data = data.replace('{{recipe.NAME}}',recipe.NAME);
var link = "/share.html#/" + encodeURIComponent(recipe._id);
notified.push(recipe.starredBy[i]._id.toString());
notify(recipe.starredBy[i]._id, data, link);
}
}
for (var i=0; i<recipe.comments.length; i++) {
//salteo q los que ya notifique por favoritos, al q comenta y al owner
if ( recipe.comments[i].user_id != user_id
&& notified.indexOf(recipe.comments[i].user_id.toString()) == -1
&& recipe.comments[i].user_id != recipe.owner ) {
var data = "<b>{{user_name}}</b> ha comentado en una receta q has comentado <b>{{recipe.NAME}}</b>";
data = data.replace('{{user_name}}',user_name);
data = data.replace('{{recipe.NAME}}',recipe.NAME);
var link = "/share.html#/" + encodeURIComponent(recipe._id);
notified.push(recipe.comments[i].user_id.toString());
notify(recipe.comments[i].user_id, data, link);
}
}
};
exports.notifyChangeFermentationStage = function(owner_id, recipe_id, recipe_name, stageFrom, stageTo) {
var data = "";
if ( stageFrom ) {
data = "Tu receta <b>{{recipe.NAME}}</b> ha pasado de la etapa de fermentacion <b>{{stageFrom}}</b> "
+ " (temperatura {{tempFrom}}º) a la <b>{{stageTo}}</b> (temperatura {{tempTo}}º).";
data = data.replace('{{recipe.NAME}}',recipe_name);
data = data.replace('{{stageFrom}}',stageFrom.title);
data = data.replace('{{stageTo}}',stageTo.title);
data = data.replace('{{tempFrom}}',stageFrom.temperatureEnd);
data = data.replace('{{tempTo}}',stageTo.temperature);
} else {
data = "Tu receta <b>{{recipe.NAME}}</b> ha comenzado las etapas de fermentacion con <b>{{stageTo}}</b> "
+ " (temperatura {{tempTo}}º).";
data = data.replace('{{recipe.NAME}}',recipe_name);
data = data.replace('{{stageTo}}',stageTo.title);
data = data.replace('{{tempTo}}',stageTo.temperature);
}
var link = "#/recipe/edit/" + encodeURIComponent(recipe_id);
notify(owner_id, data, link);
}
/**
* Cuando se ha realizado un comentario de un tercero en una receta propia
*
* @param user_id owner de la receta
* @param user_id usuario el cual comenta
* @param user_name nombre del usuario que comento la receta.
* @param recipe_id id de la receta comentada.
* @param recipe_name nombre de la receta
*/
exports.notifyCommentOnRecipe = function(owner_id, user_id,user_name , recipe_id, recipe_name) {
if (owner_id != user_id) {
var data = "<b>{{user_name}}</b> ha comentado en tu receta <b>{{recipe.NAME}}</b>";
data = data.replace('{{user_name}}',user_name);
data = data.replace('{{recipe.NAME}}',recipe_name);
var link = "#/recipe/edit/" + encodeURIComponent(recipe_id);
notify(owner_id, data, link);
}
};
var notify = function(user_id,data,link) {
model.Notification.create(new model.Notification({
user_id:user_id,
date: new Date(),
status: exports.Status.New,
data: data,
link: link
}), function(err,notification) {
if ( err) {
console.log("err",err);
console.log("notification",notification);
} else {
push.emit("NOTIFICATION_ADD_" + user_id,notification);
}
});
};
exports.notify = notify;
/**
* Elimina las notificaciones viejas.
* Por ahora limpia las mas viejas q 1 dias q ya hayan sido leidas.
* Y desde 6 dias hacia atras limpia todas.
* Luego si veo q la DB crece mucho, deberia limpiar mas.
*/
exports.removeOld = function() {
var from = new Date(new Date().getTime()-1*24*60*60*1000);
console.log("Eliminando leidas desde",from);
model.Notification.remove({date:{$lt:from},status:'read'}).exec(function() {
from = new Date(new Date().getTime()-6*24*60*60*1000);
console.log("Eliminando todas desde ",from);
model.Notification.remove({date:{$lt:from}}).exec();
});
};
/**
* Service web.
*/
exports.findAll = function(req,res) {
model.Notification.find({user_id:req.session.user_id}).sort("-date").exec(function(err,notifications) {
model.Notification.update({user_id:req.session.user_id,status:'new'},{$set:{status:'unread'}},{multi:true}).exec();
res.send(notifications);
});
};
exports.findNews = function(req, res) {
model.Notification.find({user_id:req.session.user_id,status:'new'}).sort("-date").exec(function(err,notifications) {
res.send(notifications);
});
};
exports.update = function(req,res) {
console.log("req.params.id",req.params.id);
console.log("body",req.body);
delete req.body._id;
model.Notification.findByIdAndUpdate({_id:new mongoose.Types.ObjectId(req.params.id)},req.body).exec(function(err,notification) {
res.send(notification);
});
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,482
|
Johann Adam Boost (* 8. April 1775 in Aschaffenburg; † 8. Oktober 1852 in Mainz) war ein deutscher Ökonom, Beamter und Autor.
Leben
Johann Adam Boost war der dritte von sechs Söhnen des Juristen Karl Joseph Schweikard Boost (* 9. Mai 1739 in Mainz; † 15. Oktober 1811 ebendort) und dessen Ehefrau Dorothea Vogt. Ab 1789 besuchte er die hohe Schule zu Mainz und wie sein Vater und sein Bruder Carl Joseph Boost unterstützte er 1792 die Bildung der Mainzer Republik. Als die Franzosen die Stadt am 24. Juli 1793 verließen, floh er nach Paris.
Boost wurde Forstbeamter im Elsass und ließ sich später als Ökonom in der Nähe von Mainz nieder. 1816/1817 wurde er Spezialkommissar im Großherzogtum Hessen. Von 1833 bis 1838 lebte er in Regensburg und dann bis 1847 in Darmstadt. In seinem Sterbeeintrag wird er als Großherzoglich Hessischer Kalkulator bezeichnet.
Aus seiner Feder stammen ökonomische und etliche tendenziöse Bücher mit religiösem und politischem Inhalt, die auf einen Sinneswandel des einstigen Jakobiners schließen lassen. Verheiratet war er mit Katharina Schmitt, mit der er zuletzt in Nierstein wohnte, wo das Ehepaar zwischen 1801 und 1820 zwölf Kinder taufen ließ. Am 8. Oktober 1852 starb der Witwer um sechs Uhr morgens in dem Vincenz- und Elisabethenspital zu Mainz und wurde auch in der Stadt beerdigt.
Werke
Ueber die Rheinlande in staatswirthschaftlicher und ökonomischer Beziehung. Darmstadt 1815.
Ueber die Maßregeln der älteren und neueren Staaten bei Theuerungs- und Hungersnoth. Mainz 1817.
Was waren die Rheinländer als Menschen und Bürger, und was ist aus ihnen geworden? Mainz 1819.
Ueber Productions- und Consumtionssteuern. Darmstadt 1824.
Das Jahr 1810 oder Darstellung der Revolution in ihrer Vergangenheit, Gegenwart und Zukunft. Darmstadt 1832.
Die neueste Geschichte der Menschheit. [= Die neueste Geschichte von Frankreich und Oesterreich (1789–1834)], Band 1 Regensburg 1833.
Der Gegner der Kirche, widerlegt durch die Geschichte u. die Ansichten der geistreichsten Protestanten. Augsburg 1838.
Geschichte der Reformation und Revolution von Frankreich, England und Deutschland 1517–1843. 3 Bände Augsburg 1843.
Geschichte der Reformation und Revolution von Deutschland. Von 1517–1844, Augsburg 1844, [1845 und 1846].
Neueste Geschichte Frankreich. Regensburg 1836.
Die neueste Geschichte der Menschheit. Augsburg 1836 und 1839. 1. Abth, Frankreich und Oesterreich; 1. Theil, Die neueste Geschichte von Frankreich vom Jahre 1789 bis 1836. 1. Abth., Frankreich und Oesterreich; 2. Theil: Die neueste Geschichte von Oesterreich unter den Regenten aus dem Habsburg-Lothringer Stamme.
Sammlung moralischer Erzählungen für Deutschlands Söhne und Töchter. 2 Bände, Augsburg 1841, 1843 und 1846.
Die Geschichte und die Propheten, die wahren Schlüssel zu den Pforten der Zukunft. 1. Aufl. Augsburg 1846, 2. Aufl. Augsburg 1847. 3., den jetzigen Zeitwirren gemäß sehr veränd. und verm. Aufl. Augsburg 1848, 4. Aufl. Augsburg 1848.
Die Weissagungen des Mönchs Hermann zu Lehnin über Preußen und jene des Benedictiners David Speer zu Benedictbeuern über Bayern. Ausgabe: 3., den jetzigen Zeitwirren gemäß sehr veränd. und verm. Aufl. Augsburg 1848.
mit Artraud de Montor: Geschichte der römischen Päpste. 5 Bände, Augsburg 1848–1856.
Die wunderlichen Ansichten und grossen Mißgeschicke des deutschen Michel in seinen Freiheitsschwindeleien und Nachäffungen der Pariser Revolutions-Moden. Bensheim 1850.
Die Erklärung der Offenbarung Johanni im Geiste der Geschichte und der Religion.
Literatur
Karl Georg Bockenheimer: Boost, Johann Adam. In: Allgemeine Deutsche Biographie Band 3 (1876) S. 139–140. Online-Version: URL: https://www.deutsche-biographie.de/pnd100050875.html#adbcontent.
Norbert J. Pies: Notabilia & Miscellanea oder Heimat- und familienkundliche Randnotizen Heft IV – Treiser Krankheit & Brownsche Affen. Kommentiere Edition der Streitschrift des Cochemer Arztes Carl Boost von 1807. Erftstadt-Lechenich 2021.
Einzelnachweise
Geboren 1775
Gestorben 1852
Mann
Autor
Beamter
Ökonom (19. Jahrhundert)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,026
|
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